Embed Size (px)
Citation preview
Research ArticleDynamic Analysis under Uniformly Distributed Moving Massesof Rectangular Plate with General Boundary Conditions
Emem Ayankop Andi1 and Sunday Tunbosun Oni2
1 Department of Mathematics Nigerian Defence Academy Kaduna State Kaduna 800001 Nigeria2 Department of Mathematical Sciences Federal University of Technology Ondo State Akure 340001 Nigeria
Correspondence should be addressed to Emem Ayankop Andi ememandiyahoocom
Received 3 December 2013 Revised 25 April 2014 Accepted 12 May 2014 Published 19 June 2014
Academic Editor Nam-Il Kim
Copyright copy 2014 E A Andi and S T Oni This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
The problem of the flexural vibrations of a rectangular plate having arbitrary supports at both ends is investigated The solutiontechnique which is suitable for all variants of classical boundary conditions involves using the generalized two-dimensional integraltransform to reduce the fourth order partial differential equation governing the vibration of the plate to a second order ordinarydifferential equation which is then treated with the modified asymptotic method of StrubleThe closed form solutions are obtainedand numerical analyses in plotted curves are presented It is also deduced that for the same natural frequency the critical speedfor the system traversed by uniformly distributed moving forces at constant speed is greater than that of the uniformly distributedmoving mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlier in theuniformly distributed moving mass system Furthermore for both structural parameters considered the response amplitude of themoving distributed mass system is higher than that of the moving distributed force system Thus it is established that the movingdistributed force solution is not an upper bound for an accurate solution of the moving distributed mass problem
1 Introduction
Theproblemof assessing the dynamic behaviour of structurescarrying moving loads has been almost exclusively reservedin the literature to that of one-dimensional structural mem-bers such as the response of railroad rails to moving trainsthe response of bridges and elevated roadways to movingvehicles the response of belt drives to conveyor belts and theresponse of computer tape drives to floppy disks [1ndash8]
Where two-dimensional structures such as plates havebeen considered the load acting on the structure has beensimplified as lumped mass point load or concentrated loadAmong researchers in this subject are Gbadeyan and Oni[9] who developed an analytical technique which is basedon the generalized two-dimensional integral transform theexpression of the Dirac delta function as a Fourier Cosineseries and the use of the modified Strublersquos asymptoticmethod to solve the problem of rectangular plates undermoving loads This technique was later used by Oni [10]to solve the problem of the dynamic response of an elas-tic rectangular plate under the actions of several moving
concentrated masses Huang andThambiratnam [11] studiedisotropic homogenous elastic rectangular plate resting on anelastic Winkler foundation under a single concentrated loadusing finite strip method Shadnam et al [12] investigatedthe dynamics of plates under the influence of relatively largemasses moving along an arbitrary trajectory on the platesurface In particular a rectangular plate simply supported onall its edges was presented by means of operational calculusAlisjahbana [13] presented an approximate method for thedetermination of the natural frequencies and mode shapes ofrectangular clamped orthotropic plates subjected to dynamictransverse moving loads The dynamic solution of the platewas based on orthogonality conditions of eigenfunctionsWuet al [14] analyzed by finite element method the dynamicresponses of a flat plate subjected to various moving loadsThe Newmark direct integration method was used to findthe dynamic responses of the flat plate Wang and Lin [15]presented the method of modal analysis to study the randomvibration of multispan Mindlin plates due to load moving ata constant velocity Although the above completed works onconcentrated loads are impressive they do not represent the
Hindawi Publishing CorporationJournal of Computational EngineeringVolume 2014 Article ID 140407 19 pageshttpdxdoiorg1011552014140407
2 Journal of Computational Engineering
PF(x y t)
L
Figure 1 A uniformly distributed load 119875119865(119909 119910 119905) on a plate with
both ends clamped
PF(x y t)
L
Figure 2 A uniformly distributed load 119875119865(119909 119910 119905) on a plate with
one end clamped and the other end simply supported
physical reality of the problem formulation as concentratedmasses do not exist physically since in practice moving loadsare in the form of moving distributed masses which areactually distributed over a small segment or over the entirelength of the structural member they traverse
Works on 2-dimensional structural members traversedby distributed loads have only attracted the attention of fewresearchers (Figures 1 and 2)
These include the works of Gbadeyan and Dada [16] whoinvestigated the dynamic response of plates on Pasternakfoundation to distributed moving loads Dada [17] whostudied the vibration analysis of elastic plates under uniformpartially distributed moving loads Usman [18] discussedthe dynamic response of a Bernoulli-beam on Winklerfoundation under the action of partially distributed loadThe dynamic analysis of Rayleigh beam carrying an addedmass and traversed by uniform partially distributed movingloads was carried out by Adetunde [19] More recentlyGbadeyan and Dada [20] used a finite difference algorithmto investigate the elastodynamic response of a rectangularMindlin plate subjected to a distributed moving mass Thesimply supported edge condition was used as an illustrativeexample It was found that the maximum shearing forcesbending and twisting moments occur almost at the sametime Beskou andTheodorakopoulos [21] presented a reviewof the dynamic response of road pavements modelled asa beam plate or top layer of a layered soil medium tomoving loads on their surface The loads were concentratedor distributed of finite extent varied with time and movedwith constant or variable speed The analysis was done byanalyticalnumerical and purely numerical methods suchas finite element and boundary element methods underconditions of plane strain Representative examples werepresented in order to illustrate the problem Amiri et al[22] investigated the elastodynamic response of a rectangularMindlin plate under concentrated moving loads as well asother arbitrarily selected distribution area of loads Closed
form solution for the moving force load case was derivedusing direct separation of variables and eigenfunction expan-sion method A semianalytical solution was presented formovingmass load case In these investigations however onlynumerical or semianalytical techniques have been employedto solve the governing equation due to the rigour of the load-structure interactions and complex nature of the resultingequations Nevertheless analytical solution is desirable as itoften sheds light on some vital information in the vibratingsystem This paper therefore investigates the flexural vibra-tions of a rectangular plate with general boundary conditionsunder uniformly distributed moving masses Both gravityand inertia effects of the uniformly distributed masses aretaken into consideration and the plate is taken to reston Winkler foundation The solution technique which isanalytical and suitable for all variants of classical boundaryconditions involves using the generalized two-dimensionalintegral transform the expression of the Heaviside functionas a Fourier series and the use of the modified Strublersquosasymptotic technique to solve the problem of the flexuralvibrations of a rectangular plate having arbitrary supports atboth ends In addition conditions under which resonance isreached is obtained for both uniformly distributed movingforce and uniformly distributed moving mass problems
2 Governing Equation
Consider the dynamic transverse displacement119882(119909 119910 119905) ofa rectangular flat plate of span 119871
119910along the 119910-axis and span
119871119909along the 119909-axis carrying a uniformly distributed mass
moving with constant velocity 119888 along a straight line 119910 = 1199100
parallel to the 119909-axisIf the platemodel incorporates the rotatory inertia correc-
tion factor119882(119909 119910 119905) is governed by the fourth order partialdifferential equation given by [23]
(119863(1205972
1205971199092+
1205972
1205971199102)
2
minus 1205831198770
1205972
1205971199052)(
1205972
1205971199092+
1205972
1205971199102)
2
times119882(119909 119910 119905) + 1205831205972119882(119909 119910 119905)
1205971199052+ 119870119882(119909 119910 119905)
= 119875119865(119909 119910 119905) [1 minus
Δlowast
119892[119882 (119909 119910 119905)]]
(1)
119864 is Youngrsquosmodulus ] is Poissonrsquos ratio (] lt 1) 120583 is themassper unit length (self-weight) of the plate 119909 is the positioncoordinate in 119909-direction 119910 is the position coordinate in 119910-direction 119905 is the time ℎ is the plate thickness119870 is the elasticfoundation moduli and 119877
0is the measure of rotatory inertia
correction effect 119863 = 119864ℎ312(1 minus 120592) is the bending rigidity
of the plate and is constant throughout the plate 119875119865(119909 119910 119905) is
the continuous moving uniformly distributed force acting onthe beammodel 119892 is acceleration due to gravity andΔlowast is theconvective acceleration operator defined as
Δlowast= (
1205972
1205971199052+ 2119888
1205972
120597119905120597119909+ 1198882 1205972
1205971199092) (2)
Journal of Computational Engineering 3
The continuousmoving uniformly distributed force acting onthe plate is given by
119875119865(119909 119910 119905) = 119872119892119867 (119909 minus 119888119905)119867 (119910 minus 119910
0) (3)
where 119872 is the uniformly distributed mass 119867(sdot) is theHeaviside function The time 119905 is assumed to be limited tothat interval of time within which the mass119872 is on the platethat is
0 le 119888119905 le 119871119909 (4)
Thus in view of (2) and (3) (1) can be written as
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052+ 119870119882(119909 119910 119905)
minus 1205831198770[1205974119882(119909 119910 119905)
12059711990921205971199052+1205974119882(119909 119910 119905)
12059711991021205971199052]
+119872119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times [1205972119882(119909 119910 119905)
1205971199052+ 2119888
1205972119882(119909 119910 119905)
120597119905120597119909+ 11988821205972119882(119909 119910 119905)
1205971199092]
= 119872119892119867 (119909 minus 119888119905)119867 (119910 minus 1199100)
(5)
Equation (5) is the fourth order partial differential equationgoverning the flexural vibrations of a rectangular plate underthe action of uniformly distributed loadsThe boundary con-ditions are taken to be arbitrary while the initial conditionswithout any loss of generality are given by
119882(119909 119910 0) = 0 =120597119882(119909 119910 0)
120597119905 (6)
3 Method of Solution
The analysis of the dynamic response to a uniformly dis-tributed moving mass of isotropic rectangular plates restingon aWinkler foundation and subjected to arbitrary boundaryconditions is carried out in this sectionThe generalized two-dimensional integral transform is defined as
119882(119895 119896 119905) = int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (7)
with the inverse
119882(119909 119910 119905) =
infin
sum
119895=1
infin
sum
119896=1
120583
119881119895
120583
119881119896
119882(119895 119896 119905)119882119895 (119909)119882119896 (119910) (8)
where
119881119895= int
119871119909
0
1205831198822
119895(119909) 119889119909 119881
119896= int
119871119910
0
1205831198822
119896(119910) 119889119910 (9)
119882119895(119909) and119882
119896(119910) are respectively the beam mode functions
in the 119909 and 119910 directions defined respectively as
119882119895 (119909) = Sin
120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
119882119896(119910) = Sin
120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
(10)
where 119860119895 119861119895 119862119895 119860119896 119861119896 and 119862
119896are constants and 120582
119895
120582119896are mode frequencies which are determined using the
appropriate boundary conditionsApplying the generalized two-dimensional integral trans-
form (7) (5) takes the form
1198851198601198651(0 119871119909 119871119910 119905) + 119885
1198601198652(0 119871119909 119871119910 119905) + 119882
119905119905(119895 119896 119905)
minus 1198851198611198791198611 (119905) minus 1198851198611198791198612 (119905) + 119885119862119882(119895 119896 119905)
+ 1198791198613 (119905) + 1198791198614 (119905) + 1198791198615 (119905)
=119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896(119910) 119889119909 119889119910
(11)
where
119885119860=119863
120583 119885
119861= 1198770 119885
119862=119870
120583 (12)
1198651(0 119871119909 119871119910 119905)
= int
119871119910
0
([119882119895 (119909)
1205973119882(119909 119910 119905)
1205971199093minus1198821015840
119895(119909)
1205972119882(119909 119910 119905)
1205971199092
+11988210158401015840
119895(119909)
120597119882 (119909 119910 119905)
120597119909minus119882101584010158401015840
119895(119909)119882 (119909 119910 119905)]
119871119909
0
times119882119896(119910) 119889119910
+ 2 [119882119895 (119909)
120597119882 (119909 119910 119905)
120597119909minus1198821015840
119895(119909)119882 (119909 119910 119905)]
times 11988210158401015840
119896(119910) 119889119910)
+ int
119871119909
0
(2[119882119896(119910)
120597119882 (119909 119910 119905)
120597119910minus1198821015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
4 Journal of Computational Engineering
times11988210158401015840
119895(119909) 119889119909
+ [119882119896(119910)
1205973119882(119909 119910 119905)
1205971199103
minus1198821015840
119896(119910)
1205972119882(119909 119910 119905)
1205971199102+11988210158401015840
119896(119910)
120597119882 (119909 119910 119905)
120597119910
minus119882101584010158401015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
119882119895 (119909) 119889119909)
(13)
1198652(0 119871119909 119871119910 119905)
= int
119871119910
0
int
119871119909
0
(119882 (119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910)
+ 2119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910)) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
(14)
1198791198611 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(15)
1198791198612 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199102119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (16)
1198791198613 (119905) =
119872
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199052119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(17)
1198791198614 (119905) = 2
119872119888
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
120597119905120597119909119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(18)
1198791198615 (119905) =
1198721198882
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(19)It is recalled that the equation of the free vibration of arectangular plate is given by
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052= 0
(20)
If the free vibration solution of the problem is set as119882(119909 119910 119905) = 119882
119895 (119909)119882119896 (119910) SinΩ119895119896119905 (21)
where Ω119895119896
is the natural circular frequency of a rectangularplate then substituting (21) into (20) yields
119863[1198821015840120592
119895(119909)119882119896(119910) + 2119882
10158401015840
119895(119909)119882
10158401015840
119896(119910) +119882
119895 (119909)1198821015840120592
119896(119910)]
minus 120583Ω2
119895119896119882119895 (119909)119882119896 (119910) = 0
(22)
It is well known that for a simply supported rectangular plateΩ2
119895119896is given by
Ω2
119895119896= 119863[
11989541205874
1198714119909
+ 2119895211989621205874
11987121199091198712119910
+11989641205874
1198714119910
] (23)
Multiplying (22) by119882(119909 119910 119905) and integrating with respect to119909 and 119910 between the limits 0 119871
119909and 0 119871
119910 respectively we
get
int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910) 119889119909 119889119910
+ 2int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
=120583
119863Ω2
119895119896int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(24)
In view of (14) we have
1198652(0 119871119909 119871119910) =
120583
119863Ω2
119895119896119882(119895 119896 119905) (25)
Since119882(119901 119902 119905) is just the coefficient of the generalized two-dimensional finite integral transform
119882(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)119882119901 (119909)119882119902 (119910) (26)
it follows that
11988210158401015840(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)11988210158401015840
119901(119909)119882119902 (119910)
(27)
Therefore the integrals (15) and (16) can be rewritten as
1198791198611 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867
2(119901 119895) 119865 (119902 119896) (28)
where
1198672(119901 119895) = int
119871119909
0
11988210158401015840
119901(119909)119882119895 (119909) 119889119909
119865 (119902 119896) = int
119871119910
0
119882119902(119910)119882
119896(119910) 119889119910
(29)
1198791198612 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896) (30)
Journal of Computational Engineering 5
where
119867(119901 119895) = int
119871119909
0
119882119901 (119909)119882119895 (119909) 119889119909
1198652(119902 119896) = int
119871119910
0
11988210158401015840
119902(119910)119882
119896(119910) 119889119910
(31)
In order to evaluate the integrals (17) (18) and (19) use ismade of the Fourier series representation of the Heavisidefunction namely
119867(119909 minus 119888119905) =1
4+1
120587
infin
sum
119899=0
Sin ((2119899 + 1) 120587 (119909 minus 119888119905))2119899 + 1
(32)
Similarly
119867(119910 minus 1199100) =
1
4+1
120587
infin
sum
119898=0
Sin ((2119898 + 1) 120587 (119910 minus 1199100))
2119898 + 1 (33)
Using (30) and (31) one obtains
1198791198613 (119905)
=119872
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905) 119865 (119902 119896)119867 (119901 119895)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119901 119895) 119865 (119902 119896)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119901 119895) 119865 (119902 119896)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119902 119896)119867 (119901 119895)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119902 119896)119867 (119901 119895)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119888(119898 119902 119896)
(34)
where
119867119904(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119867119888(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119865119904(119898 119902 119896) = int
119871119910
0
Sin (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
119865119888(119898 119902 119896) = int
119871119910
0
Cos (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
(35)
6 Journal of Computational Engineering
Similarly
1198791198614 (119905)
=2119872119888
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905(119901 119902 119905)119867
1(119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119901 119895) 119865 (119902 119896)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119888(119898 119902 119896)
(36)
where
1198671(119901 119895) = int
119871119909
0
1198821015840
119901(119909)119882119895 (119909) 119889119909
119867119904
1(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
119867119888
1(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
1198791198615 (119905)
=1198721198882
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882(119901 119902 119905)1198672(119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119901 119895) 119865 (119902 119896)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
2 Journal of Computational Engineering
PF(x y t)
L
Figure 1 A uniformly distributed load 119875119865(119909 119910 119905) on a plate with
both ends clamped
PF(x y t)
L
Figure 2 A uniformly distributed load 119875119865(119909 119910 119905) on a plate with
one end clamped and the other end simply supported
physical reality of the problem formulation as concentratedmasses do not exist physically since in practice moving loadsare in the form of moving distributed masses which areactually distributed over a small segment or over the entirelength of the structural member they traverse
Works on 2-dimensional structural members traversedby distributed loads have only attracted the attention of fewresearchers (Figures 1 and 2)
These include the works of Gbadeyan and Dada [16] whoinvestigated the dynamic response of plates on Pasternakfoundation to distributed moving loads Dada [17] whostudied the vibration analysis of elastic plates under uniformpartially distributed moving loads Usman [18] discussedthe dynamic response of a Bernoulli-beam on Winklerfoundation under the action of partially distributed loadThe dynamic analysis of Rayleigh beam carrying an addedmass and traversed by uniform partially distributed movingloads was carried out by Adetunde [19] More recentlyGbadeyan and Dada [20] used a finite difference algorithmto investigate the elastodynamic response of a rectangularMindlin plate subjected to a distributed moving mass Thesimply supported edge condition was used as an illustrativeexample It was found that the maximum shearing forcesbending and twisting moments occur almost at the sametime Beskou andTheodorakopoulos [21] presented a reviewof the dynamic response of road pavements modelled asa beam plate or top layer of a layered soil medium tomoving loads on their surface The loads were concentratedor distributed of finite extent varied with time and movedwith constant or variable speed The analysis was done byanalyticalnumerical and purely numerical methods suchas finite element and boundary element methods underconditions of plane strain Representative examples werepresented in order to illustrate the problem Amiri et al[22] investigated the elastodynamic response of a rectangularMindlin plate under concentrated moving loads as well asother arbitrarily selected distribution area of loads Closed
form solution for the moving force load case was derivedusing direct separation of variables and eigenfunction expan-sion method A semianalytical solution was presented formovingmass load case In these investigations however onlynumerical or semianalytical techniques have been employedto solve the governing equation due to the rigour of the load-structure interactions and complex nature of the resultingequations Nevertheless analytical solution is desirable as itoften sheds light on some vital information in the vibratingsystem This paper therefore investigates the flexural vibra-tions of a rectangular plate with general boundary conditionsunder uniformly distributed moving masses Both gravityand inertia effects of the uniformly distributed masses aretaken into consideration and the plate is taken to reston Winkler foundation The solution technique which isanalytical and suitable for all variants of classical boundaryconditions involves using the generalized two-dimensionalintegral transform the expression of the Heaviside functionas a Fourier series and the use of the modified Strublersquosasymptotic technique to solve the problem of the flexuralvibrations of a rectangular plate having arbitrary supports atboth ends In addition conditions under which resonance isreached is obtained for both uniformly distributed movingforce and uniformly distributed moving mass problems
2 Governing Equation
Consider the dynamic transverse displacement119882(119909 119910 119905) ofa rectangular flat plate of span 119871
119910along the 119910-axis and span
119871119909along the 119909-axis carrying a uniformly distributed mass
moving with constant velocity 119888 along a straight line 119910 = 1199100
parallel to the 119909-axisIf the platemodel incorporates the rotatory inertia correc-
tion factor119882(119909 119910 119905) is governed by the fourth order partialdifferential equation given by [23]
(119863(1205972
1205971199092+
1205972
1205971199102)
2
minus 1205831198770
1205972
1205971199052)(
1205972
1205971199092+
1205972
1205971199102)
2
times119882(119909 119910 119905) + 1205831205972119882(119909 119910 119905)
1205971199052+ 119870119882(119909 119910 119905)
= 119875119865(119909 119910 119905) [1 minus
Δlowast
119892[119882 (119909 119910 119905)]]
(1)
119864 is Youngrsquosmodulus ] is Poissonrsquos ratio (] lt 1) 120583 is themassper unit length (self-weight) of the plate 119909 is the positioncoordinate in 119909-direction 119910 is the position coordinate in 119910-direction 119905 is the time ℎ is the plate thickness119870 is the elasticfoundation moduli and 119877
0is the measure of rotatory inertia
correction effect 119863 = 119864ℎ312(1 minus 120592) is the bending rigidity
of the plate and is constant throughout the plate 119875119865(119909 119910 119905) is
the continuous moving uniformly distributed force acting onthe beammodel 119892 is acceleration due to gravity andΔlowast is theconvective acceleration operator defined as
Δlowast= (
1205972
1205971199052+ 2119888
1205972
120597119905120597119909+ 1198882 1205972
1205971199092) (2)
Journal of Computational Engineering 3
The continuousmoving uniformly distributed force acting onthe plate is given by
119875119865(119909 119910 119905) = 119872119892119867 (119909 minus 119888119905)119867 (119910 minus 119910
0) (3)
where 119872 is the uniformly distributed mass 119867(sdot) is theHeaviside function The time 119905 is assumed to be limited tothat interval of time within which the mass119872 is on the platethat is
0 le 119888119905 le 119871119909 (4)
Thus in view of (2) and (3) (1) can be written as
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052+ 119870119882(119909 119910 119905)
minus 1205831198770[1205974119882(119909 119910 119905)
12059711990921205971199052+1205974119882(119909 119910 119905)
12059711991021205971199052]
+119872119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times [1205972119882(119909 119910 119905)
1205971199052+ 2119888
1205972119882(119909 119910 119905)
120597119905120597119909+ 11988821205972119882(119909 119910 119905)
1205971199092]
= 119872119892119867 (119909 minus 119888119905)119867 (119910 minus 1199100)
(5)
Equation (5) is the fourth order partial differential equationgoverning the flexural vibrations of a rectangular plate underthe action of uniformly distributed loadsThe boundary con-ditions are taken to be arbitrary while the initial conditionswithout any loss of generality are given by
119882(119909 119910 0) = 0 =120597119882(119909 119910 0)
120597119905 (6)
3 Method of Solution
The analysis of the dynamic response to a uniformly dis-tributed moving mass of isotropic rectangular plates restingon aWinkler foundation and subjected to arbitrary boundaryconditions is carried out in this sectionThe generalized two-dimensional integral transform is defined as
119882(119895 119896 119905) = int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (7)
with the inverse
119882(119909 119910 119905) =
infin
sum
119895=1
infin
sum
119896=1
120583
119881119895
120583
119881119896
119882(119895 119896 119905)119882119895 (119909)119882119896 (119910) (8)
where
119881119895= int
119871119909
0
1205831198822
119895(119909) 119889119909 119881
119896= int
119871119910
0
1205831198822
119896(119910) 119889119910 (9)
119882119895(119909) and119882
119896(119910) are respectively the beam mode functions
in the 119909 and 119910 directions defined respectively as
119882119895 (119909) = Sin
120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
119882119896(119910) = Sin
120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
(10)
where 119860119895 119861119895 119862119895 119860119896 119861119896 and 119862
119896are constants and 120582
119895
120582119896are mode frequencies which are determined using the
appropriate boundary conditionsApplying the generalized two-dimensional integral trans-
form (7) (5) takes the form
1198851198601198651(0 119871119909 119871119910 119905) + 119885
1198601198652(0 119871119909 119871119910 119905) + 119882
119905119905(119895 119896 119905)
minus 1198851198611198791198611 (119905) minus 1198851198611198791198612 (119905) + 119885119862119882(119895 119896 119905)
+ 1198791198613 (119905) + 1198791198614 (119905) + 1198791198615 (119905)
=119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896(119910) 119889119909 119889119910
(11)
where
119885119860=119863
120583 119885
119861= 1198770 119885
119862=119870
120583 (12)
1198651(0 119871119909 119871119910 119905)
= int
119871119910
0
([119882119895 (119909)
1205973119882(119909 119910 119905)
1205971199093minus1198821015840
119895(119909)
1205972119882(119909 119910 119905)
1205971199092
+11988210158401015840
119895(119909)
120597119882 (119909 119910 119905)
120597119909minus119882101584010158401015840
119895(119909)119882 (119909 119910 119905)]
119871119909
0
times119882119896(119910) 119889119910
+ 2 [119882119895 (119909)
120597119882 (119909 119910 119905)
120597119909minus1198821015840
119895(119909)119882 (119909 119910 119905)]
times 11988210158401015840
119896(119910) 119889119910)
+ int
119871119909
0
(2[119882119896(119910)
120597119882 (119909 119910 119905)
120597119910minus1198821015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
4 Journal of Computational Engineering
times11988210158401015840
119895(119909) 119889119909
+ [119882119896(119910)
1205973119882(119909 119910 119905)
1205971199103
minus1198821015840
119896(119910)
1205972119882(119909 119910 119905)
1205971199102+11988210158401015840
119896(119910)
120597119882 (119909 119910 119905)
120597119910
minus119882101584010158401015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
119882119895 (119909) 119889119909)
(13)
1198652(0 119871119909 119871119910 119905)
= int
119871119910
0
int
119871119909
0
(119882 (119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910)
+ 2119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910)) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
(14)
1198791198611 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(15)
1198791198612 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199102119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (16)
1198791198613 (119905) =
119872
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199052119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(17)
1198791198614 (119905) = 2
119872119888
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
120597119905120597119909119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(18)
1198791198615 (119905) =
1198721198882
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(19)It is recalled that the equation of the free vibration of arectangular plate is given by
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052= 0
(20)
If the free vibration solution of the problem is set as119882(119909 119910 119905) = 119882
119895 (119909)119882119896 (119910) SinΩ119895119896119905 (21)
where Ω119895119896
is the natural circular frequency of a rectangularplate then substituting (21) into (20) yields
119863[1198821015840120592
119895(119909)119882119896(119910) + 2119882
10158401015840
119895(119909)119882
10158401015840
119896(119910) +119882
119895 (119909)1198821015840120592
119896(119910)]
minus 120583Ω2
119895119896119882119895 (119909)119882119896 (119910) = 0
(22)
It is well known that for a simply supported rectangular plateΩ2
119895119896is given by
Ω2
119895119896= 119863[
11989541205874
1198714119909
+ 2119895211989621205874
11987121199091198712119910
+11989641205874
1198714119910
] (23)
Multiplying (22) by119882(119909 119910 119905) and integrating with respect to119909 and 119910 between the limits 0 119871
119909and 0 119871
119910 respectively we
get
int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910) 119889119909 119889119910
+ 2int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
=120583
119863Ω2
119895119896int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(24)
In view of (14) we have
1198652(0 119871119909 119871119910) =
120583
119863Ω2
119895119896119882(119895 119896 119905) (25)
Since119882(119901 119902 119905) is just the coefficient of the generalized two-dimensional finite integral transform
119882(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)119882119901 (119909)119882119902 (119910) (26)
it follows that
11988210158401015840(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)11988210158401015840
119901(119909)119882119902 (119910)
(27)
Therefore the integrals (15) and (16) can be rewritten as
1198791198611 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867
2(119901 119895) 119865 (119902 119896) (28)
where
1198672(119901 119895) = int
119871119909
0
11988210158401015840
119901(119909)119882119895 (119909) 119889119909
119865 (119902 119896) = int
119871119910
0
119882119902(119910)119882
119896(119910) 119889119910
(29)
1198791198612 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896) (30)
Journal of Computational Engineering 5
where
119867(119901 119895) = int
119871119909
0
119882119901 (119909)119882119895 (119909) 119889119909
1198652(119902 119896) = int
119871119910
0
11988210158401015840
119902(119910)119882
119896(119910) 119889119910
(31)
In order to evaluate the integrals (17) (18) and (19) use ismade of the Fourier series representation of the Heavisidefunction namely
119867(119909 minus 119888119905) =1
4+1
120587
infin
sum
119899=0
Sin ((2119899 + 1) 120587 (119909 minus 119888119905))2119899 + 1
(32)
Similarly
119867(119910 minus 1199100) =
1
4+1
120587
infin
sum
119898=0
Sin ((2119898 + 1) 120587 (119910 minus 1199100))
2119898 + 1 (33)
Using (30) and (31) one obtains
1198791198613 (119905)
=119872
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905) 119865 (119902 119896)119867 (119901 119895)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119901 119895) 119865 (119902 119896)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119901 119895) 119865 (119902 119896)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119902 119896)119867 (119901 119895)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119902 119896)119867 (119901 119895)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119888(119898 119902 119896)
(34)
where
119867119904(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119867119888(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119865119904(119898 119902 119896) = int
119871119910
0
Sin (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
119865119888(119898 119902 119896) = int
119871119910
0
Cos (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
(35)
6 Journal of Computational Engineering
Similarly
1198791198614 (119905)
=2119872119888
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905(119901 119902 119905)119867
1(119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119901 119895) 119865 (119902 119896)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119888(119898 119902 119896)
(36)
where
1198671(119901 119895) = int
119871119909
0
1198821015840
119901(119909)119882119895 (119909) 119889119909
119867119904
1(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
119867119888
1(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
1198791198615 (119905)
=1198721198882
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882(119901 119902 119905)1198672(119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119901 119895) 119865 (119902 119896)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 3
The continuousmoving uniformly distributed force acting onthe plate is given by
119875119865(119909 119910 119905) = 119872119892119867 (119909 minus 119888119905)119867 (119910 minus 119910
0) (3)
where 119872 is the uniformly distributed mass 119867(sdot) is theHeaviside function The time 119905 is assumed to be limited tothat interval of time within which the mass119872 is on the platethat is
0 le 119888119905 le 119871119909 (4)
Thus in view of (2) and (3) (1) can be written as
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052+ 119870119882(119909 119910 119905)
minus 1205831198770[1205974119882(119909 119910 119905)
12059711990921205971199052+1205974119882(119909 119910 119905)
12059711991021205971199052]
+119872119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times [1205972119882(119909 119910 119905)
1205971199052+ 2119888
1205972119882(119909 119910 119905)
120597119905120597119909+ 11988821205972119882(119909 119910 119905)
1205971199092]
= 119872119892119867 (119909 minus 119888119905)119867 (119910 minus 1199100)
(5)
Equation (5) is the fourth order partial differential equationgoverning the flexural vibrations of a rectangular plate underthe action of uniformly distributed loadsThe boundary con-ditions are taken to be arbitrary while the initial conditionswithout any loss of generality are given by
119882(119909 119910 0) = 0 =120597119882(119909 119910 0)
120597119905 (6)
3 Method of Solution
The analysis of the dynamic response to a uniformly dis-tributed moving mass of isotropic rectangular plates restingon aWinkler foundation and subjected to arbitrary boundaryconditions is carried out in this sectionThe generalized two-dimensional integral transform is defined as
119882(119895 119896 119905) = int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (7)
with the inverse
119882(119909 119910 119905) =
infin
sum
119895=1
infin
sum
119896=1
120583
119881119895
120583
119881119896
119882(119895 119896 119905)119882119895 (119909)119882119896 (119910) (8)
where
119881119895= int
119871119909
0
1205831198822
119895(119909) 119889119909 119881
119896= int
119871119910
0
1205831198822
119896(119910) 119889119910 (9)
119882119895(119909) and119882
119896(119910) are respectively the beam mode functions
in the 119909 and 119910 directions defined respectively as
119882119895 (119909) = Sin
120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
119882119896(119910) = Sin
120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
(10)
where 119860119895 119861119895 119862119895 119860119896 119861119896 and 119862
119896are constants and 120582
119895
120582119896are mode frequencies which are determined using the
appropriate boundary conditionsApplying the generalized two-dimensional integral trans-
form (7) (5) takes the form
1198851198601198651(0 119871119909 119871119910 119905) + 119885
1198601198652(0 119871119909 119871119910 119905) + 119882
119905119905(119895 119896 119905)
minus 1198851198611198791198611 (119905) minus 1198851198611198791198612 (119905) + 119885119862119882(119895 119896 119905)
+ 1198791198613 (119905) + 1198791198614 (119905) + 1198791198615 (119905)
=119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896(119910) 119889119909 119889119910
(11)
where
119885119860=119863
120583 119885
119861= 1198770 119885
119862=119870
120583 (12)
1198651(0 119871119909 119871119910 119905)
= int
119871119910
0
([119882119895 (119909)
1205973119882(119909 119910 119905)
1205971199093minus1198821015840
119895(119909)
1205972119882(119909 119910 119905)
1205971199092
+11988210158401015840
119895(119909)
120597119882 (119909 119910 119905)
120597119909minus119882101584010158401015840
119895(119909)119882 (119909 119910 119905)]
119871119909
0
times119882119896(119910) 119889119910
+ 2 [119882119895 (119909)
120597119882 (119909 119910 119905)
120597119909minus1198821015840
119895(119909)119882 (119909 119910 119905)]
times 11988210158401015840
119896(119910) 119889119910)
+ int
119871119909
0
(2[119882119896(119910)
120597119882 (119909 119910 119905)
120597119910minus1198821015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
4 Journal of Computational Engineering
times11988210158401015840
119895(119909) 119889119909
+ [119882119896(119910)
1205973119882(119909 119910 119905)
1205971199103
minus1198821015840
119896(119910)
1205972119882(119909 119910 119905)
1205971199102+11988210158401015840
119896(119910)
120597119882 (119909 119910 119905)
120597119910
minus119882101584010158401015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
119882119895 (119909) 119889119909)
(13)
1198652(0 119871119909 119871119910 119905)
= int
119871119910
0
int
119871119909
0
(119882 (119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910)
+ 2119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910)) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
(14)
1198791198611 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(15)
1198791198612 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199102119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (16)
1198791198613 (119905) =
119872
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199052119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(17)
1198791198614 (119905) = 2
119872119888
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
120597119905120597119909119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(18)
1198791198615 (119905) =
1198721198882
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(19)It is recalled that the equation of the free vibration of arectangular plate is given by
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052= 0
(20)
If the free vibration solution of the problem is set as119882(119909 119910 119905) = 119882
119895 (119909)119882119896 (119910) SinΩ119895119896119905 (21)
where Ω119895119896
is the natural circular frequency of a rectangularplate then substituting (21) into (20) yields
119863[1198821015840120592
119895(119909)119882119896(119910) + 2119882
10158401015840
119895(119909)119882
10158401015840
119896(119910) +119882
119895 (119909)1198821015840120592
119896(119910)]
minus 120583Ω2
119895119896119882119895 (119909)119882119896 (119910) = 0
(22)
It is well known that for a simply supported rectangular plateΩ2
119895119896is given by
Ω2
119895119896= 119863[
11989541205874
1198714119909
+ 2119895211989621205874
11987121199091198712119910
+11989641205874
1198714119910
] (23)
Multiplying (22) by119882(119909 119910 119905) and integrating with respect to119909 and 119910 between the limits 0 119871
119909and 0 119871
119910 respectively we
get
int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910) 119889119909 119889119910
+ 2int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
=120583
119863Ω2
119895119896int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(24)
In view of (14) we have
1198652(0 119871119909 119871119910) =
120583
119863Ω2
119895119896119882(119895 119896 119905) (25)
Since119882(119901 119902 119905) is just the coefficient of the generalized two-dimensional finite integral transform
119882(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)119882119901 (119909)119882119902 (119910) (26)
it follows that
11988210158401015840(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)11988210158401015840
119901(119909)119882119902 (119910)
(27)
Therefore the integrals (15) and (16) can be rewritten as
1198791198611 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867
2(119901 119895) 119865 (119902 119896) (28)
where
1198672(119901 119895) = int
119871119909
0
11988210158401015840
119901(119909)119882119895 (119909) 119889119909
119865 (119902 119896) = int
119871119910
0
119882119902(119910)119882
119896(119910) 119889119910
(29)
1198791198612 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896) (30)
Journal of Computational Engineering 5
where
119867(119901 119895) = int
119871119909
0
119882119901 (119909)119882119895 (119909) 119889119909
1198652(119902 119896) = int
119871119910
0
11988210158401015840
119902(119910)119882
119896(119910) 119889119910
(31)
In order to evaluate the integrals (17) (18) and (19) use ismade of the Fourier series representation of the Heavisidefunction namely
119867(119909 minus 119888119905) =1
4+1
120587
infin
sum
119899=0
Sin ((2119899 + 1) 120587 (119909 minus 119888119905))2119899 + 1
(32)
Similarly
119867(119910 minus 1199100) =
1
4+1
120587
infin
sum
119898=0
Sin ((2119898 + 1) 120587 (119910 minus 1199100))
2119898 + 1 (33)
Using (30) and (31) one obtains
1198791198613 (119905)
=119872
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905) 119865 (119902 119896)119867 (119901 119895)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119901 119895) 119865 (119902 119896)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119901 119895) 119865 (119902 119896)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119902 119896)119867 (119901 119895)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119902 119896)119867 (119901 119895)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119888(119898 119902 119896)
(34)
where
119867119904(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119867119888(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119865119904(119898 119902 119896) = int
119871119910
0
Sin (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
119865119888(119898 119902 119896) = int
119871119910
0
Cos (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
(35)
6 Journal of Computational Engineering
Similarly
1198791198614 (119905)
=2119872119888
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905(119901 119902 119905)119867
1(119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119901 119895) 119865 (119902 119896)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119888(119898 119902 119896)
(36)
where
1198671(119901 119895) = int
119871119909
0
1198821015840
119901(119909)119882119895 (119909) 119889119909
119867119904
1(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
119867119888
1(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
1198791198615 (119905)
=1198721198882
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882(119901 119902 119905)1198672(119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119901 119895) 119865 (119902 119896)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 Journal of Computational Engineering
times11988210158401015840
119895(119909) 119889119909
+ [119882119896(119910)
1205973119882(119909 119910 119905)
1205971199103
minus1198821015840
119896(119910)
1205972119882(119909 119910 119905)
1205971199102+11988210158401015840
119896(119910)
120597119882 (119909 119910 119905)
120597119910
minus119882101584010158401015840
119896(119910)119882 (119909 119910 119905)]
119871119910
0
119882119895 (119909) 119889119909)
(13)
1198652(0 119871119909 119871119910 119905)
= int
119871119910
0
int
119871119909
0
(119882 (119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910)
+ 2119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910)) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
(14)
1198791198611 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(15)
1198791198612 (119905) = int
119871119910
0
int
119871119909
0
1205974119882(119909 119910 119905)
12059711990521205971199102119882119895 (119909)119882119896 (119910) 119889119909 119889119910 (16)
1198791198613 (119905) =
119872
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199052119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(17)
1198791198614 (119905) = 2
119872119888
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
120597119905120597119909119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(18)
1198791198615 (119905) =
1198721198882
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)
times1205972119882(119909 119910 119905)
1205971199092119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(19)It is recalled that the equation of the free vibration of arectangular plate is given by
119863[1205974119882(119909 119910 119905)
1205971199094+ 2
1205974119882(119909 119910 119905)
12059711990921205971199102+1205974119882(119909 119910 119905)
1205971199104]
+ 1205831205972119882(119909 119910 119905)
1205971199052= 0
(20)
If the free vibration solution of the problem is set as119882(119909 119910 119905) = 119882
119895 (119909)119882119896 (119910) SinΩ119895119896119905 (21)
where Ω119895119896
is the natural circular frequency of a rectangularplate then substituting (21) into (20) yields
119863[1198821015840120592
119895(119909)119882119896(119910) + 2119882
10158401015840
119895(119909)119882
10158401015840
119896(119910) +119882
119895 (119909)1198821015840120592
119896(119910)]
minus 120583Ω2
119895119896119882119895 (119909)119882119896 (119910) = 0
(22)
It is well known that for a simply supported rectangular plateΩ2
119895119896is given by
Ω2
119895119896= 119863[
11989541205874
1198714119909
+ 2119895211989621205874
11987121199091198712119910
+11989641205874
1198714119910
] (23)
Multiplying (22) by119882(119909 119910 119905) and integrating with respect to119909 and 119910 between the limits 0 119871
119909and 0 119871
119910 respectively we
get
int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)1198821015840120592
119895(119909)119882119896 (119910) 119889119909 119889119910
+ 2int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)11988210158401015840
119895(119909)119882
10158401015840
119896(119910) 119889119909 119889119910
+ int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882
1015840120592
119896(119910) 119889119909 119889119910
=120583
119863Ω2
119895119896int
119871119910
0
int
119871119909
0
119882(119909 119910 119905)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
(24)
In view of (14) we have
1198652(0 119871119909 119871119910) =
120583
119863Ω2
119895119896119882(119895 119896 119905) (25)
Since119882(119901 119902 119905) is just the coefficient of the generalized two-dimensional finite integral transform
119882(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)119882119901 (119909)119882119902 (119910) (26)
it follows that
11988210158401015840(119909 119910 119905) =
infin
sum
119901=1
infin
sum
119902=1
120583
119881119901
120583
119881119902
119882(119901 119902 119905)11988210158401015840
119901(119909)119882119902 (119910)
(27)
Therefore the integrals (15) and (16) can be rewritten as
1198791198611 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867
2(119901 119895) 119865 (119902 119896) (28)
where
1198672(119901 119895) = int
119871119909
0
11988210158401015840
119901(119909)119882119895 (119909) 119889119909
119865 (119902 119896) = int
119871119910
0
119882119902(119910)119882
119896(119910) 119889119910
(29)
1198791198612 (119905) =
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896) (30)
Journal of Computational Engineering 5
where
119867(119901 119895) = int
119871119909
0
119882119901 (119909)119882119895 (119909) 119889119909
1198652(119902 119896) = int
119871119910
0
11988210158401015840
119902(119910)119882
119896(119910) 119889119910
(31)
In order to evaluate the integrals (17) (18) and (19) use ismade of the Fourier series representation of the Heavisidefunction namely
119867(119909 minus 119888119905) =1
4+1
120587
infin
sum
119899=0
Sin ((2119899 + 1) 120587 (119909 minus 119888119905))2119899 + 1
(32)
Similarly
119867(119910 minus 1199100) =
1
4+1
120587
infin
sum
119898=0
Sin ((2119898 + 1) 120587 (119910 minus 1199100))
2119898 + 1 (33)
Using (30) and (31) one obtains
1198791198613 (119905)
=119872
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905) 119865 (119902 119896)119867 (119901 119895)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119901 119895) 119865 (119902 119896)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119901 119895) 119865 (119902 119896)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119902 119896)119867 (119901 119895)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119902 119896)119867 (119901 119895)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119888(119898 119902 119896)
(34)
where
119867119904(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119867119888(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119865119904(119898 119902 119896) = int
119871119910
0
Sin (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
119865119888(119898 119902 119896) = int
119871119910
0
Cos (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
(35)
6 Journal of Computational Engineering
Similarly
1198791198614 (119905)
=2119872119888
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905(119901 119902 119905)119867
1(119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119901 119895) 119865 (119902 119896)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119888(119898 119902 119896)
(36)
where
1198671(119901 119895) = int
119871119909
0
1198821015840
119901(119909)119882119895 (119909) 119889119909
119867119904
1(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
119867119888
1(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
1198791198615 (119905)
=1198721198882
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882(119901 119902 119905)1198672(119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119901 119895) 119865 (119902 119896)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 5
where
119867(119901 119895) = int
119871119909
0
119882119901 (119909)119882119895 (119909) 119889119909
1198652(119902 119896) = int
119871119910
0
11988210158401015840
119902(119910)119882
119896(119910) 119889119910
(31)
In order to evaluate the integrals (17) (18) and (19) use ismade of the Fourier series representation of the Heavisidefunction namely
119867(119909 minus 119888119905) =1
4+1
120587
infin
sum
119899=0
Sin ((2119899 + 1) 120587 (119909 minus 119888119905))2119899 + 1
(32)
Similarly
119867(119910 minus 1199100) =
1
4+1
120587
infin
sum
119898=0
Sin ((2119898 + 1) 120587 (119910 minus 1199100))
2119898 + 1 (33)
Using (30) and (31) one obtains
1198791198613 (119905)
=119872
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905119905(119901 119902 119905) 119865 (119902 119896)119867 (119901 119895)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119901 119895) 119865 (119902 119896)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119901 119895) 119865 (119902 119896)
+119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119902 119896)119867 (119901 119895)
minus119872
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119902 119896)119867 (119901 119895)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+119872
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119888(119898 119902 119896)
(34)
where
119867119904(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119867119888(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 120587119909119882119901 (119909)119882119895 (119909) 119889119909
119865119904(119898 119902 119896) = int
119871119910
0
Sin (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
119865119888(119898 119902 119896) = int
119871119910
0
Cos (2119898 + 1) 120587119910119882119902 (119910)119882119896 (119910) 119889119910
(35)
6 Journal of Computational Engineering
Similarly
1198791198614 (119905)
=2119872119888
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905(119901 119902 119905)119867
1(119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119901 119895) 119865 (119902 119896)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119888(119898 119902 119896)
(36)
where
1198671(119901 119895) = int
119871119909
0
1198821015840
119901(119909)119882119895 (119909) 119889119909
119867119904
1(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
119867119888
1(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
1198791198615 (119905)
=1198721198882
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882(119901 119902 119905)1198672(119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119901 119895) 119865 (119902 119896)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 Journal of Computational Engineering
Similarly
1198791198614 (119905)
=2119872119888
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882119905(119901 119902 119905)119867
1(119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119901 119895) 119865 (119902 119896)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119901 119895) 119865 (119902 119896)
+2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minus2119872119888
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+2119872119888
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119888(119898 119902 119896)
(36)
where
1198671(119901 119895) = int
119871119909
0
1198821015840
119901(119909)119882119895 (119909) 119889119909
119867119904
1(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
119867119888
1(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 1205871199091198821015840119901(119909)119882119895 (119909) 119889119909
1198791198615 (119905)
=1198721198882
16120583
infin
sum
119901=1
infin
sum
119902=1
1205832
119881119901119881119902
119882(119901 119902 119905)1198672(119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119901 119895) 119865 (119902 119896)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119901 119895) 119865 (119902 119896)
+1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesCos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minus1198721198882
4120583120587
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
timesSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 7
minus1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+1198721198882
1205831205872
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
1205832
119881119901119881119902
119882119905(119901 119902 119905)
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)
(37)
where
119867119904
2(119899 119901 119895) = int
119871119909
0
Sin (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119867119888
2(119899 119901 119895) = int
119871119909
0
Cos (2119899 + 1) 12058711990911988210158401015840119901(119909)119882119895 (119909) 119889119909
119872119892
120583int
119871119910
0
int
119871119909
0
119867(119909 minus 119888119905)119867 (119910 minus 1199100)119882119895 (119909)119882119896 (119910) 119889119909 119889119910
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(38)
where
119861119891(120582 119895) = minusCos 120582
119895+ 119860119895Sin 120582119895+ 119861119895Cos ℎ120582
119895+ 119862119895Sin ℎ120582
119895
119861119891 (120582 119896) = minusCos 120582119896 + 119860119896 Sin 120582119896 + 119861119896 Cos ℎ120582119896 + 119862119896 Sin ℎ120582119896
(39)
Substituting the above expressions in (11) and simplifying weobtain
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905) +
119870
120583119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905) 119867
2(119901 119895) 119865 (119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
119882119905119905(119901 119902 119905)119867 (119901 119895) 119865
2(119902 119896)]
+ Γ1119871119909119871119910
1
16[
infin
sum
119901=1
infin
sum
119902=1
119865 (119902 119896)119867 (119901 119895)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times119867119888(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867 (119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867 (119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119901 119895)
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
8 Journal of Computational Engineering
times119865119888(119898 119902 119896) ]
times119882119905119905(119901 119902 119905)
+ 21198881
16[
infin
sum
119901=1
infin
sum
119902=1
1198671(119901 119895) 119865 (119902 119896)] +
1
4120587
times [
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
1(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
1(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
1(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
1(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888
1(119899 119901 119895)119865
119888(119898 119902 119896)]
times119882119905(119901 119902 119905)
+ 11988821
16[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896)]
+1
4120587[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119904
2(119899 119901 119895) 119865 (119902 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1
times 119867119888
2(119899 119901 119895) 119865 (119902 119896))
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1
times 119865119904(119898 119902 119896)119867
2(119901 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1
times 119865119888(119898 119902 119896)119867
2(119901 119895))]
+1
1205872[
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119904(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119901 119895) 119865
119888(119898 119902 119896)
minus
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119901 119895) 119865
119904(119898 119902 119896)
+
infin
sum
119901=1
infin
sum
119902=1
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 9
times119867119888
2(119899 119901 119895) 119865
119888(119898 119902 119896)]
times119882(119901 119902 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(40)
where
Γ1=
119872
120583119871119909119871119910
(41)
Equation (40) is the generalized transformed coupled non-homogeneous second order ordinary differential equationdescribing the flexural vibration of the rectangular plateunder the action of uniformly distributed loads travellingat constant velocity It is now the fundamental equation ofour dynamical problem and holds for all arbitrary boundaryconditions The initial conditions are given by
119882(119895 119896 0) = 0 119882119905(119895 119896 0) = 0 (42)
In what follows two special cases of (40) are discussed
4 Solution of the Transformed Equation
41 Isotropic Rectangular Plate Traversed by UniformlyDistributed Moving Force An approximate model whichassumes the inertia effect of the uniformly distributedmovingmass119872 as negligible is obtained when the mass ratio Γ
1is set
to zero in (40) Thus setting Γ1= 0 in (40) one obtains
119882119905119905(119895 119896 119905) + Ω
2
119895119896119882(119895 119896 119905)
minus 1198770[
infin
sum
119901=1
infin
sum
119902=1
1198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)]
times119882119905119905(119901 119902 119905) +
119870
120583119882(119895 119896 119905)
=
119872119892119871119909119871119910
120583120582119895120582119896
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
sdot [119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus119861119896Cos ℎ
1205821198961199100
119871119910
minus 119862119896Sin ℎ
1205821198961199100
119871119910
]
(43)
This represents the classical case of a uniformly distributedmoving force problem associated with our dynamical systemEvidently an analytical solution to (43) is not possible Conse-quently we resort to the modified asymptotic technique dueto Struble discussed inNayfeh [24] By this technique we seekthemodified frequency corresponding to the frequency of thefree system due to the presence of the effect of the rotatoryinertia An equivalent free system operator defined by themodified frequency then replaces (43) To this end (43) isrearranged to take the form
119882119905119905(119895 119896 119905)
+ (
Ωlowast2
119898119891
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
)
times119882(119895 119896 119905)
minus1205761
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119901 119895) 119865 (119902 119896) + 119867 (119901 119895) 119865
2(119902 119896)
sdot 119882119905119905(119901 119902 119905)
=1198751
(1 minus 1205761119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
sdot [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(44)
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 Journal of Computational Engineering
where
Ωlowast2
119898119891= (Ω
2
119895119896+119870
120583) 120576
1=
1198770
119871119909119871119910
1198751=
119872119892119871119909119871119910
120583120582119895120582119896
119882(120582119896 1199100)
119882 (120582119896 1199100)
= 119861119891 (120582 119896) + Cos
1205821198961199100
119871119910
minus 119860119896Sin
1205821198961199100
119871119910
minus 119861119896Cos ℎ
1205821198961199100
119871119910
minus119862119896Sin ℎ
1205821198961199100
119871119910
(45)
We now set the right hand side of (44) to zero and consider aparameter 120578 lt 1 for any arbitrary ratio 120576
1 defined as
120578 =1205761
1 + 1205761
(46)
Then
1205761= 120578 + 119900 (120578
2) (47)
And consequently we have that
1
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
= 1 + 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
+ 119900 (1205782)
(48)
where10038161003816100381610038161003816120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]10038161003816100381610038161003816lt 1 (49)
Substituting (47) and (48) into the homogeneous part of (44)one obtains
119882119905119905(119895 119896 119905)
+ (Ωlowast2
119898119891+ 120578Ωlowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])119882 (119895 119896 119905)
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]
times119882119905119905(119901 119902 119905) = 0
(50)
When 120578 is set to zero in (50) one obtains a situationcorresponding to the case in which the rotatory inertia effecton the vibrations of the plate is regarded as negligible In sucha case the solution of (50) can be obtained as
119882(119895 119896 119905) = 119862119898119891
Cos [Ωlowast119898119891119905 minus 120601119898119891] (51)
where 119862119898119891
and 120601119898119891
are constants
Since 120578 lt 1 Strublersquos technique requires that the solutionto (50) be of the form [24]
119882(119895 119896 119905) = 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
+ 1205781198821(119895 119896 119905) + 119900 (120578
2)
(52)
where 119860(119895 119896 119905) and 120601(119895 119896 119905) are slowly varying functions oftime To obtain the modified frequency (52) and its deriva-tives are substituted into (50) After some simplifications andrearrangements one obtains
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]]
times 119860 (119895 119896 119905)Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119895 119896 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119895 119896 119905)]
minus 120578
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
1198711199091198711199101198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)
sdot 2Ωlowast
119898119891120601 (119901 119902 119905) minus Ω
lowast2
119898119891119860 (119901 119902 119905)
times Cos [Ωlowast119898119891119905 minus 120601 (119895 119896 119905)]
minus 2 (119901 119902 119905)Ωlowast
119898119891Sin [Ωlowast
119898119891119905 minus 120601 (119901 119902 119905)]
(53)
neglecting terms to 119900(1205782)The variational equations are obtained by equating the
coefficients of Sin[Ωlowast119898119891119905minus120601(119895 119896 119905)] and Cos[Ωlowast
119898119891119905minus120601(119895 119896 119905)]
on both sides of (53) to zero thus one obtains
minus 2 (119895 119896 119905) Ωlowast
119898119891= 0
[2Ωlowast
119898119891120601 (119895 119896 119905) + 120578Ω
lowast2
119898119891119871119909119871119910
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)]] = 0
(54)
Evaluating (54) one obtains
119860 (119895 119896 119905) = 119860119895119896
120601 (119895 119896 119905) = minus 120578Ωlowast
119898119891
119871119909119871119910
2
times [1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)] 119905 + 120601119891
(55)
where 119860119895119896and 120601
119891are constants
Therefore when the effect of the rotatory inertia cor-rection factor is considered the first approximation to thehomogeneous system is given by
119881 (119895 119896 119905) = 119860119895119896Cos [120574
119901119898119891119905 minus 120601119891] (56)
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 11
where
120574119901119898119891
= Ωlowast
119898119891
times 1 +120578
2(119871119909119871119910[1198672(119895 119895)119865 (119896 119896) + 119867(119895 119895)1198652(119896 119896)])
(57)
represents themodified natural frequency due to the presenceof rotatory inertia correction factor It is observed that when120578 = 0 we recover the frequency of the moving force problemwhen the rotatory inertia effect of the plate is neglectedThusto solve the nonhomogeneous equation (44) the differentialoperator which acts on 119882(119895 119896 119905) and 119882(119901 119902 119905) is nowreplaced by the equivalent free systemoperator defined by themodified frequency 120574
119901119898119891 that is
119882119905119905(119895 119896 119905) + 120574
2
119901119898119891119882(119895 119896 119905)
= 119875119901119898119891
[119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(58)
where
119875119901119898119891
=1198751
(1 minus 120578119871119909119871119910[1198672(119895 119895) 119865 (119896 119896) + 119867 (119895 119895) 119865
2 (119896 119896)])
(59)
To obtain the solution to (58) it is subjected to a Laplacetransform Thus solving (58) in conjunction with the initialcondition the solution is given by
119881 (119895 119896 119905)
= 119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722119888+ 1205742
119901119898119891)
]
]
(60)
where
120572119895=
120582119895119888
119871119909
(61)
Substituting (60) into (8) we have
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
119875119901119898119891
[
[
119861119891(120582 119895) (1 minus Cos 120574
119901119898119891119905)
120574119901119898119891
+
Cos120572119895119905 minus Cos 120574
119901119898119891119905
1205742
119901119898119891minus 1205722
119895
minus
119860119895(120574119901119898119891
Sin120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205742
119901119898119891minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119901119898119891119905)
1205722
119895+ 1205742
119901119898119891
minus
119862119895(120574119901119898119891
Sin ℎ120572119895119905 minus 120572119895Sin 120574119901119898119891
119905)
120574119901119898119891
(1205722
119895+ 1205742
119901119898119891)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+ 119860119896Cos
120582119896119910
119871119910
+ 119861119896Sin ℎ
120582119896119910
119871119910
+ 119862119896Cos ℎ
120582119896119910
119871119910
)
(62)
Equation (62) represents the transverse displacementresponse to a uniformly distributed force moving at constantvelocity for arbitrary boundary conditions of a rectangularplate incorporating the effects of rotatory inertia correctionfactor and resting on elastic foundation
42 Isotropic Rectangular Plate Traversed by Uniformly Dis-tributed Moving Mass In this section we consider the casein which the mass of the structure and that of the load areof comparable magnitude Under such condition the inertiaeffect of the uniformly distributedmovingmass is not negligi-ble Thus Γ
1= 0 and the solution to the entire equation (40)
is required This gives the moving mass problem An exactanalytical solution to this dynamical problem is not possibleAgain we resort to the modified asymptotic technique due toStruble discussed in Nayfeh [24] Evidently the homogenouspart of (40) can be replaced by a free system operator definedby the modified frequency due to the presence of rotatoryinertia correction factor 119877
0 To this end (40) can now be
rearranged to take the form
119882119905119905(119895 119896 119905) +
2119888Γ11198711199091198711199101198662(119895 119896 119905)
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882119905(119895 119896 119905)
+
1198882Γ11198711199091198711199101198663(119895 119896 119905) + 120574
2
119901119898119891
1 + Γ11198711199091198711199101198661(119895 119896 119905)
119882 (119895 119896 119905)
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
12 Journal of Computational Engineering
+
Γ1119871119909119871119910
1 + Γ11198711199091198711199101198661(119895 119896 119905)
times
infin
sum
119901=1
119901 = 119895
infin
sum
119902=1
119902 = 119896
119866119886(119901 119902 119905)119882
119905119905(119901 119902 119905) + 2119888119866
119887(119901 119902 119905)
times 119882119905(119901 119902 119905) + 119888
2119866119888(119901 119902 119905)119882 (119901 119902 119905)
=
119872119892119871119909119871119910119882(120582119896 1199100)
120583120582119895120582119896[1 + Γ
11198711199091198711199101198661(119895 119896 119905)]
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(63)
where
1198661(119895 119896 119905)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895119895))]
minus1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times119867119888(119899 119901 119895) 119865
119888(119898 119902 119896) ]
1198662(119895 119896 119905)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
1(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
1(119899 119895 119895) 119865
119888(119898 119896 119896)]
1198663(119895 119896 119905)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898119896119896)1198672(119895 119895))]
+1
1205872[
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119904
2(119899 119895 119895) 119865
119904(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Cos (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 13
times 119867119904
2(119899 119895 119895) 119865
119888(119898 119896 119896)
minus
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotCos (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119904(119898 119896 119896)
+
infin
sum
119899=0
infin
sum
119898=0
Sin (2119899 + 1) 1205871198881199052119899 + 1
sdotSin (2119898 + 1) 1205871199100
2119898 + 1
times 119867119888
2(119899 119895 119895) 119865
119888(119898 119896 119896) ]
119866119886(119901 119902 119905) = 119866
1(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119887(119901 119902 119905) = 119866
2(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
119866119888(119901 119902 119905) = 119866
3(119895 119896 119905)
1003816100381610038161003816119895=119901119896=119902
(64)
Going through similar argument as in the previous sectionwe obtain the first approximation to the homogenous systemwhen the effect of the mass of the uniformly distributed loadis considered as
119882(119895 119896 119905) = Δ119895119896119890minus120573(119895119896)119905 Cos [120574
119898119898119905 minus 120601119898] (65)
where 120601119898and Δ
119895119896are constants Consider
120573 (119895 119896) 119905 = 119888Γ119886119871119909119871119910119867119888(119895 119896) Γ
119886=
Γ1
1 + Γ1
(66)
where
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]
(67)
is called the modified natural frequency representing thefrequency of the free system due to the presence of theuniformly distributed moving mass and
119867119886(119895 119896)
=1
16119865 (119896 119896)1198672 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
2(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
2(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198672 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198672(119895 119895))]
119867119887(119895 119896)
=1
16119865 (119896 119896)119867 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)119867 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)119867 (119895 119895))]
119867119888(119895 119896)
=1
16119865 (119896 119896)1198671 (119895 119895)
+1
4120587[
infin
sum
119899=0
(Cos (2119899 + 1) 120587119888119905
2119899 + 1119867119904
1(119899 119895 119895) 119865 (119896 119896)
minusSin (2119899 + 1) 120587119888119905
2119899 + 1119867119888
1(119899 119895 119895) 119865 (119896 119896))
+
infin
sum
119898=0
(Cos (2119898 + 1) 1205871199100
2119898 + 1119865119904(119898 119896 119896)1198671 (119895 119895)
minusSin (2119898 + 1) 1205871199100
2119898 + 1119865119888(119898 119896 119896)1198671 (119895 119895))]
(68)
To solve the nonhomogeneous equation (63) the differentialoperator which acts on119882(119895 119896 119905) and119882(119901 119902 119905) is replaced bythe equivalent free system operator defined by the modifiedfrequency 120574
119898119898 that is
119882119905119905(119895 119896 119905) + 120574
2
119898119898119882(119895 119896 119905)
=
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
14 Journal of Computational Engineering
times [119861119891(120582 119895) + Cos
120582119895119888119905
119871119909
minus 119860119895Sin
120582119895119888119905
119871119909
minus 119861119895Cos ℎ
120582119895119888119905
119871119909
minus 119862119895Sin ℎ
120582119895119888119905
119871119909
]
(69)
Evidently (69) is analogous to (58) Thus using similarargument as in the previous section solution to (69) can beobtained as
119882(119909 119910 119905)
=
infin
sum
119895=1
infin
sum
119896=1
1
119881119895119881119896
Γ1198861198921198712
1199091198712
119910119882(120582119896 1199100)
120582119895120582119896
times [
[
119861119891(120582 119895) (1 minus Cos 120574
119898119898119905)
120574119898119898
+
Cos120572119895119905 minus Cos 120574
119898119898119905
1205742119898119898
minus 1205722
119895
minus
119860119895(120574119898119898
Sin120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205742119898119898
minus 1205722
119895)
minus
119861119895(Cos ℎ120572
119895119905 minus Cos 120574
119898119898119905)
1205722
119895+ 1205742119898119898
minus
119862119895(120574119898119898
Sin ℎ120572119895119905 minus 120572119895Sin 120574119898119898
119905)
120574119898119898
(1205722
119895+ 1205742119898119898
)
]
]
sdot (Sin120582119895119909
119871119909
+ 119860119895Cos
120582119895119909
119871119909
+ 119861119895Sin ℎ
120582119895119909
119871119909
+ 119862119895Cos ℎ
120582119895119909
119871119909
)
sdot (Sin120582119896119910
119871119910
+119860119896Cos
120582119896119910
119871119910
+119861119896Sin ℎ
120582119896119910
119871119910
+119862119896Cos ℎ
120582119896119910
119871119910
)
(70)
Equation (70) represents the transverse displacementresponse to uniformly distributed masses moving at constantvelocity of an isotropic rectangular plate resting on elasticfoundation for various end conditions
5 Illustrative Examples
In this section practical examples of classical boundaryconditions are selected to illustrate the analyses presented inthis paper
51 Rectangular Plate Clamped at Edge 119909 = 0 119909 = 119871119909with
Simple Supports at 119910 = 0 119910 = 119871119910 In this example the
rectangular plate has simple supports at the edges 119910 = 0
119910 = 119871119910and is clamped at edges 119909 = 0 119909 = 119871
119909 The boundary
conditions at such opposite edges are
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
1205972119882(119909 0 119905)
1205971199102= 0
1205972119882(119909 119871
119910 119905)
1205971199102= 0
(71)
and hence for normal modes
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
1205972119882119896 (0)
1205971199102= 0
1205972119882119896(119871119910)
1205971199102= 0
(72)
Using the boundary conditions (71) in (10) the followingvalues of the constants are obtained for the clamped edges
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(73)
119861119895= minus1 997904rArr 119861
119901= minus1
119862119895= minus119860
119895997904rArr 119862
119901= minus119860
119901
(74)
The frequency equation for the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (75)
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 15
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (76)
119881119895=120583119871119909
2
times 1 + 1198602
119895+ 1198612
119895+ 1198622
119895+1
120582119895
times [2119862119895minus 2119860119895119861119895minus 119861119895119862119895minus1
2(1 minus 119860
2
119895) Sin 2120582
119895
+ (1198612
119895+ 1198622
119895) Sin ℎ120582
119895sdot Cos ℎ120582
119895+ 2119860119895Sin2120582
119895
+ 2 (119861119895+ 119860119895119862119895) Sin 120582
119895Cos ℎ120582
119895
+ 2 (minus119861119895+ 119860119895119862119895) Sin ℎ120582
119895Cos 120582
119895
+ 2 (119862119895+ 119860119895119861119895) Sin ℎ120582
119895Sin 120582119895
+ 2 (minus119862119895+ 119860119895119861119895)Cos 120582
119895Cos ℎ120582
119895
+119861119895119862119895Cos ℎ2120582
119895]
(77)
119881119901is obtained by replacing subscript 119895 with 119901 in (77) For the
simple edges it is readily shown that
119860119896= 0 997904rArr 119860
119902= 0 119861
119896= 0 997904rArr 119861
119902= 0
119862119896= 0 997904rArr 119862
119902= 0
(78)
The corresponding frequency equation is
120582119896= 119896120587 997904rArr 120582
119902= 119902120587 (79)
119881119896=
120583119871119910
2 119881
119902=
120583119871119910
2 (80)
Thus the general solutions of the associated uniformlydistributed moving force and uniformly distributed movingmass problems of the simple-clamped rectangular plate areobtained by substituting the above results in (73) to (80) into(62) and (70)
52 Rectangular Plate Clamped at All Edges For a rectangularplate clamped at all edges both the deflection and the slopevanish at such endsThus the following boundary conditionspertain
119882(0 119910 119905) = 0 119882 (119871119909 119910 119905) = 0
119882 (119909 0 119905) = 0 119882 (119909 119871119910 119905) = 0
120597119882 (0 119910 119905)
120597119909= 0
120597119882 (119871119909 119910 119905)
120597119909= 0
120597119882 (119909 0 119905)
120597119910= 0
120597119882(119909 119871119910 119905)
120597119910= 0
(81)
and hence for normal modes we have
119882119895 (0) = 0 119882
119895(119871119909) = 0
119882119896 (0) = 0 119882
119896(119871119910) = 0
120597119882119895 (0)
120597119909= 0
120597119882119895(119871119909)
120597119909= 0
120597119882119896 (0)
120597119910= 0
120597119882119896(119871119910)
120597119910= 0
(82)
Using the boundary conditions (81)-(82) in (10) one obtainsthe following values of the constants for the clamped edges at119909 = 0 and 119909 = 119871
119909
119860119895=
Sin ℎ120582119895minus Sin 120582
119895
Cos 120582119895minus Cos ℎ120582
119895
997904rArr 119860119901=
Sin ℎ120582119901minus Sin 120582
119901
Cos 120582119901minus Cos ℎ120582
119901
(83)
119861119895= minus1 997904rArr 119861
119901= minus1 119862
119895= minus119860
119895997904rArr 119862
119901= minus119860
119901 (84)
The frequency equation of the clamped edges is given by
Cos 120582119895Cos ℎ120582
119895minus 1 = 0 (85)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (86)
119881119895is defined by (77) and119881
119901is obtained by replacing subscript
119895 with 119901For the clamped edges at 119910 = 0 and 119910 = 119871
119910 the same
process is followed to obtain
119860119896=
Sin ℎ120582119896minus Sin 120582
119896
Cos 120582119896minus Cos ℎ120582
119896
997904rArr 119860119902=
Sin ℎ120582119902minus Sin 120582
119902
Cos 120582119902minus Cos ℎ120582
119902
119861119896= minus1 997904rArr 119861
119902= minus1 119862
119896= minus119860
119896997904rArr 119862
119902= minus119860
119902
(87)
The frequency equation of the clamped edges is given by
Cos 120582119896Cos ℎ120582
119896minus 1 = 0 (88)
such that
1205821= 473004 120582
2= 785320 120582
3= 1099561 (89)
119881119896and 119881
119902are obtained by replacing subscript 119895 with 119896 and 119902
in (77) respectivelyThus the general solutions of the associated uniformly
distributed moving force and uniformly distributed movingmass problems of the clamped-clamped rectangular plate areobtained by substituting the above results in (83) to (89) into(62) and (70)
6 Discussion of the Analytical Solutions
When an undamped system such as this is studied one isinterested in the resonance conditions of the vibrating system
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
16 Journal of Computational Engineering
because the transverse displacement of the elastic rectangularplate may increase without bound Thus for both illustrativeexamples we observe that the isotropic rectangular platetraversed by a uniformly distributed moving force reaches astate of resonance whenever
120574119901119898119891
=
120582119895119888
119871119909
(90)
while the same plate under the action of a uniformly dis-tributedmovingmass experiences resonance effect whenever
120574119898119898
=
120582119895119888
119871119909
(91)
but
120574119898119898
= 120574119901119898119891
1 minus
Γ119886119871119909119871119910
2[119867119887(119895 119896) minus
1198882119867119886(119895 119896)
1205742
119901119898119891
]=
120582119895119888
119871119909
(92)
Equations (90) and (92) show that for the same natural fre-quency the critical speed for the same system consisting of anisotropic rectangular plate resting on a constant foundationand traversed by a uniformly distributed moving force isgreater than that traversed by a uniformly distributedmovingmass Thus resonance is reached earlier in the movingdistributed mass system than in the moving distributed forcesystem
7 Numerical Results and Discussion
In order to illustrate the foregoing analysis a rectangular plateof length 119871
119909= 457m and width 119871
119910= 914m is considered
The mass per unit length of the plate is 120583 = 2758291Kgmmass ratio Γ
1= 02 and bending rigidity 119863 = 10000 The
velocity of the travelling distributed load is 119888 = 15msThe values of the foundation modulus 119870 are varied between0Nm3 and 4000000Nm3 and the values of the rotatoryinertia correction factor 119877
0are varied between 005 and 95
In Figure 3 the transverse displacement response ofthe simple-clamped rectangular plate to moving distributedforces for various values of Foundation modulli 119870 and fixedvalue of rotatory inertia correction factor 119877
0= 05 is
displayed It is seen from the figure that as the values ofthe foundation modulli increase the response amplitude ofthe simple-clamped rectangular plate under the action ofdistributed forces decreases The same results are obtainedwhen the same simple-clamped rectangular plate is traversedby moving distributed masses as depicted in Figure 5 Theresponse of the simple-clamped rectangular plate to mov-ing distributed forces for various values of rotatory inertiacorrection factor 119877
0and fixed value of foundation modulli
119870 = 4000 is shown in Figure 4 It is seen that the displacementof the plate decreases with increase in the rotatory inertiacorrection factor The same behaviour characterizes thedeflection profile of the same simple-clamped rectangularplate when it is traversed by moving distributed massesas shown in Figure 6 In Figure 7 the transverse displace-ment response of the clamped-clamped rectangular plate to
00
1 2 3 4 5
Plat
e disp
lace
men
t (m
)
K = 0
K = 4000
K = 40000
K = 400000
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
2E minus 10
4E minus 10
Travelling time t (s)
Figure 3 Displacement response to distributed forces of simple-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus12E minus 09
minus1E minus 09
minus8E minus 10
minus6E minus 10
minus4E minus 10
minus2E minus 10
0
2E minus 10
4E minus 10
6E minus 10
R0 = 05
R0 = 25
R0 = 55
Travelling time t (s)
Figure 4 Displacement response to distributed forces of simple-clamped plate for various values of rotatory inertia correction factor1198770
moving distributed forces for various values of Foundationmodulli119870 and fixed value of rotatory inertia correction factor1198770= 05 is displayed It is seen from the figure that as
the values of the foundation modulli increase the responseamplitude of the clamped-clamped rectangular plate underthe action of distributed forces decreasesThe same behaviourcharacterizes the displacement profile of the same clamped-clamped rectangular plate when it is traversed by movingdistributed masses as shown in Figure 9 The response ofthe clamped-clamped rectangular plate tomoving distributedforces for various values of rotatory inertia correction factor1198770and fixed value of foundation modulli 119870 = 4000 is
shown in Figure 8 It is seen that the displacement of theplate decreaseswith increase in the rotatory inertia correctionfactor
In Figure 10 the transverse displacement response of theclamped-clamped rectangular plate to moving distributedmasses for various values of rotatory inertia correction factor
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 17
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus15E minus 08
minus1E minus 08
minus5E minus 09
0
5E minus 09
1E minus 08
K = 0
K = 4000
K = 400000
K = 4000000
Travelling time t (s)
Figure 5 Displacement response to distributed masses of simple-clamped plate for various values of foundation moduli 119870
Plat
e disp
lace
men
t (m
)
0 2 4 6
00000001
12E minus 07
14E minus 07
0
2E minus 08
4E minus 08
6E minus 08
8E minus 08
R0 = 05
R0 = 095
R0 = 15
Travelling time t (s)
Figure 6 Displacement response to distributed masses of simple-clamped plate for various values of rotatory inertia correction factor1198770
1198770and fixed value of foundation modulli119870 = 4000 is shown
It is seen that the displacement of the plate decreases withincrease in the rotatory inertia correction factor
Figure 11 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the simple-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000
and rotatory inertia correction factor 1198770
= 05 Clearlythe response amplitude of the moving distributed mass isgreater than that of the moving distributed force problemWhile Figure 12 depicts the comparison of the transverse dis-placement response for moving distributed force andmovingdistributed mass cases of the clamped-clamped rectangularplate for fixed values of foundation modulli 119870 = 4000 androtatory inertia correction factor 119877
0= 05 it is observed
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 50
2E minus 09
4E minus 09
6E minus 09
8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
1E minus 08
K = 0
K = 4000
K = 40000
K = 400000
Travelling time t (s)
Figure 7 Displacement response to distributed forces of clamped-clamped plate for various values of foundation moduli 119870
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
minus4E minus 09
minus2E minus 09
minus3E minus 09
minus5E minus 09
minus1E minus 09
0
2E minus 09
4E minus 09
3E minus 09
1E minus 09
5E minus 09
6E minus 09
R0 = 05
R0 = 35
R0 = 55
Travelling time t (s)
Figure 8 Displacement response to distributed forces of clamped-clamped plate for various values of rotatory inertia correction factor1198770
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
K = 0
K = 40000
K = 400000
K = 4000000
Travelling time t (s)
Figure 9 Displacement response to distributed masses of clamped-clamped plate for various values of foundation moduli 119870
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
18 Journal of Computational EngineeringPl
ate d
ispla
cem
ent (
m)
0 1 2 3 4 5
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
25E minus 07
00000003
R0 = 05
R0 = 45
R0 = 95
Travelling time t (s)
Figure 10Displacement response to distributedmasses of clamped-clamped plate for various values of rotatory inertia correction factor1198770
0 1 2 3 4 5
Plat
e disp
lace
men
t (m
)
Moving forceMoving mass
minus12E minus 08
minus1E minus 08
minus8E minus 09
minus6E minus 09
minus4E minus 09
minus2E minus 09
0
2E minus 09
4E minus 09
6E minus 09
8E minus 09
Travelling time t (s)
Figure 11 Comparison of displacement response of distributedforce and distributed mass of simple-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
that the response amplitude of the moving distributed massis greater than that of the moving distributed force problem
8 Conclusion
In this paper the problem of the flexural vibrations of arectangular plate having arbitrary supports at both ends ispresented The solution technique suitable for all variants ofclassical boundary conditions involves using the generalizedtwo-dimensional integral transform to reduce the fourthorder partial differential equation governing the vibrationof the plate to a second order ordinary differential equationwhich is then treated with the modified asymptotic methodof Struble The closed form solutions obtained are analyzedand numerical analyses in plotted curves were presentedResults show that as the foundation moduli 119870 and rotatory
Plat
e disp
lace
men
t (m
)
0 1 2 3 4 5
Moving forceMoving mass
minus15E minus 07
minus1E minus 07
minus5E minus 08
0
5E minus 08
00000001
15E minus 07
00000002
Travelling time t (s)
Figure 12 Comparison of displacement response of distributedforce and distributedmass of clamped-clamped rectangular plate forfixed values of 119870 = 4000 and 119877
0= 05
inertia correction factor1198770increase the response amplitudes
of the dynamic system decrease for all illustrative examplesconsidered Analytical solutions further show that for thesame natural frequency the critical speed for the systemtraversed by uniformly distributed moving forces at constantspeed is greater than that of the uniformly distributed mov-ing mass problem for both clamped-clamped and simple-clamped end conditions Hence resonance is reached earlierin the moving distributed mass system It is clearly seen thatfor both end conditions under consideration the responseamplitude of the uniformly distributed moving mass systemis higher than that of the uniformly distributed movingforce system for fixed values of rotatory inertia correctionfactor and foundationmoduliThus higher values of rotatoryinertia correction factor and foundation moduli reduce therisk factor of resonance in a vibrating system
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] S Sadiku and H H E Leipholz ldquoOn the dynamics of elasticsystems with moving concentrated massesrdquo Ingenieur-Archivvol 57 no 3 pp 223ndash242 1987
[2] G V Rao ldquoLinear dynamics of an elastic beam under movingloadsrdquo Journal of Vibration andAcoustics vol 122 no 3 pp 281ndash289 2000
[3] Y-H Chen and C-Y Li ldquoDynamic response of elevated high-speed railwayrdquo Journal of Bridge Engineering vol 5 no 2 pp124ndash130 2000
[4] E Savin ldquoDynamic amplification factor and response spectrumfor the evaluation of vibrations of beams under successivemoving loadsrdquo Journal of Sound and Vibration vol 248 no 2pp 267ndash288 2001
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
Journal of Computational Engineering 19
[5] S T Oni and T O Awodola ldquoVibrations under a moving loadof a non-uniformRayleigh beamonvariable elastic foundationrdquoJournal of the Nigerian Association of Mathematical Physics vol7 pp 191ndash206 2003
[6] S T Oni and B Omolofe ldquoDynamic Analysis of a prestressedelastic beam with general boundary conditions under movingloads at varying velocitiesrdquo Journal of Engineering and Engineer-ing Technology FUTA vol 4 no 1 pp 55ndash72 2005
[7] B Omolofe and S O Ajibola ldquoOn the transversemotions underheavy loads of thin beams with variable prestressrdquo Journal of theNigerian Association ofMathematical Physics vol 9 pp 127ndash1422005
[8] G Muscolino and A Palmeri ldquoResponse of beams restingon viscoelastically damped foundation to moving oscillatorsrdquoInternational Journal of Solids and Structures vol 44 no 5 pp1317ndash1336 2007
[9] J A Gbadeyan and S T Oni ldquoDynamic response to movingconcentrated masses of elastic plates on a non-Winkler elasticfoundationrdquo Journal of Sound and Vibration vol 154 no 2 pp343ndash358 1992
[10] S T Oni ldquoFlexural vibrations under Moving Loads of Isotropicrectangular plates on a Non-Winkler elastic foundationrdquo Jour-nal of the Nigerian Society of Engineers vol 35 no 1 pp 18ndash272000
[11] M-H Huang and D P Thambiratnam ldquoDeflection responseof plate on winkler foundation to moving accelerated loadsrdquoEngineering Structures vol 23 no 9 pp 1134ndash1141 2001
[12] M R Shadnam M Mofid and J E Akin ldquoOn the dynamicresponse of rectangular plate with moving massrdquo Thin-WalledStructures vol 39 no 9 pp 797ndash806 2001
[13] S W Alisjahbana ldquoDynamic response of clamped orthotropicplates to dynamic moving loadsrdquo in Proceedings of the 13thWorld Conference on Earthquake Engineering Paper no 3176Vancouver BC Canada August 2004
[14] J-S Wu M-L Lee and T-S Lai ldquoThe Dynamic analysis of aflat plate under a moving load by the finite element methodrdquoInternational Journal for Numerical Methods in Engineering vol24 no 4 pp 743ndash762 1987
[15] R-T Wang and T-Y Lin ldquoRandom vibration of multi-spanmindlin plate due to moving loadrdquo Journal of the ChineseInstitute of Engineers vol 21 no 3 pp 347ndash356 1998
[16] J A Gbadeyan and M S Dada ldquoDynamic response of plateson Pasternak foundation to distributed moving loadsrdquo Journalof Nigerian Association of Mathematical Physics vol 5 pp 184ndash200 2001
[17] M S Dada Vibration analysis of elastic plates under uniformpartially distributed moving loads [PhD thesis] University ofIlorin Ilorin Nigeria 2002
[18] M A Usman ldquoDynamic response of a Bernoulli beam onWinkler foundation under the action of partially distributedloadrdquoNigerian Journal of Mathematics and Applications vol 16no 1 pp 128ndash150 2003
[19] I A Adetunde Dynamic analysis of Rayleigh beam carryingan added mass and traversed by uniform partially distributedmoving loads [PhD thesis] University of Ilorin Ilorin Nigeria2003
[20] J AGbadeyan andM SDada ldquoDynamic response of aMindlinelastic rectangular plate under a distributed moving massrdquoInternational Journal of Mechanical Sciences vol 48 no 3 pp323ndash340 2006
[21] N D Beskou and D DTheodorakopoulos ldquoDynamic effects ofmoving loads on road pavements a reviewrdquo Soil Dynamics andEarthquake Engineering vol 31 no 4 pp 547ndash567 2011
[22] J V Amiri A NikkhooM R Davoodi andM E HassanabadildquoVibration analysis of a Mindlin elastic plate under a movingmass excitation by eigenfunction expansion methodrdquo Thin-Walled Structures vol 62 pp 53ndash64 2013
[23] L Fryba Vibrations of Solids and Structures Under MovingLoads Noordhoff Groningen The Netherlands 1972
[24] A H Nayfey Perturbation Methods John Wiley amp Sons NewYork NY USA 1973
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
