Research ArticleOn the Dynamic Analysis of a Beam Carrying MultipleMass-Spring-Mass-Damper System

O Barry1 D C D Oguamanam2 and J W Zu1

1 Department of Mechanical and Industrial Engineering University of Toronto Toronto ON Canada M5S 3G82Department of Mechanical and Industrial Engineering Ryerson University Toronto ON Canada M5B 2K3

Correspondence should be addressed to O Barry rafioulabegmailcom

Received 25 July 2013 Accepted 6 October 2013 Published 23 February 2014

Academic Editor Jeong-Hoi Koo

Copyright copy 2014 O Barry et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The exact natural frequencies mode shapes and the corresponding orthogonality relations are important in forced vibrationanalysis via modal expansion In the present paper a free vibration analysis is conducted to determine the exact natural frequenciesand mode shapes of an axially loaded beam carrying several absorbers An explicit expression is presented for the generalizedorthogonality relations These generalized orthogonality conditions are employed along with the assumed modes method toperform forced vibration analysis The present approach is compared to other approximate methods in the literature withthe classical orthogonality relations and different choice of mode shapes The results indicate that the use of the generalizedorthogonality relation with the exact mode shapes is required for a precise investigation of the dynamic response of a beam withmass-spring-mass-damper system

1 Introduction

The study of beam vibrations is attracting continued interestall these years because of the wide range of engineering appli-cations involving beams Beam models are used to idealizebuildings components bridges overhead transmission linesmicroelectromechanical systems andmanymore In the caseof overhead transmission lines a single conductor may bemodeled as a beam that is subjected to a tensile load with thedamper represented as a mass-spring-damper-mass system

Numerous authors have studied the free andor forcedresponse of beams carrying in-span mass or spring-masssystems In most investigations on the free vibration differ-ent authors have used analytical methods to determine theexact solutions for the natural frequencies and mode shapes(eg [1ndash13] and references mentioned therein)

There are however few published exact solutions ofthe forced vibration problems The most common analyt-ical approach used in forced vibration analysis is basedon the assumed modes method in conjunction with theclassical orthogonality conditions which may be expressed asint119871

0119884119894119884119895119889119909 = 120575

119894119895 where 119871 is the length of the beam 119884

119894and 119884

119895

are the 119894th and 119895th normal mode shapes respectively and 120575119894119895

is the Kronecker delta function Hereinafter a beam withoutmass or spring-mass system is referred to as a bare beam orelse it is called a loaded beam

In [14ndash16] the authors employed the mode shapes ofthe bare beam and the classical orthogonality condition tostudy the forced response of loaded beams Combinationsof the mode shapes of the loaded beam with the classicalorthogonality relation were employed in [17ndash23] These twoapproaches are simple and easy to implement and in somecases give very good approximations However it was shownin [21 24] that themode shapes of the loaded beam and thoseof the bare beam are different Further Hassanpour et al [25]showed that the mode shapes of the loaded beam were notorthonormal under the classical orthogonality conditionsThey also presented the generalized orthogonality conditionsfor an axially loaded beam with in-span concentrated massand springs The study ignored the case where a suspendedmass is connected to the beam via a spring-dashpot system

The majority of the studies reported in the literaturefocused on the study of beams with either an in-span massor spring-mass system A series combination of each type

Hindawi Publishing CorporationShock and VibrationVolume 2014 Article ID 485630 9 pageshttpdxdoiorg1011552014485630

2 Shock and Vibration

was found in some casesHowever works that simultaneouslyexamined the free and forced vibration of a beam with multi-ple in-spanmass-spring-damper-mass systems and subjectedto an axial force were not found The investigation of thisproblem is reported in the present paper it is an extensionof the work presented in [13] The exact natural frequenciesand mode shapes are determined and explicit expressionsare presented for the orthogonality relations Parametricstudies are used to examine the effect of the magnitude andlocation of the in-span mass-spring-damper-mass system onthe natural frequencies and vibrational response The effectof the use of the classical orthogonality condition is alsoexamined The results are validated using the finite elementmethod and results from the literature

2 Equations of Motion

A schematic of a beam with several in-span mass-spring-damper-mass systems is depicted in Figure 1 The systemkinetic energy T and potential energy V may be expressedas

T =119873

sum119894=1

(12119898int119871 119894

119871 119894minus1

2119894(119909 119905) 119889119909 +

121198721198891198942119894+1211987211988811989420119894)

V =119873

sum119894=1

(12119864119868int119871 119894

119871 119894minus1

119908101584010158402119894(119909 119905) 119889119909 +

12119888119889119894(0119894minus 119894)2

+12119896119894(1199110119894minus 119911119894)2 +

12119879int119871 119894

119871 119894minus1

11990810158402119894(119909 119905) 119889119909)

(1)

where 1199110119894(119905) = 119908

119894(119871119894 119905) 119864119868 is the beam flexural rigidity 119898

is the mass per unit length of the beam and 119879 is the beampretension The overdots and primes denote temporal andspatial derivatives respectively The subscript ldquo119894rdquo denotes theposition of the mass-spring-damper-mass system and ldquo119873rdquois the total number of attached mass-spring-damper-masssystems The beam is assumed to be uniform and the preten-sion is constant

Introducing these energies equations into Hamiltonrsquosprinciple and adding the forcing term 119865(119909 119905) yield the equa-tions of motion and continuity conditions

1198641198681199081015840101584010158401015840119894+ 119898

119894minus 11987911990810158401015840119894= 119865 (119909 119905) (2)

119872119889119894 + 119896119894(119911119894minus 1199110119894) + 119888119889119894(119894minus 0119894) = 0 (3)

119908119894(119871119894 119905) = 119908

119894+1(119871119894 119905) (4)

1199081015840119894(119871119894 119905) = 1199081015840

119894+1(119871119894 119905) (5)

11990810158401015840119894(119871119894 119905) = 11990810158401015840

119894+1(119871119894 119905) minus 119872

1198881198940119894+ 119864119868119908101584010158401015840

119894(119871119894 119905)

minus 1198791199081015840119894(119871119894 119905) minus 119896

119894(1199110119894minus 119911119894) minus 119888119889119894(0119894minus 119894) (6)

minus119864119868119908101584010158401015840119894+1(119871119894 119905) + 1198791199081015840

119894+1(119871119894 119905) = 0 (7)

3 Frequency Equation and Mode Shapes

The deformations 119908119894(119909119894 119905) and displacements 119911(119905) are

expressed as

119908119894(119909119894 119905) = 119871119882

119894 (120577) e120596119905

119911119894 (119905) = 119871119860 119894e

120596119905(8)

where 119882119894(120577119894) and 119860

119894are the respective nondimensional

amplitudes of119908119894(119909119894 119905) and 119911

119894(119905) and 120596 is the circular complex

natural frequency of the system Substituting (8) into (2)ndash(7)yields the following nondimensional system equations

1198821015840101584010158401015840119894(120577119894) minus 119904211988210158401015840

119894(120577119894) + Ω4119882

119894120577119894= 0 (9)

119860119894minus 119870119894119882119894(120585119894) = 0 (10)

where

119882119894(120577119894) = 1198881119894sin120572120577

119894+ 1198882119894cos120572120577

119894+ 1198883119894sinh120573120577

119894+ 1198884119894cosh120573120577

119894

(11)

Substituting (8) into (3)ndash(7) yields

119882119894(120585119894) = 119882

119894+1(120585119894) (12)

1198821015840119894(120585119894) = 1198821015840

119894+1(120585119894) (13)

11988210158401015840119894(120585119894) = 11988210158401015840

119894+1(120585119894) (14)

119882101584010158401015840119894(120585119894) minus 119882101584010158401015840

119894+1(120585119894) minus 120578119894119882119894(120585119894) + 120574119894119860119894= 0 (15)

where the following nondimensional variables are used

120585119894=119871119894

119871 119860

119894= 119870119882

119894(120585119894)

119870119894=

119896119894+ 120596119888119889119894

119896119894+1198721198891198941205962 + 120596119888

119889119894

1199042 =1198791198712

119864119868

Ω4 =1198981205962

1198641198681198714 120578

119894=119896119894+ 1205962119872

119888119894+ 120596119888119889119894

1198641198681198713

120574119894=119896119894+ 120596119888119889119894

1198641198681198713 120577 =

119909119871

120572 = radicminus1199042

2+ radic

1199044

4+ Ω4 120573 = radic

1199042

2+ radic

1199044

4+ Ω4

(16)

Substituting (10) into (15) yields

119882101584010158401015840119894(120585119894) minus 119882101584010158401015840

119894(120585119894) + 119882

119894(120585119894) (119870119894120574119894minus 120578119894) = 0 (17)

The use of any classical boundary conditions at each end ofthe beam along with (12)ndash(14) and (17) yields a set of 4 +4119873 algebraic homogeneous equations (4 equations from theboundary condition at the ends and 4119873 equations from thecontinuity relations) These algebraic equations are linear inthe unknown coefficients (119862rsquos) and they can be presented inmatrix format as

[F](4+4119873)119883(4+4119873)119862(4+4119873)119883(1) = 0(4+4119873)119883(1) (18)

Shock and Vibration 3

af

LN

Mc1 Mc2

Z1 Z2 ZNMd1 Md2 MdN

F(x t)

L1

L2

McN

k1 k2 kNcd1 cd2 cdN

Figure 1 Schematic of an axial beam carrying several mass-spring-damper-mass systems

where the elements of the matrix F are listed in theAppendix A nontrivial solution is obtained when matrix Fis singular Hence the characteristic or frequency equation isobtained as

det ([F](4+4119873)119883(4+4119873)) = 0 (19)

The mode shapes associated with each beam segment areobtained by substituting the integration constants from (18)into (11) An example of a matrix that yields the frequencyequation is provided in the Appendix for the case of a can-tilever beam with one in-span mass-spring-damper-masssupport

4 Generalized Orthogonality Relations

The solution of the equations of motion (2) and (3) can beexpressed as

119908119894 (119909 119905) = 119884119894(119909)

(119903)e120596119905

119911119894 (119905) = 119885

(119903)

119894e120596119905

(20)

where the superscript ldquo119903rdquo denotes the mode number and

119884119894 (119909) = 119871119882119894 (120577)

119885119894= 119871119860119894

(21)

Substituting (20) into (3) and multiplying the resulting equa-tion by 119885(119904) yield

119896119894119885(119903)119894119885(119904)119894+1198721198891198941205962119903119885(119903)119894119885(119904)119894+ 120596119903119888119889119894119885(119903)119894119885(119904)119894

= 119884lowast(119903)119894119885(119904)119894(119896119894+ 120596119903119888119889119894)

(22)

where 119884lowast(119903)119894

= 119884(119903)119894(1205851) Interchanging ldquo119903rdquo and ldquo119904rdquo in (22) and

subtracting the resulting equation from (22) yield

(1205962119903minus 1205962119904)119872119889119894119885(119903)119894119885(119904)119894+ 119888119889119894(120596119903minus 120596119904) 119885(119903)119894119885(119904)119894

= 119896119894(119884lowast(119903)119894119885(119904)119894minus 119884lowast(119904)119894119885(119903)119894)

+ 119888119889119894(120596119903119884lowast(119903)119894119885(119904)119894minus 120596119904119884lowast(119904)119894119885(119903)119894)

(23)

With reference to the equations of motion of the beam sub-stituting (20) into (2) thenmultiplying the resulting equationby 119884(119904) and integrating over the entire length of the beam aswell as applying the continuity conditions (12)ndash(15) with anyclassical boundary conditions except those for free ends yield

1205962119903

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894)

+ 120596119903

119873

sum119894=1

119888119889119894119884lowast(119903)119894119884lowast(119904)119894

= minus119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909

+ 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909 + 119896

119894119884lowast(119903)119894119884lowast(119904)119894

minus 119896119894119885(119903)119894119884lowast(119904)119894

minus 119888119889119894120596119903119885(119903)119894119884lowast(119904)119894)

(24)

Rewriting (24) and interchanging ldquo119903rdquo and ldquo119904rdquo yield

1205962119904

119873

sum119894=1

(119898int119871 119894

0

119884(119904)119894119884(119903)119894119889119909 +119872

119888119894119884lowast(119904)119894119884lowast(119903)119894) + 120596

119904

119873

sum119894=1

119888119889119894119884lowast(119904)119894119884lowast(119903)119894

= minus119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119904)119894

11988410158401015840(119903)119894

119889119909

+ 119879int119871 119894

0

1198841015840(119904)1198941198841015840(119903)119894119889119909 + 119896

119894119884lowast(119903)119894119884lowast(119904)119894

minus 119896119894119885(119904)119894119884lowast(119903)119894

minus 119888119889119894120596119904119885(119904)119894119884lowast(119903)119894)

(25)

Subtracting (25) from (24) and substituting (23) into theresulting equation yield

(1205962119903minus 1205962119904)119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894)

+ (120596119903minus 120596119904)119873

sum119894=1

119888119889119894(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 0

(26)

4 Shock and Vibration

From (26) the first set of orthogonality relation is obtainedas

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894) = 120575

119903119904

(27)

where 120575119903119904is the Kronecker delta The second set of orthogo-

nality relation is expressed as

119873

sum119894=1

(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 120575119903119904 (28)

The use of (22) and (25) with the aid of some algebraicmanipulation yields the third and fourth set of orthogonalityrelation This may be written as

119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909 + 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909)

+119873

sum119894=1

119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 120575119903119904

(29)

119873

sum119894=1

(119884lowast(119903)119894119885lowast(119904)119894

+ 119885(119903)119894119884(119904)119894) = 120575119903119904 (30)

5 Forced Vibration

Assume a harmonic force 119865(119909 119905) is arbitrarily applied alongthe span of the beam as depicted in Figure 1 The responseof the loaded beam is now derived using the generalizedorthogonality relations (27)ndash(30) Let the excitation force beapplied at a location 119886

119891from the left-hand end on the beam

the governing equations of motion are now

119898119894+ 1198641198681199081015840101584010158401015840

119894minus 11987911990810158401015840119894= 119865 (119909 119905) 120575 (119909 minus 119886119891) (31)

119872119889119894119894+ 119896119894(119911119894minus 119908119894(119871119894)) + 119888119889119894(119894minus 119894(119871119894)) = 0 (32)

Using the assumed mode method the transverse displace-ment of the beam and the displacement of vibration absorbermay be expressed as

119908119894=119873119903

sum119903=1

119902119903 (119905) 119884

(119903)

119894(119909) (33)

119911119894=119873119903

sum119903=1

119902119903 (119905) 119885

(119903)

119894 (34)

where 119873119903is the number of retained modes 119884(119903)

119894(119909) is the

mode shape corresponding to the 119903th mode 119885(119903)119894

is thedisplacement amplitude of the absorber and 119902

119903(119905) is the 119903th

generalized coordinate Substituting (33) and (34) into (31)and (32) respectively yields

119898119888

119873119903

sum119903=1

119902119903119884(119903)119894+ 119864119868119873119903

sum119903=1

1199021199031198841015840101584010158401015840(119903)119894

minus 119879119873119903

sum119903=1

11990211990311988410158401015840(119903)119894

= 119865 (119905) 120575 (119909 minus 119886119891)

119872119889119894

119873119903

sum119903=1

119902119903119884(119903)119898119894+ 119896119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894))

+ 119888119889119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894)) = 0

(35)

Multiplying (35) by 119884(119904)119894

and119885(119904)119894 respectively adding the two

resulting equations integrating over the entire length of thebeam and applying the continuity conditions (12)ndash(15) withany classical boundary conditions (except for free ends) yield

119873119903

sum119903=1

119902119903119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894

+ 119888119889119894

119873119903

sum119903=1

119902119903119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894

+119873119903

sum119903=1

119902119903int119871 119894

0

(11986411986811988410158401015840(119903)119894

11988410158401015840(119904)119894

+ 1198791198841015840(119903)1198941198841015840(119904)119894) 119889119909

+ 119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 119865 (119909 119905) 119884(119904)119894(119886119891)

(36)

Use of the orthogonality relations (27)ndash(30) yields the fol-lowing uncoupled differential equation

[119872119903119903] 119902119903 + [119862

119903119903] 119902119903 + [119870

119903119903] 119902119903 = 119865

119903 (37)

where the matrices119872119903119903 119862119903 and119870

119903119903are expressed as

119872119903119903=119873

sum119894=1

(119898int119871 119894

0

119884(119903)2

119894119889119909 +119872

119888119894119884lowast(119903)

2

119894+119872119889119894119885(119903)2

119894)

119862119903119903=119873

sum119894=1

119888119889119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119870119903=119873

sum119894=1

int119871 119894

0

(11986411986811988410158401015840(119903)2

119894+ 1198791198841015840(119903)

2

119894) 119889119909 + 119896

119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119865119903= 119865 (119909 119905) 119884(119903)

119894(119886119891)

(38)

Shock and Vibration 5

Table 1 Material properties and parameters

Parameter Reference [1] Reference [16] UnitsEI 634761 364583 Nm2

119871 10 10 m119898 15385 0675 kgm1198981

30775 01 kg1198982

4614 01 kg1198983

769 01 kg1198984

9997 mdash kg1198985

1538 mdash kg1198961

190428 01 Nm1198962

222166 01 Nm1198963

28564245 01 Nm1198964

317380 mdash Nm1198965

380856 mdash Nm1198881198891 1198881198892 1198881198893

mdash 01 Nsm

The amplitude of the vibration absorber can be readilyexpressed as

119885(119903)119894= 120581119894119884lowast(119903)119894

120581119894= (

1 + (2120589119894119903119894)2

(1 minus 1199032119894)2 + (2120589

119894119903119894)2)12

120589119894=

119888119889119894

2119872119889119894120596119904119894

119903119894=Ω120596119904119894

120596119904119894= radic

119896119894

119872119889119894

(39)

6 Numerical Simulation

The sets of parameters employed in the numerical examplesare taken from [1 16] they are tabulated in Table 1The valid-ity of the free vibration of the present model is inferred fromthe results tabulated in Table 2 In the case of a cantileveredbeam with three spring-mass-damper systems the first fivenatural frequencies of the present model are compared tothose of [16] The results show very good agreement with amaximum error of 028

With respect to a pinned-pinned beam with attachedthree and five spring-mass systems the validation is done via[1] and shows excellent agreement with four decimal placesThe correspondingmode shapes for a beamcarrying the threein-span spring-mass systems are depicted in Figure 2 Thisfigure is identical to Figure 3 from [1]

For the forced vibration simulations an excitation forceof 119865(119905) = 10 sin(Ω119905) is used with zero initial conditions Theforce is first applied at the free end of a cantilevered beamwith three identical spring-mass-damper systems located at01 05 and 09m from the clamp end The time history ofthe vertical displacement of the free end is presented to servefor comparison between the proposed approach (ie thecombination of the generalized orthogonality conditionswiththe exact mode shapes of the loaded beam) the finite elementmethod (FEM) and the approach used in [16]The results aredepicted in Figures 3 and 4 for an excitation frequency of 5and 10 rads respectively All threemethods are in agreement

0 1Nondimensional axial coordinates

Vert

ical

mod

e disp

lace

men

t

Mode 1Mode 2Mode 3

Mode 4Mode 5

08 0907060504030201

15

1

05

0

minus05

minus1

minus15

Figure 2 Mode shapes of a simply supported beam carrying threein-span spring-mass systems using the same parameters from [1]

Time (s)Vert

ical

disp

lace

men

t at t

he fr

ee en

d (m

)

4353252151050minus02

minus015

minus01

minus005

0

005

01

015

Present for Ω = 5 rads

FEA for Ω = 5 radsRef [16] for Ω = 5 rads

Figure 3 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 5 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

but the FEM results agree better with the present approachthan that of [16]

The frequency response curves for a simply supportedbeam with one in-span mass-spring-damper-mass systemattached at 01m from one end using three approachesare depicted in Figure 5 The first approach which is themost accurate [25] employs the generalized orthogonalityconditions along with the mode shapes of the loaded beamThe classical orthogonality relations and the mode shapesof the loaded beam are employed in the second approachsuch as in [17ndash20] The third method also uses the classicalorthogonality conditions but the mode shapes are those ofthe bare beam (see [14ndash16]) All three approaches yield verydifferent midspan vertical displacements The first approachis the exact solution and hence the error associated with thedifference is defined with respect to the first approach Thefrequency response amplitude errors are depicted in Figure 6where the error can be as high as 104

6 Shock and Vibration

Table 2 Natural frequency validation (rads)

ModeCantilevered Pinned-pinned

3 spring-mass-damper 3 spring-mass 5 spring-massReference [16] Present Reference [1] Present Reference [1] Present

1 minus0367 plusmn 258769119894 minus0367 plusmn 258544119894 1527339 1527339 1509571 15095712 minus0274 plusmn 1617655119894 minus0275 plusmn 1619399119894 1850949 1850949 1694728 16947283 minus0031 plusmn 4521367119894 minus0031 plusmn 4524320119894 2478313 2478313 1879146 18791464 minus0171 plusmn 8872157119894 minus0174 plusmn 8873469119894 6775959 6775959 2171278 21712785 minus0081 plusmn 14675170119894 minus0082 plusmn 14688356119894 25486572 25486572 2479867 2479867

Time (s)4353252151050

minus02

minus015

minus01

minus005

0

005

01

015

Vert

ical

disp

lace

men

t (m

)

Present for Ω = 10 rads

FEA for Ω = 10 radsRef [16] for Ω = 5 rads

Figure 4 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 10 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

Generalized orthogonality (present model)Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 5 Frequency response of a simply supported beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2 119888

119889=

01Nsm and 119871119889= 01m

Figure 7 is an identical plot to Figure 5 with the pinned-pinned configuration replaced by a guided-guided one Thecorresponding frequency response amplitude error is shown

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

105

104

103

102

101

100

Figure 6 Frequency response amplitude error for a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N

119864119868 = 36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

in Figure 8 The maximum amplitude error for the guided-guided beam is higher than that for the simply supportedbeam A similar observation with regard to the discrepancyassociated with the use of the classical orthogonality wasreported in [25] for a simply supported beamcarrying a heavymass An error as high as 105 was reported

The three approaches are also examined using the param-eters taken from [1] for a simply supported beam with onein-span mass-spring-damper system and no tension Figures9 and 10 show the frequency response of the midspan dis-placement for a damping coefficient 119888

119889119894= 01Nsm and

119888119889119894= 100Nsm respectively It can be observed that all three

methods yield very similar results and the plots are barelydistinguishable The amplitude errors associated with theresults presented in Figures 9 and 10 are depicted in Figures11 and 12 respectively But for excitation frequencies closerto 600 rads the results indicate very minimal error Thediscrepancy is more pronounced in the case of the classicalorthogonality with the combination of the bare beam modeshapes

7 Conclusions

A common approach to studying the vibrational response ofan elastically loaded beam involves the use of the classical

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

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Shock and Vibration

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

Propagation

International Journal of

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DistributedSensor Networks

International Journal of

2 Shock and Vibration

was found in some casesHowever works that simultaneouslyexamined the free and forced vibration of a beam with multi-ple in-spanmass-spring-damper-mass systems and subjectedto an axial force were not found The investigation of thisproblem is reported in the present paper it is an extensionof the work presented in [13] The exact natural frequenciesand mode shapes are determined and explicit expressionsare presented for the orthogonality relations Parametricstudies are used to examine the effect of the magnitude andlocation of the in-span mass-spring-damper-mass system onthe natural frequencies and vibrational response The effectof the use of the classical orthogonality condition is alsoexamined The results are validated using the finite elementmethod and results from the literature

2 Equations of Motion

A schematic of a beam with several in-span mass-spring-damper-mass systems is depicted in Figure 1 The systemkinetic energy T and potential energy V may be expressedas

T =119873

sum119894=1

(12119898int119871 119894

119871 119894minus1

2119894(119909 119905) 119889119909 +

121198721198891198942119894+1211987211988811989420119894)

V =119873

sum119894=1

(12119864119868int119871 119894

119871 119894minus1

119908101584010158402119894(119909 119905) 119889119909 +

12119888119889119894(0119894minus 119894)2

+12119896119894(1199110119894minus 119911119894)2 +

12119879int119871 119894

119871 119894minus1

11990810158402119894(119909 119905) 119889119909)

(1)

where 1199110119894(119905) = 119908

119894(119871119894 119905) 119864119868 is the beam flexural rigidity 119898

is the mass per unit length of the beam and 119879 is the beampretension The overdots and primes denote temporal andspatial derivatives respectively The subscript ldquo119894rdquo denotes theposition of the mass-spring-damper-mass system and ldquo119873rdquois the total number of attached mass-spring-damper-masssystems The beam is assumed to be uniform and the preten-sion is constant

Introducing these energies equations into Hamiltonrsquosprinciple and adding the forcing term 119865(119909 119905) yield the equa-tions of motion and continuity conditions

1198641198681199081015840101584010158401015840119894+ 119898

119894minus 11987911990810158401015840119894= 119865 (119909 119905) (2)

119872119889119894 + 119896119894(119911119894minus 1199110119894) + 119888119889119894(119894minus 0119894) = 0 (3)

119908119894(119871119894 119905) = 119908

119894+1(119871119894 119905) (4)

1199081015840119894(119871119894 119905) = 1199081015840

119894+1(119871119894 119905) (5)

11990810158401015840119894(119871119894 119905) = 11990810158401015840

119894+1(119871119894 119905) minus 119872

1198881198940119894+ 119864119868119908101584010158401015840

119894(119871119894 119905)

minus 1198791199081015840119894(119871119894 119905) minus 119896

119894(1199110119894minus 119911119894) minus 119888119889119894(0119894minus 119894) (6)

minus119864119868119908101584010158401015840119894+1(119871119894 119905) + 1198791199081015840

119894+1(119871119894 119905) = 0 (7)

3 Frequency Equation and Mode Shapes

The deformations 119908119894(119909119894 119905) and displacements 119911(119905) are

expressed as

119908119894(119909119894 119905) = 119871119882

119894 (120577) e120596119905

119911119894 (119905) = 119871119860 119894e

120596119905(8)

where 119882119894(120577119894) and 119860

119894are the respective nondimensional

amplitudes of119908119894(119909119894 119905) and 119911

119894(119905) and 120596 is the circular complex

natural frequency of the system Substituting (8) into (2)ndash(7)yields the following nondimensional system equations

1198821015840101584010158401015840119894(120577119894) minus 119904211988210158401015840

119894(120577119894) + Ω4119882

119894120577119894= 0 (9)

119860119894minus 119870119894119882119894(120585119894) = 0 (10)

where

119882119894(120577119894) = 1198881119894sin120572120577

119894+ 1198882119894cos120572120577

119894+ 1198883119894sinh120573120577

119894+ 1198884119894cosh120573120577

119894

(11)

Substituting (8) into (3)ndash(7) yields

119882119894(120585119894) = 119882

119894+1(120585119894) (12)

1198821015840119894(120585119894) = 1198821015840

119894+1(120585119894) (13)

11988210158401015840119894(120585119894) = 11988210158401015840

119894+1(120585119894) (14)

119882101584010158401015840119894(120585119894) minus 119882101584010158401015840

119894+1(120585119894) minus 120578119894119882119894(120585119894) + 120574119894119860119894= 0 (15)

where the following nondimensional variables are used

120585119894=119871119894

119871 119860

119894= 119870119882

119894(120585119894)

119870119894=

119896119894+ 120596119888119889119894

119896119894+1198721198891198941205962 + 120596119888

119889119894

1199042 =1198791198712

119864119868

Ω4 =1198981205962

1198641198681198714 120578

119894=119896119894+ 1205962119872

119888119894+ 120596119888119889119894

1198641198681198713

120574119894=119896119894+ 120596119888119889119894

1198641198681198713 120577 =

119909119871

120572 = radicminus1199042

2+ radic

1199044

4+ Ω4 120573 = radic

1199042

2+ radic

1199044

4+ Ω4

(16)

Substituting (10) into (15) yields

119882101584010158401015840119894(120585119894) minus 119882101584010158401015840

119894(120585119894) + 119882

119894(120585119894) (119870119894120574119894minus 120578119894) = 0 (17)

The use of any classical boundary conditions at each end ofthe beam along with (12)ndash(14) and (17) yields a set of 4 +4119873 algebraic homogeneous equations (4 equations from theboundary condition at the ends and 4119873 equations from thecontinuity relations) These algebraic equations are linear inthe unknown coefficients (119862rsquos) and they can be presented inmatrix format as

[F](4+4119873)119883(4+4119873)119862(4+4119873)119883(1) = 0(4+4119873)119883(1) (18)

Shock and Vibration 3

af

LN

Mc1 Mc2

Z1 Z2 ZNMd1 Md2 MdN

F(x t)

L1

L2

McN

k1 k2 kNcd1 cd2 cdN

Figure 1 Schematic of an axial beam carrying several mass-spring-damper-mass systems

where the elements of the matrix F are listed in theAppendix A nontrivial solution is obtained when matrix Fis singular Hence the characteristic or frequency equation isobtained as

det ([F](4+4119873)119883(4+4119873)) = 0 (19)

The mode shapes associated with each beam segment areobtained by substituting the integration constants from (18)into (11) An example of a matrix that yields the frequencyequation is provided in the Appendix for the case of a can-tilever beam with one in-span mass-spring-damper-masssupport

4 Generalized Orthogonality Relations

The solution of the equations of motion (2) and (3) can beexpressed as

119908119894 (119909 119905) = 119884119894(119909)

(119903)e120596119905

119911119894 (119905) = 119885

(119903)

119894e120596119905

(20)

where the superscript ldquo119903rdquo denotes the mode number and

119884119894 (119909) = 119871119882119894 (120577)

119885119894= 119871119860119894

(21)

Substituting (20) into (3) and multiplying the resulting equa-tion by 119885(119904) yield

119896119894119885(119903)119894119885(119904)119894+1198721198891198941205962119903119885(119903)119894119885(119904)119894+ 120596119903119888119889119894119885(119903)119894119885(119904)119894

= 119884lowast(119903)119894119885(119904)119894(119896119894+ 120596119903119888119889119894)

(22)

where 119884lowast(119903)119894

= 119884(119903)119894(1205851) Interchanging ldquo119903rdquo and ldquo119904rdquo in (22) and

subtracting the resulting equation from (22) yield

(1205962119903minus 1205962119904)119872119889119894119885(119903)119894119885(119904)119894+ 119888119889119894(120596119903minus 120596119904) 119885(119903)119894119885(119904)119894

= 119896119894(119884lowast(119903)119894119885(119904)119894minus 119884lowast(119904)119894119885(119903)119894)

+ 119888119889119894(120596119903119884lowast(119903)119894119885(119904)119894minus 120596119904119884lowast(119904)119894119885(119903)119894)

(23)

With reference to the equations of motion of the beam sub-stituting (20) into (2) thenmultiplying the resulting equationby 119884(119904) and integrating over the entire length of the beam aswell as applying the continuity conditions (12)ndash(15) with anyclassical boundary conditions except those for free ends yield

1205962119903

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894)

+ 120596119903

119873

sum119894=1

119888119889119894119884lowast(119903)119894119884lowast(119904)119894

= minus119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909

+ 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909 + 119896

119894119884lowast(119903)119894119884lowast(119904)119894

minus 119896119894119885(119903)119894119884lowast(119904)119894

minus 119888119889119894120596119903119885(119903)119894119884lowast(119904)119894)

(24)

Rewriting (24) and interchanging ldquo119903rdquo and ldquo119904rdquo yield

1205962119904

119873

sum119894=1

(119898int119871 119894

0

119884(119904)119894119884(119903)119894119889119909 +119872

119888119894119884lowast(119904)119894119884lowast(119903)119894) + 120596

119904

119873

sum119894=1

119888119889119894119884lowast(119904)119894119884lowast(119903)119894

= minus119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119904)119894

11988410158401015840(119903)119894

119889119909

+ 119879int119871 119894

0

1198841015840(119904)1198941198841015840(119903)119894119889119909 + 119896

119894119884lowast(119903)119894119884lowast(119904)119894

minus 119896119894119885(119904)119894119884lowast(119903)119894

minus 119888119889119894120596119904119885(119904)119894119884lowast(119903)119894)

(25)

Subtracting (25) from (24) and substituting (23) into theresulting equation yield

(1205962119903minus 1205962119904)119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894)

+ (120596119903minus 120596119904)119873

sum119894=1

119888119889119894(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 0

(26)

4 Shock and Vibration

From (26) the first set of orthogonality relation is obtainedas

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894) = 120575

119903119904

(27)

where 120575119903119904is the Kronecker delta The second set of orthogo-

nality relation is expressed as

119873

sum119894=1

(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 120575119903119904 (28)

The use of (22) and (25) with the aid of some algebraicmanipulation yields the third and fourth set of orthogonalityrelation This may be written as

119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909 + 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909)

+119873

sum119894=1

119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 120575119903119904

(29)

119873

sum119894=1

(119884lowast(119903)119894119885lowast(119904)119894

+ 119885(119903)119894119884(119904)119894) = 120575119903119904 (30)

5 Forced Vibration

Assume a harmonic force 119865(119909 119905) is arbitrarily applied alongthe span of the beam as depicted in Figure 1 The responseof the loaded beam is now derived using the generalizedorthogonality relations (27)ndash(30) Let the excitation force beapplied at a location 119886

119891from the left-hand end on the beam

the governing equations of motion are now

119898119894+ 1198641198681199081015840101584010158401015840

119894minus 11987911990810158401015840119894= 119865 (119909 119905) 120575 (119909 minus 119886119891) (31)

119872119889119894119894+ 119896119894(119911119894minus 119908119894(119871119894)) + 119888119889119894(119894minus 119894(119871119894)) = 0 (32)

Using the assumed mode method the transverse displace-ment of the beam and the displacement of vibration absorbermay be expressed as

119908119894=119873119903

sum119903=1

119902119903 (119905) 119884

(119903)

119894(119909) (33)

119911119894=119873119903

sum119903=1

119902119903 (119905) 119885

(119903)

119894 (34)

where 119873119903is the number of retained modes 119884(119903)

119894(119909) is the

mode shape corresponding to the 119903th mode 119885(119903)119894

is thedisplacement amplitude of the absorber and 119902

119903(119905) is the 119903th

generalized coordinate Substituting (33) and (34) into (31)and (32) respectively yields

119898119888

119873119903

sum119903=1

119902119903119884(119903)119894+ 119864119868119873119903

sum119903=1

1199021199031198841015840101584010158401015840(119903)119894

minus 119879119873119903

sum119903=1

11990211990311988410158401015840(119903)119894

= 119865 (119905) 120575 (119909 minus 119886119891)

119872119889119894

119873119903

sum119903=1

119902119903119884(119903)119898119894+ 119896119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894))

+ 119888119889119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894)) = 0

(35)

Multiplying (35) by 119884(119904)119894

and119885(119904)119894 respectively adding the two

resulting equations integrating over the entire length of thebeam and applying the continuity conditions (12)ndash(15) withany classical boundary conditions (except for free ends) yield

119873119903

sum119903=1

119902119903119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894

+ 119888119889119894

119873119903

sum119903=1

119902119903119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894

+119873119903

sum119903=1

119902119903int119871 119894

0

(11986411986811988410158401015840(119903)119894

11988410158401015840(119904)119894

+ 1198791198841015840(119903)1198941198841015840(119904)119894) 119889119909

+ 119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 119865 (119909 119905) 119884(119904)119894(119886119891)

(36)

Use of the orthogonality relations (27)ndash(30) yields the fol-lowing uncoupled differential equation

[119872119903119903] 119902119903 + [119862

119903119903] 119902119903 + [119870

119903119903] 119902119903 = 119865

119903 (37)

where the matrices119872119903119903 119862119903 and119870

119903119903are expressed as

119872119903119903=119873

sum119894=1

(119898int119871 119894

0

119884(119903)2

119894119889119909 +119872

119888119894119884lowast(119903)

2

119894+119872119889119894119885(119903)2

119894)

119862119903119903=119873

sum119894=1

119888119889119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119870119903=119873

sum119894=1

int119871 119894

0

(11986411986811988410158401015840(119903)2

119894+ 1198791198841015840(119903)

2

119894) 119889119909 + 119896

119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119865119903= 119865 (119909 119905) 119884(119903)

119894(119886119891)

(38)

Shock and Vibration 5

Table 1 Material properties and parameters

Parameter Reference [1] Reference [16] UnitsEI 634761 364583 Nm2

119871 10 10 m119898 15385 0675 kgm1198981

30775 01 kg1198982

4614 01 kg1198983

769 01 kg1198984

9997 mdash kg1198985

1538 mdash kg1198961

190428 01 Nm1198962

222166 01 Nm1198963

28564245 01 Nm1198964

317380 mdash Nm1198965

380856 mdash Nm1198881198891 1198881198892 1198881198893

mdash 01 Nsm

The amplitude of the vibration absorber can be readilyexpressed as

119885(119903)119894= 120581119894119884lowast(119903)119894

120581119894= (

1 + (2120589119894119903119894)2

(1 minus 1199032119894)2 + (2120589

119894119903119894)2)12

120589119894=

119888119889119894

2119872119889119894120596119904119894

119903119894=Ω120596119904119894

120596119904119894= radic

119896119894

119872119889119894

(39)

6 Numerical Simulation

The sets of parameters employed in the numerical examplesare taken from [1 16] they are tabulated in Table 1The valid-ity of the free vibration of the present model is inferred fromthe results tabulated in Table 2 In the case of a cantileveredbeam with three spring-mass-damper systems the first fivenatural frequencies of the present model are compared tothose of [16] The results show very good agreement with amaximum error of 028

With respect to a pinned-pinned beam with attachedthree and five spring-mass systems the validation is done via[1] and shows excellent agreement with four decimal placesThe correspondingmode shapes for a beamcarrying the threein-span spring-mass systems are depicted in Figure 2 Thisfigure is identical to Figure 3 from [1]

For the forced vibration simulations an excitation forceof 119865(119905) = 10 sin(Ω119905) is used with zero initial conditions Theforce is first applied at the free end of a cantilevered beamwith three identical spring-mass-damper systems located at01 05 and 09m from the clamp end The time history ofthe vertical displacement of the free end is presented to servefor comparison between the proposed approach (ie thecombination of the generalized orthogonality conditionswiththe exact mode shapes of the loaded beam) the finite elementmethod (FEM) and the approach used in [16]The results aredepicted in Figures 3 and 4 for an excitation frequency of 5and 10 rads respectively All threemethods are in agreement

0 1Nondimensional axial coordinates

Vert

ical

mod

e disp

lace

men

t

Mode 1Mode 2Mode 3

Mode 4Mode 5

08 0907060504030201

15

1

05

0

minus05

minus1

minus15

Figure 2 Mode shapes of a simply supported beam carrying threein-span spring-mass systems using the same parameters from [1]

Time (s)Vert

ical

disp

lace

men

t at t

he fr

ee en

d (m

)

4353252151050minus02

minus015

minus01

minus005

0

005

01

015

Present for Ω = 5 rads

FEA for Ω = 5 radsRef [16] for Ω = 5 rads

Figure 3 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 5 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

but the FEM results agree better with the present approachthan that of [16]

The frequency response curves for a simply supportedbeam with one in-span mass-spring-damper-mass systemattached at 01m from one end using three approachesare depicted in Figure 5 The first approach which is themost accurate [25] employs the generalized orthogonalityconditions along with the mode shapes of the loaded beamThe classical orthogonality relations and the mode shapesof the loaded beam are employed in the second approachsuch as in [17ndash20] The third method also uses the classicalorthogonality conditions but the mode shapes are those ofthe bare beam (see [14ndash16]) All three approaches yield verydifferent midspan vertical displacements The first approachis the exact solution and hence the error associated with thedifference is defined with respect to the first approach Thefrequency response amplitude errors are depicted in Figure 6where the error can be as high as 104

6 Shock and Vibration

Table 2 Natural frequency validation (rads)

ModeCantilevered Pinned-pinned

3 spring-mass-damper 3 spring-mass 5 spring-massReference [16] Present Reference [1] Present Reference [1] Present

1 minus0367 plusmn 258769119894 minus0367 plusmn 258544119894 1527339 1527339 1509571 15095712 minus0274 plusmn 1617655119894 minus0275 plusmn 1619399119894 1850949 1850949 1694728 16947283 minus0031 plusmn 4521367119894 minus0031 plusmn 4524320119894 2478313 2478313 1879146 18791464 minus0171 plusmn 8872157119894 minus0174 plusmn 8873469119894 6775959 6775959 2171278 21712785 minus0081 plusmn 14675170119894 minus0082 plusmn 14688356119894 25486572 25486572 2479867 2479867

Time (s)4353252151050

minus02

minus015

minus01

minus005

0

005

01

015

Vert

ical

disp

lace

men

t (m

)

Present for Ω = 10 rads

FEA for Ω = 10 radsRef [16] for Ω = 5 rads

Figure 4 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 10 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

Generalized orthogonality (present model)Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 5 Frequency response of a simply supported beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2 119888

119889=

01Nsm and 119871119889= 01m

Figure 7 is an identical plot to Figure 5 with the pinned-pinned configuration replaced by a guided-guided one Thecorresponding frequency response amplitude error is shown

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

105

104

103

102

101

100

Figure 6 Frequency response amplitude error for a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N

119864119868 = 36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

in Figure 8 The maximum amplitude error for the guided-guided beam is higher than that for the simply supportedbeam A similar observation with regard to the discrepancyassociated with the use of the classical orthogonality wasreported in [25] for a simply supported beamcarrying a heavymass An error as high as 105 was reported

The three approaches are also examined using the param-eters taken from [1] for a simply supported beam with onein-span mass-spring-damper system and no tension Figures9 and 10 show the frequency response of the midspan dis-placement for a damping coefficient 119888

119889119894= 01Nsm and

119888119889119894= 100Nsm respectively It can be observed that all three

methods yield very similar results and the plots are barelydistinguishable The amplitude errors associated with theresults presented in Figures 9 and 10 are depicted in Figures11 and 12 respectively But for excitation frequencies closerto 600 rads the results indicate very minimal error Thediscrepancy is more pronounced in the case of the classicalorthogonality with the combination of the bare beam modeshapes

7 Conclusions

A common approach to studying the vibrational response ofan elastically loaded beam involves the use of the classical

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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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

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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

Shock and Vibration 3

af

LN

Mc1 Mc2

Z1 Z2 ZNMd1 Md2 MdN

F(x t)

L1

L2

McN

k1 k2 kNcd1 cd2 cdN

Figure 1 Schematic of an axial beam carrying several mass-spring-damper-mass systems

where the elements of the matrix F are listed in theAppendix A nontrivial solution is obtained when matrix Fis singular Hence the characteristic or frequency equation isobtained as

det ([F](4+4119873)119883(4+4119873)) = 0 (19)

The mode shapes associated with each beam segment areobtained by substituting the integration constants from (18)into (11) An example of a matrix that yields the frequencyequation is provided in the Appendix for the case of a can-tilever beam with one in-span mass-spring-damper-masssupport

4 Generalized Orthogonality Relations

The solution of the equations of motion (2) and (3) can beexpressed as

119908119894 (119909 119905) = 119884119894(119909)

(119903)e120596119905

119911119894 (119905) = 119885

(119903)

119894e120596119905

(20)

where the superscript ldquo119903rdquo denotes the mode number and

119884119894 (119909) = 119871119882119894 (120577)

119885119894= 119871119860119894

(21)

Substituting (20) into (3) and multiplying the resulting equa-tion by 119885(119904) yield

119896119894119885(119903)119894119885(119904)119894+1198721198891198941205962119903119885(119903)119894119885(119904)119894+ 120596119903119888119889119894119885(119903)119894119885(119904)119894

= 119884lowast(119903)119894119885(119904)119894(119896119894+ 120596119903119888119889119894)

(22)

where 119884lowast(119903)119894

= 119884(119903)119894(1205851) Interchanging ldquo119903rdquo and ldquo119904rdquo in (22) and

subtracting the resulting equation from (22) yield

(1205962119903minus 1205962119904)119872119889119894119885(119903)119894119885(119904)119894+ 119888119889119894(120596119903minus 120596119904) 119885(119903)119894119885(119904)119894

= 119896119894(119884lowast(119903)119894119885(119904)119894minus 119884lowast(119904)119894119885(119903)119894)

+ 119888119889119894(120596119903119884lowast(119903)119894119885(119904)119894minus 120596119904119884lowast(119904)119894119885(119903)119894)

(23)

With reference to the equations of motion of the beam sub-stituting (20) into (2) thenmultiplying the resulting equationby 119884(119904) and integrating over the entire length of the beam aswell as applying the continuity conditions (12)ndash(15) with anyclassical boundary conditions except those for free ends yield

1205962119903

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894)

+ 120596119903

119873

sum119894=1

119888119889119894119884lowast(119903)119894119884lowast(119904)119894

= minus119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909

+ 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909 + 119896

119894119884lowast(119903)119894119884lowast(119904)119894

minus 119896119894119885(119903)119894119884lowast(119904)119894

minus 119888119889119894120596119903119885(119903)119894119884lowast(119904)119894)

(24)

Rewriting (24) and interchanging ldquo119903rdquo and ldquo119904rdquo yield

1205962119904

119873

sum119894=1

(119898int119871 119894

0

119884(119904)119894119884(119903)119894119889119909 +119872

119888119894119884lowast(119904)119894119884lowast(119903)119894) + 120596

119904

119873

sum119894=1

119888119889119894119884lowast(119904)119894119884lowast(119903)119894

= minus119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119904)119894

11988410158401015840(119903)119894

119889119909

+ 119879int119871 119894

0

1198841015840(119904)1198941198841015840(119903)119894119889119909 + 119896

119894119884lowast(119903)119894119884lowast(119904)119894

minus 119896119894119885(119904)119894119884lowast(119903)119894

minus 119888119889119894120596119904119885(119904)119894119884lowast(119903)119894)

(25)

Subtracting (25) from (24) and substituting (23) into theresulting equation yield

(1205962119903minus 1205962119904)119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894)

+ (120596119903minus 120596119904)119873

sum119894=1

119888119889119894(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 0

(26)

4 Shock and Vibration

From (26) the first set of orthogonality relation is obtainedas

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894) = 120575

119903119904

(27)

where 120575119903119904is the Kronecker delta The second set of orthogo-

nality relation is expressed as

119873

sum119894=1

(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 120575119903119904 (28)

The use of (22) and (25) with the aid of some algebraicmanipulation yields the third and fourth set of orthogonalityrelation This may be written as

119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909 + 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909)

+119873

sum119894=1

119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 120575119903119904

(29)

119873

sum119894=1

(119884lowast(119903)119894119885lowast(119904)119894

+ 119885(119903)119894119884(119904)119894) = 120575119903119904 (30)

5 Forced Vibration

Assume a harmonic force 119865(119909 119905) is arbitrarily applied alongthe span of the beam as depicted in Figure 1 The responseof the loaded beam is now derived using the generalizedorthogonality relations (27)ndash(30) Let the excitation force beapplied at a location 119886

119891from the left-hand end on the beam

the governing equations of motion are now

119898119894+ 1198641198681199081015840101584010158401015840

119894minus 11987911990810158401015840119894= 119865 (119909 119905) 120575 (119909 minus 119886119891) (31)

119872119889119894119894+ 119896119894(119911119894minus 119908119894(119871119894)) + 119888119889119894(119894minus 119894(119871119894)) = 0 (32)

Using the assumed mode method the transverse displace-ment of the beam and the displacement of vibration absorbermay be expressed as

119908119894=119873119903

sum119903=1

119902119903 (119905) 119884

(119903)

119894(119909) (33)

119911119894=119873119903

sum119903=1

119902119903 (119905) 119885

(119903)

119894 (34)

where 119873119903is the number of retained modes 119884(119903)

119894(119909) is the

mode shape corresponding to the 119903th mode 119885(119903)119894

is thedisplacement amplitude of the absorber and 119902

119903(119905) is the 119903th

generalized coordinate Substituting (33) and (34) into (31)and (32) respectively yields

119898119888

119873119903

sum119903=1

119902119903119884(119903)119894+ 119864119868119873119903

sum119903=1

1199021199031198841015840101584010158401015840(119903)119894

minus 119879119873119903

sum119903=1

11990211990311988410158401015840(119903)119894

= 119865 (119905) 120575 (119909 minus 119886119891)

119872119889119894

119873119903

sum119903=1

119902119903119884(119903)119898119894+ 119896119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894))

+ 119888119889119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894)) = 0

(35)

Multiplying (35) by 119884(119904)119894

and119885(119904)119894 respectively adding the two

resulting equations integrating over the entire length of thebeam and applying the continuity conditions (12)ndash(15) withany classical boundary conditions (except for free ends) yield

119873119903

sum119903=1

119902119903119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894

+ 119888119889119894

119873119903

sum119903=1

119902119903119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894

+119873119903

sum119903=1

119902119903int119871 119894

0

(11986411986811988410158401015840(119903)119894

11988410158401015840(119904)119894

+ 1198791198841015840(119903)1198941198841015840(119904)119894) 119889119909

+ 119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 119865 (119909 119905) 119884(119904)119894(119886119891)

(36)

Use of the orthogonality relations (27)ndash(30) yields the fol-lowing uncoupled differential equation

[119872119903119903] 119902119903 + [119862

119903119903] 119902119903 + [119870

119903119903] 119902119903 = 119865

119903 (37)

where the matrices119872119903119903 119862119903 and119870

119903119903are expressed as

119872119903119903=119873

sum119894=1

(119898int119871 119894

0

119884(119903)2

119894119889119909 +119872

119888119894119884lowast(119903)

2

119894+119872119889119894119885(119903)2

119894)

119862119903119903=119873

sum119894=1

119888119889119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119870119903=119873

sum119894=1

int119871 119894

0

(11986411986811988410158401015840(119903)2

119894+ 1198791198841015840(119903)

2

119894) 119889119909 + 119896

119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119865119903= 119865 (119909 119905) 119884(119903)

119894(119886119891)

(38)

Shock and Vibration 5

Table 1 Material properties and parameters

Parameter Reference [1] Reference [16] UnitsEI 634761 364583 Nm2

119871 10 10 m119898 15385 0675 kgm1198981

30775 01 kg1198982

4614 01 kg1198983

769 01 kg1198984

9997 mdash kg1198985

1538 mdash kg1198961

190428 01 Nm1198962

222166 01 Nm1198963

28564245 01 Nm1198964

317380 mdash Nm1198965

380856 mdash Nm1198881198891 1198881198892 1198881198893

mdash 01 Nsm

The amplitude of the vibration absorber can be readilyexpressed as

119885(119903)119894= 120581119894119884lowast(119903)119894

120581119894= (

1 + (2120589119894119903119894)2

(1 minus 1199032119894)2 + (2120589

119894119903119894)2)12

120589119894=

119888119889119894

2119872119889119894120596119904119894

119903119894=Ω120596119904119894

120596119904119894= radic

119896119894

119872119889119894

(39)

6 Numerical Simulation

The sets of parameters employed in the numerical examplesare taken from [1 16] they are tabulated in Table 1The valid-ity of the free vibration of the present model is inferred fromthe results tabulated in Table 2 In the case of a cantileveredbeam with three spring-mass-damper systems the first fivenatural frequencies of the present model are compared tothose of [16] The results show very good agreement with amaximum error of 028

With respect to a pinned-pinned beam with attachedthree and five spring-mass systems the validation is done via[1] and shows excellent agreement with four decimal placesThe correspondingmode shapes for a beamcarrying the threein-span spring-mass systems are depicted in Figure 2 Thisfigure is identical to Figure 3 from [1]

For the forced vibration simulations an excitation forceof 119865(119905) = 10 sin(Ω119905) is used with zero initial conditions Theforce is first applied at the free end of a cantilevered beamwith three identical spring-mass-damper systems located at01 05 and 09m from the clamp end The time history ofthe vertical displacement of the free end is presented to servefor comparison between the proposed approach (ie thecombination of the generalized orthogonality conditionswiththe exact mode shapes of the loaded beam) the finite elementmethod (FEM) and the approach used in [16]The results aredepicted in Figures 3 and 4 for an excitation frequency of 5and 10 rads respectively All threemethods are in agreement

0 1Nondimensional axial coordinates

Vert

ical

mod

e disp

lace

men

t

Mode 1Mode 2Mode 3

Mode 4Mode 5

08 0907060504030201

15

1

05

0

minus05

minus1

minus15

Figure 2 Mode shapes of a simply supported beam carrying threein-span spring-mass systems using the same parameters from [1]

Time (s)Vert

ical

disp

lace

men

t at t

he fr

ee en

d (m

)

4353252151050minus02

minus015

minus01

minus005

0

005

01

015

Present for Ω = 5 rads

FEA for Ω = 5 radsRef [16] for Ω = 5 rads

Figure 3 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 5 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

but the FEM results agree better with the present approachthan that of [16]

The frequency response curves for a simply supportedbeam with one in-span mass-spring-damper-mass systemattached at 01m from one end using three approachesare depicted in Figure 5 The first approach which is themost accurate [25] employs the generalized orthogonalityconditions along with the mode shapes of the loaded beamThe classical orthogonality relations and the mode shapesof the loaded beam are employed in the second approachsuch as in [17ndash20] The third method also uses the classicalorthogonality conditions but the mode shapes are those ofthe bare beam (see [14ndash16]) All three approaches yield verydifferent midspan vertical displacements The first approachis the exact solution and hence the error associated with thedifference is defined with respect to the first approach Thefrequency response amplitude errors are depicted in Figure 6where the error can be as high as 104

6 Shock and Vibration

Table 2 Natural frequency validation (rads)

ModeCantilevered Pinned-pinned

3 spring-mass-damper 3 spring-mass 5 spring-massReference [16] Present Reference [1] Present Reference [1] Present

1 minus0367 plusmn 258769119894 minus0367 plusmn 258544119894 1527339 1527339 1509571 15095712 minus0274 plusmn 1617655119894 minus0275 plusmn 1619399119894 1850949 1850949 1694728 16947283 minus0031 plusmn 4521367119894 minus0031 plusmn 4524320119894 2478313 2478313 1879146 18791464 minus0171 plusmn 8872157119894 minus0174 plusmn 8873469119894 6775959 6775959 2171278 21712785 minus0081 plusmn 14675170119894 minus0082 plusmn 14688356119894 25486572 25486572 2479867 2479867

Time (s)4353252151050

minus02

minus015

minus01

minus005

0

005

01

015

Vert

ical

disp

lace

men

t (m

)

Present for Ω = 10 rads

FEA for Ω = 10 radsRef [16] for Ω = 5 rads

Figure 4 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 10 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

Generalized orthogonality (present model)Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 5 Frequency response of a simply supported beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2 119888

119889=

01Nsm and 119871119889= 01m

Figure 7 is an identical plot to Figure 5 with the pinned-pinned configuration replaced by a guided-guided one Thecorresponding frequency response amplitude error is shown

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

105

104

103

102

101

100

Figure 6 Frequency response amplitude error for a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N

119864119868 = 36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

in Figure 8 The maximum amplitude error for the guided-guided beam is higher than that for the simply supportedbeam A similar observation with regard to the discrepancyassociated with the use of the classical orthogonality wasreported in [25] for a simply supported beamcarrying a heavymass An error as high as 105 was reported

The three approaches are also examined using the param-eters taken from [1] for a simply supported beam with onein-span mass-spring-damper system and no tension Figures9 and 10 show the frequency response of the midspan dis-placement for a damping coefficient 119888

119889119894= 01Nsm and

119888119889119894= 100Nsm respectively It can be observed that all three

methods yield very similar results and the plots are barelydistinguishable The amplitude errors associated with theresults presented in Figures 9 and 10 are depicted in Figures11 and 12 respectively But for excitation frequencies closerto 600 rads the results indicate very minimal error Thediscrepancy is more pronounced in the case of the classicalorthogonality with the combination of the bare beam modeshapes

7 Conclusions

A common approach to studying the vibrational response ofan elastically loaded beam involves the use of the classical

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

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Submit your manuscripts athttpwwwhindawicom

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Shock and Vibration

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Civil EngineeringAdvances in

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Electrical and Computer Engineering

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Advances inOptoElectronics

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Volume 2014

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Chemical EngineeringInternational Journal of Antennas and

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International Journal of

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International Journal of

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DistributedSensor Networks

International Journal of

4 Shock and Vibration

From (26) the first set of orthogonality relation is obtainedas

119873

sum119894=1

(119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894) = 120575

119903119904

(27)

where 120575119903119904is the Kronecker delta The second set of orthogo-

nality relation is expressed as

119873

sum119894=1

(119884lowast(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894) = 120575119903119904 (28)

The use of (22) and (25) with the aid of some algebraicmanipulation yields the third and fourth set of orthogonalityrelation This may be written as

119873

sum119894=1

(119864119868int119871 119894

0

11988410158401015840(119903)119894

11988410158401015840(119904)119894

119889119909 + 119879int119871 119894

0

1198841015840(119903)1198941198841015840(119904)119894119889119909)

+119873

sum119894=1

119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 120575119903119904

(29)

119873

sum119894=1

(119884lowast(119903)119894119885lowast(119904)119894

+ 119885(119903)119894119884(119904)119894) = 120575119903119904 (30)

5 Forced Vibration

Assume a harmonic force 119865(119909 119905) is arbitrarily applied alongthe span of the beam as depicted in Figure 1 The responseof the loaded beam is now derived using the generalizedorthogonality relations (27)ndash(30) Let the excitation force beapplied at a location 119886

119891from the left-hand end on the beam

the governing equations of motion are now

119898119894+ 1198641198681199081015840101584010158401015840

119894minus 11987911990810158401015840119894= 119865 (119909 119905) 120575 (119909 minus 119886119891) (31)

119872119889119894119894+ 119896119894(119911119894minus 119908119894(119871119894)) + 119888119889119894(119894minus 119894(119871119894)) = 0 (32)

Using the assumed mode method the transverse displace-ment of the beam and the displacement of vibration absorbermay be expressed as

119908119894=119873119903

sum119903=1

119902119903 (119905) 119884

(119903)

119894(119909) (33)

119911119894=119873119903

sum119903=1

119902119903 (119905) 119885

(119903)

119894 (34)

where 119873119903is the number of retained modes 119884(119903)

119894(119909) is the

mode shape corresponding to the 119903th mode 119885(119903)119894

is thedisplacement amplitude of the absorber and 119902

119903(119905) is the 119903th

generalized coordinate Substituting (33) and (34) into (31)and (32) respectively yields

119898119888

119873119903

sum119903=1

119902119903119884(119903)119894+ 119864119868119873119903

sum119903=1

1199021199031198841015840101584010158401015840(119903)119894

minus 119879119873119903

sum119903=1

11990211990311988410158401015840(119903)119894

= 119865 (119905) 120575 (119909 minus 119886119891)

119872119889119894

119873119903

sum119903=1

119902119903119884(119903)119898119894+ 119896119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894))

+ 119888119889119894

119873119903

sum119903=1

119902119903(119885(119903)119894minus 119884(119903)119888119894(119871119894)) = 0

(35)

Multiplying (35) by 119884(119904)119894

and119885(119904)119894 respectively adding the two

resulting equations integrating over the entire length of thebeam and applying the continuity conditions (12)ndash(15) withany classical boundary conditions (except for free ends) yield

119873119903

sum119903=1

119902119903119898int119871 119894

0

119884(119903)119894119884(119904)119894119889119909 +119872

119888119894119884lowast(119903)119894119884lowast(119904)119894

+119872119889119894119885(119903)119894119885(119904)119894

+ 119888119889119894

119873119903

sum119903=1

119902119903119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus 119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894

+119873119903

sum119903=1

119902119903int119871 119894

0

(11986411986811988410158401015840(119903)119894

11988410158401015840(119904)119894

+ 1198791198841015840(119903)1198941198841015840(119904)119894) 119889119909

+ 119896119894(119884lowast(119903)119894119884lowast(119904)119894

minus 119885(119904)119894119884lowast(119903)119894

minus119885(119903)119894119884lowast(119904)119894

+ 119885(119903)119894119885(119904)119894)

= 119865 (119909 119905) 119884(119904)119894(119886119891)

(36)

Use of the orthogonality relations (27)ndash(30) yields the fol-lowing uncoupled differential equation

[119872119903119903] 119902119903 + [119862

119903119903] 119902119903 + [119870

119903119903] 119902119903 = 119865

119903 (37)

where the matrices119872119903119903 119862119903 and119870

119903119903are expressed as

119872119903119903=119873

sum119894=1

(119898int119871 119894

0

119884(119903)2

119894119889119909 +119872

119888119894119884lowast(119903)

2

119894+119872119889119894119885(119903)2

119894)

119862119903119903=119873

sum119894=1

119888119889119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119870119903=119873

sum119894=1

int119871 119894

0

(11986411986811988410158401015840(119903)2

119894+ 1198791198841015840(119903)

2

119894) 119889119909 + 119896

119894(119884lowast(119903)119894

minus 119885(119903)119894)2

119865119903= 119865 (119909 119905) 119884(119903)

119894(119886119891)

(38)

Shock and Vibration 5

Table 1 Material properties and parameters

Parameter Reference [1] Reference [16] UnitsEI 634761 364583 Nm2

119871 10 10 m119898 15385 0675 kgm1198981

30775 01 kg1198982

4614 01 kg1198983

769 01 kg1198984

9997 mdash kg1198985

1538 mdash kg1198961

190428 01 Nm1198962

222166 01 Nm1198963

28564245 01 Nm1198964

317380 mdash Nm1198965

380856 mdash Nm1198881198891 1198881198892 1198881198893

mdash 01 Nsm

The amplitude of the vibration absorber can be readilyexpressed as

119885(119903)119894= 120581119894119884lowast(119903)119894

120581119894= (

1 + (2120589119894119903119894)2

(1 minus 1199032119894)2 + (2120589

119894119903119894)2)12

120589119894=

119888119889119894

2119872119889119894120596119904119894

119903119894=Ω120596119904119894

120596119904119894= radic

119896119894

119872119889119894

(39)

6 Numerical Simulation

The sets of parameters employed in the numerical examplesare taken from [1 16] they are tabulated in Table 1The valid-ity of the free vibration of the present model is inferred fromthe results tabulated in Table 2 In the case of a cantileveredbeam with three spring-mass-damper systems the first fivenatural frequencies of the present model are compared tothose of [16] The results show very good agreement with amaximum error of 028

With respect to a pinned-pinned beam with attachedthree and five spring-mass systems the validation is done via[1] and shows excellent agreement with four decimal placesThe correspondingmode shapes for a beamcarrying the threein-span spring-mass systems are depicted in Figure 2 Thisfigure is identical to Figure 3 from [1]

For the forced vibration simulations an excitation forceof 119865(119905) = 10 sin(Ω119905) is used with zero initial conditions Theforce is first applied at the free end of a cantilevered beamwith three identical spring-mass-damper systems located at01 05 and 09m from the clamp end The time history ofthe vertical displacement of the free end is presented to servefor comparison between the proposed approach (ie thecombination of the generalized orthogonality conditionswiththe exact mode shapes of the loaded beam) the finite elementmethod (FEM) and the approach used in [16]The results aredepicted in Figures 3 and 4 for an excitation frequency of 5and 10 rads respectively All threemethods are in agreement

0 1Nondimensional axial coordinates

Vert

ical

mod

e disp

lace

men

t

Mode 1Mode 2Mode 3

Mode 4Mode 5

08 0907060504030201

15

1

05

0

minus05

minus1

minus15

Figure 2 Mode shapes of a simply supported beam carrying threein-span spring-mass systems using the same parameters from [1]

Time (s)Vert

ical

disp

lace

men

t at t

he fr

ee en

d (m

)

4353252151050minus02

minus015

minus01

minus005

0

005

01

015

Present for Ω = 5 rads

FEA for Ω = 5 radsRef [16] for Ω = 5 rads

Figure 3 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 5 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

but the FEM results agree better with the present approachthan that of [16]

The frequency response curves for a simply supportedbeam with one in-span mass-spring-damper-mass systemattached at 01m from one end using three approachesare depicted in Figure 5 The first approach which is themost accurate [25] employs the generalized orthogonalityconditions along with the mode shapes of the loaded beamThe classical orthogonality relations and the mode shapesof the loaded beam are employed in the second approachsuch as in [17ndash20] The third method also uses the classicalorthogonality conditions but the mode shapes are those ofthe bare beam (see [14ndash16]) All three approaches yield verydifferent midspan vertical displacements The first approachis the exact solution and hence the error associated with thedifference is defined with respect to the first approach Thefrequency response amplitude errors are depicted in Figure 6where the error can be as high as 104

6 Shock and Vibration

Table 2 Natural frequency validation (rads)

ModeCantilevered Pinned-pinned

3 spring-mass-damper 3 spring-mass 5 spring-massReference [16] Present Reference [1] Present Reference [1] Present

1 minus0367 plusmn 258769119894 minus0367 plusmn 258544119894 1527339 1527339 1509571 15095712 minus0274 plusmn 1617655119894 minus0275 plusmn 1619399119894 1850949 1850949 1694728 16947283 minus0031 plusmn 4521367119894 minus0031 plusmn 4524320119894 2478313 2478313 1879146 18791464 minus0171 plusmn 8872157119894 minus0174 plusmn 8873469119894 6775959 6775959 2171278 21712785 minus0081 plusmn 14675170119894 minus0082 plusmn 14688356119894 25486572 25486572 2479867 2479867

Time (s)4353252151050

minus02

minus015

minus01

minus005

0

005

01

015

Vert

ical

disp

lace

men

t (m

)

Present for Ω = 10 rads

FEA for Ω = 10 radsRef [16] for Ω = 5 rads

Figure 4 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 10 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

Generalized orthogonality (present model)Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 5 Frequency response of a simply supported beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2 119888

119889=

01Nsm and 119871119889= 01m

Figure 7 is an identical plot to Figure 5 with the pinned-pinned configuration replaced by a guided-guided one Thecorresponding frequency response amplitude error is shown

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

105

104

103

102

101

100

Figure 6 Frequency response amplitude error for a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N

119864119868 = 36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

in Figure 8 The maximum amplitude error for the guided-guided beam is higher than that for the simply supportedbeam A similar observation with regard to the discrepancyassociated with the use of the classical orthogonality wasreported in [25] for a simply supported beamcarrying a heavymass An error as high as 105 was reported

The three approaches are also examined using the param-eters taken from [1] for a simply supported beam with onein-span mass-spring-damper system and no tension Figures9 and 10 show the frequency response of the midspan dis-placement for a damping coefficient 119888

119889119894= 01Nsm and

119888119889119894= 100Nsm respectively It can be observed that all three

methods yield very similar results and the plots are barelydistinguishable The amplitude errors associated with theresults presented in Figures 9 and 10 are depicted in Figures11 and 12 respectively But for excitation frequencies closerto 600 rads the results indicate very minimal error Thediscrepancy is more pronounced in the case of the classicalorthogonality with the combination of the bare beam modeshapes

7 Conclusions

A common approach to studying the vibrational response ofan elastically loaded beam involves the use of the classical

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

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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

Shock and Vibration 5

Table 1 Material properties and parameters

Parameter Reference [1] Reference [16] UnitsEI 634761 364583 Nm2

119871 10 10 m119898 15385 0675 kgm1198981

30775 01 kg1198982

4614 01 kg1198983

769 01 kg1198984

9997 mdash kg1198985

1538 mdash kg1198961

190428 01 Nm1198962

222166 01 Nm1198963

28564245 01 Nm1198964

317380 mdash Nm1198965

380856 mdash Nm1198881198891 1198881198892 1198881198893

mdash 01 Nsm

The amplitude of the vibration absorber can be readilyexpressed as

119885(119903)119894= 120581119894119884lowast(119903)119894

120581119894= (

1 + (2120589119894119903119894)2

(1 minus 1199032119894)2 + (2120589

119894119903119894)2)12

120589119894=

119888119889119894

2119872119889119894120596119904119894

119903119894=Ω120596119904119894

120596119904119894= radic

119896119894

119872119889119894

(39)

6 Numerical Simulation

The sets of parameters employed in the numerical examplesare taken from [1 16] they are tabulated in Table 1The valid-ity of the free vibration of the present model is inferred fromthe results tabulated in Table 2 In the case of a cantileveredbeam with three spring-mass-damper systems the first fivenatural frequencies of the present model are compared tothose of [16] The results show very good agreement with amaximum error of 028

With respect to a pinned-pinned beam with attachedthree and five spring-mass systems the validation is done via[1] and shows excellent agreement with four decimal placesThe correspondingmode shapes for a beamcarrying the threein-span spring-mass systems are depicted in Figure 2 Thisfigure is identical to Figure 3 from [1]

For the forced vibration simulations an excitation forceof 119865(119905) = 10 sin(Ω119905) is used with zero initial conditions Theforce is first applied at the free end of a cantilevered beamwith three identical spring-mass-damper systems located at01 05 and 09m from the clamp end The time history ofthe vertical displacement of the free end is presented to servefor comparison between the proposed approach (ie thecombination of the generalized orthogonality conditionswiththe exact mode shapes of the loaded beam) the finite elementmethod (FEM) and the approach used in [16]The results aredepicted in Figures 3 and 4 for an excitation frequency of 5and 10 rads respectively All threemethods are in agreement

0 1Nondimensional axial coordinates

Vert

ical

mod

e disp

lace

men

t

Mode 1Mode 2Mode 3

Mode 4Mode 5

08 0907060504030201

15

1

05

0

minus05

minus1

minus15

Figure 2 Mode shapes of a simply supported beam carrying threein-span spring-mass systems using the same parameters from [1]

Time (s)Vert

ical

disp

lace

men

t at t

he fr

ee en

d (m

)

4353252151050minus02

minus015

minus01

minus005

0

005

01

015

Present for Ω = 5 rads

FEA for Ω = 5 radsRef [16] for Ω = 5 rads

Figure 3 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 5 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

but the FEM results agree better with the present approachthan that of [16]

The frequency response curves for a simply supportedbeam with one in-span mass-spring-damper-mass systemattached at 01m from one end using three approachesare depicted in Figure 5 The first approach which is themost accurate [25] employs the generalized orthogonalityconditions along with the mode shapes of the loaded beamThe classical orthogonality relations and the mode shapesof the loaded beam are employed in the second approachsuch as in [17ndash20] The third method also uses the classicalorthogonality conditions but the mode shapes are those ofthe bare beam (see [14ndash16]) All three approaches yield verydifferent midspan vertical displacements The first approachis the exact solution and hence the error associated with thedifference is defined with respect to the first approach Thefrequency response amplitude errors are depicted in Figure 6where the error can be as high as 104

6 Shock and Vibration

Table 2 Natural frequency validation (rads)

ModeCantilevered Pinned-pinned

3 spring-mass-damper 3 spring-mass 5 spring-massReference [16] Present Reference [1] Present Reference [1] Present

1 minus0367 plusmn 258769119894 minus0367 plusmn 258544119894 1527339 1527339 1509571 15095712 minus0274 plusmn 1617655119894 minus0275 plusmn 1619399119894 1850949 1850949 1694728 16947283 minus0031 plusmn 4521367119894 minus0031 plusmn 4524320119894 2478313 2478313 1879146 18791464 minus0171 plusmn 8872157119894 minus0174 plusmn 8873469119894 6775959 6775959 2171278 21712785 minus0081 plusmn 14675170119894 minus0082 plusmn 14688356119894 25486572 25486572 2479867 2479867

Time (s)4353252151050

minus02

minus015

minus01

minus005

0

005

01

015

Vert

ical

disp

lace

men

t (m

)

Present for Ω = 10 rads

FEA for Ω = 10 radsRef [16] for Ω = 5 rads

Figure 4 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 10 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

Generalized orthogonality (present model)Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 5 Frequency response of a simply supported beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2 119888

119889=

01Nsm and 119871119889= 01m

Figure 7 is an identical plot to Figure 5 with the pinned-pinned configuration replaced by a guided-guided one Thecorresponding frequency response amplitude error is shown

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

105

104

103

102

101

100

Figure 6 Frequency response amplitude error for a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N

119864119868 = 36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

in Figure 8 The maximum amplitude error for the guided-guided beam is higher than that for the simply supportedbeam A similar observation with regard to the discrepancyassociated with the use of the classical orthogonality wasreported in [25] for a simply supported beamcarrying a heavymass An error as high as 105 was reported

The three approaches are also examined using the param-eters taken from [1] for a simply supported beam with onein-span mass-spring-damper system and no tension Figures9 and 10 show the frequency response of the midspan dis-placement for a damping coefficient 119888

119889119894= 01Nsm and

119888119889119894= 100Nsm respectively It can be observed that all three

methods yield very similar results and the plots are barelydistinguishable The amplitude errors associated with theresults presented in Figures 9 and 10 are depicted in Figures11 and 12 respectively But for excitation frequencies closerto 600 rads the results indicate very minimal error Thediscrepancy is more pronounced in the case of the classicalorthogonality with the combination of the bare beam modeshapes

7 Conclusions

A common approach to studying the vibrational response ofan elastically loaded beam involves the use of the classical

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

International Journal of

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Active and Passive Electronic Components

Control Scienceand Engineering

Journal of

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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 Shock and Vibration

Table 2 Natural frequency validation (rads)

ModeCantilevered Pinned-pinned

3 spring-mass-damper 3 spring-mass 5 spring-massReference [16] Present Reference [1] Present Reference [1] Present

1 minus0367 plusmn 258769119894 minus0367 plusmn 258544119894 1527339 1527339 1509571 15095712 minus0274 plusmn 1617655119894 minus0275 plusmn 1619399119894 1850949 1850949 1694728 16947283 minus0031 plusmn 4521367119894 minus0031 plusmn 4524320119894 2478313 2478313 1879146 18791464 minus0171 plusmn 8872157119894 minus0174 plusmn 8873469119894 6775959 6775959 2171278 21712785 minus0081 plusmn 14675170119894 minus0082 plusmn 14688356119894 25486572 25486572 2479867 2479867

Time (s)4353252151050

minus02

minus015

minus01

minus005

0

005

01

015

Vert

ical

disp

lace

men

t (m

)

Present for Ω = 10 rads

FEA for Ω = 10 radsRef [16] for Ω = 5 rads

Figure 4 Time history of the free-end vertical displacement for acantilevered beam carrying three mass-spring-damper systems fora forcing frequency Ω = 10 rads 119896 = 01Nm 119879 = 0119872

119889= 01 kg

119872119888= 0 kg 119888

119889= 01Nsm and 119871

123= 01 05 09m

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

Generalized orthogonality (present model)Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 5 Frequency response of a simply supported beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2 119888

119889=

01Nsm and 119871119889= 01m

Figure 7 is an identical plot to Figure 5 with the pinned-pinned configuration replaced by a guided-guided one Thecorresponding frequency response amplitude error is shown

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

105

104

103

102

101

100

Figure 6 Frequency response amplitude error for a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N

119864119868 = 36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

in Figure 8 The maximum amplitude error for the guided-guided beam is higher than that for the simply supportedbeam A similar observation with regard to the discrepancyassociated with the use of the classical orthogonality wasreported in [25] for a simply supported beamcarrying a heavymass An error as high as 105 was reported

The three approaches are also examined using the param-eters taken from [1] for a simply supported beam with onein-span mass-spring-damper system and no tension Figures9 and 10 show the frequency response of the midspan dis-placement for a damping coefficient 119888

119889119894= 01Nsm and

119888119889119894= 100Nsm respectively It can be observed that all three

methods yield very similar results and the plots are barelydistinguishable The amplitude errors associated with theresults presented in Figures 9 and 10 are depicted in Figures11 and 12 respectively But for excitation frequencies closerto 600 rads the results indicate very minimal error Thediscrepancy is more pronounced in the case of the classicalorthogonality with the combination of the bare beam modeshapes

7 Conclusions

A common approach to studying the vibrational response ofan elastically loaded beam involves the use of the classical

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Navigation and Observation

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

DistributedSensor Networks

International Journal of

Shock and Vibration 7

Generalized orthogonalityClassical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)

Mid

span

ver

tical

disp

lace

men

t (m

)

101

102

103

104

10minus1

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

Figure 7 Frequency response of a guided-guided beam carryingone in-span mass-spring-damper-mass system for 119896 = 10Nm119872119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 = 36458Nm2

119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

101

102

103

104

10minus1

101

102

103

104

105

106

100

Figure 8 Frequency response amplitude error of a guided-guidedbeam carrying one in-span mass-spring-damper-mass system for119896 = 10Nm 119872

119889= 01 kg 119872

119888= 005 kg 119879 = 360N 119864119868 =

36458Nm2 119888119889= 01Nsm and 119871

119889= 01m

Frequency (rads)

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

101

102

103

104

10minus2

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 9 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

01Nsm and 119871119889= 075m

Vert

ical

disp

lace

men

t (m

)

Classical orthogonality with bare beam modesGeneralized orthogonality (present model)Classical orthogonality with exact modes

Frequency (rads)10

110

210

310

4

10minus3

10minus4

10minus5

10minus6

10minus7

10minus8

10minus9

Figure 10 Frequency response of a simply supported beam carryingone in-spanmass-spring-damper-mass system for 119896 = 190428Nm119872119889= 30775 kg 119872

119888= 0 kg 119879 = 0N 119864119868 = 634761Nm2 119888

119889=

100Nsm and 119871119889= 075m

Am

plitu

de er

ror (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 11 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 01Nsm and 119871

119889= 075m

orthogonality relations along with the mode shapes of thebare beam It has been shown however that the modeshapes of the bare beam can be quite different from thoseof the loaded beam In the present paper the exact naturalfrequencies and mode shapes were presented for an axiallyloaded beam carrying several vibration absorbers Explicitexpressions were presented for the generalized orthogonalitycondition The obtained generalized orthogonality relationwas employed along with the assumed modes method tostudy the forced vibrational response The numerical sim-ulations indicated that using the common approach couldproduce erroneous results The combination of the exactmode shapes of the loaded beam and the correspondinggeneralized orthogonality relation is necessary for accuratedynamic modeling of a beam carrying elastically mountedmasses with dampers

8 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

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 Shock and VibrationA

mpl

itude

err

or (

)

Classical orthogonality with exact modesClassical orthogonality with bare beam modes

Frequency (rads)10

110

210

310

4

103

102

101

100

10minus1

Figure 12 Frequency response amplitude error of a simply sup-ported beam carrying one in-span mass-spring-damper-mass sys-tem for 119896 = 190428Nm 119872

119889= 30775 kg 119872

119888= 0 kg 119879 = 0N

119864119868 = 634761Nm2 119888119889= 100Nsm and 119871

119889= 075m

Appendix

For the sake of simplicity the following notations are used

119904120572= sin120572 119888

120572= cos120572

119904ℎ120573= sinh120573 119888ℎ

120573= cosh120573

119904120572119894= sin120572 (120577

119894) 119888

120572119894= cos120572 (120577

119894)

119904ℎ120573119894= sinh120573 (120577

119894) 119888ℎ

120573119894= cosh120573 (120577

119894)

120598 = 11987011205741minus 1205781

(A1)

The elements of the matrixF are expressed as

F(4119894minus14119894minus3)

= 119904120572119894 F

(4119894minus14119894minus2)= 119888120572119894

F(4119894minus14119894minus1)

= 119904ℎ120573119894 F

(4119894minus14119894)= 119888ℎ120573119894

F(4119894minus14119894+1)

= minus119904120572119894 F

(4119894minus14119894+2)= minus119888120572119894

F(4119894minus14119894+3)

= minus119904ℎ120573119894 F

(4119894minus14119894+4)= minus119888ℎ

120573119894

F(41198944119894minus3)

= 120572119888120572119894 F

(41198944119894minus2)= minus120572119904

120572119894

F(41198944119894minus1)

= 120573119888ℎ120573119894 F

(41198944119894)= 120573119904ℎ

120573119894

F(41198944119894+1)

= minus120572119888120572119894 F

(41198944119894+2)= 120572119904120572119894

F(41198944119894+3)

= minus120573119888ℎ120573119894 F

(41198944119894+4)= minus120573119904ℎ

120573119894

F(4119894+14119894minus3)

= 1205722119904120572119894 F

(4119894+14119894minus2)= minus1205722119888

120572119894

F(4119894+14119894minus1)

= 1205732119904ℎ120573119894 F

(4119894+14119894)= 1205732119888ℎ

120573119894

F(4119894+14119894+1)

= 1205722119904120572119894 F

(4119894+14119894+2)= 1205722119888120572119894

F(4119894+14119894+3)

= minus1205732119904ℎ120573119894 F

(4119894+14119894+4)= minus1205732119888ℎ

120573119894

F(4119894+24119894minus3)

= minus1205723119888120572119894+ 120598119904120572119894 F

(4119894+24119894minus2)= 1205723119904120572119894+ 120598119888120572119894

F(4119894+24119894minus1)

= 1205733119888ℎ120573119894+ 120598119904ℎ120573119894 F

(4119894+24119894)= 1205733119904ℎ

120573119894+ 120598119888ℎ120573119894

F(4119894+24119894+1)

= 1205723119888120572119894 F

(4119894+24119894+2)= minus1205723119904

120572119894

F(4119894+24119894+3)

= minus1205733119888ℎ120573119894 F

(4119894+24119894+4)= minus1205733119904ℎ

120573119894(A2)

F(11)

F(12)

F(13)

F(14)

F(21)

F(22)

F(23)

and F(24)

depend on the boundary conditions at the origin 119909 =0 F

(119902minus1119901+1) F(119902minus1119901+2)

F(119902minus1119901+3)

F(119902minus1119901+4)

F(119902119901+1)

F(119902119901+2)

F(119902119901+3)

andF(119902119901+4)

are obtained from the bound-ary conditions at 119909 = 119871 where 119902 = 4119873 + 4 and 119901 = 4119873

The frequency equation for a cantilevered beam with onein-span mass-spring-mass support is obtained by taking thedeterminant of the following matrix

[[[[[[[[[[[[[[[

[

0 1 0 1 0 0 0 0120572 0 120573 0 0 0 0 01199041205721

1198881205721

119904ℎ1205731

119888ℎ1205731

minus1199041205721

minus1198881205721

minus119904ℎ1205731

minus119888ℎ1205731

1205721198881205721

minus1205721199041205721

120573119888ℎ1205731

120573119904ℎ1205731

minus1205721198881205721

1205721199041205721

minus120573119888ℎ1205731

minus120573119904ℎ1205731

minus12057221199041205721

minus12057221198881205721

1205732119904ℎ1205731

1205732119888ℎ1205731

12057221199041205721

12057221198881205721

minus1205732119904ℎ1205731

minus1205732119888ℎ1205731

minus12057231198881205721+ 120598119904120572112057231199041205721+ 12059811988812057211205733119888ℎ1205731+ 120598119904ℎ12057311205733119904ℎ1205731+ 120598119888ℎ1205731

12057231198881205721

minus12057231199041205721minus1205733119888ℎ

1205731minus1205733119904ℎ

1205731

0 0 0 0 119904120572

119888120572

119904ℎ120573

119888ℎ120573

0 0 0 0 120572119888120572

minus120572119904120572

120573119888ℎ120573

120573119904ℎ120573

]]]]]]]]]]]]]]]

]

(A3)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

References

[1] H-Y Lin and Y-C Tsai ldquoFree vibration analysis of a uniformmulti-span beam carrying multiple spring-mass systemsrdquo Jour-nal of Sound and Vibration vol 302 no 3 pp 442ndash456 2007

[2] P A Hassanpour W L Cleghorn J K Mills and E Esmail-zadeh ldquoExact solution of the oscillatory behavior under axialforce of a beam with a concentrated mass within its intervalrdquoJournal of Vibration and Control vol 13 no 12 pp 1723ndash17392007

[3] G Sakar ldquoThe effect of axial force on the free vibration of anEuler-Bernoulli beam carrying a number of various concen-trated elementsrdquo Shock andVibration vol 20 no 3 pp 357ndash3672013

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

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

Shock and Vibration 9

[4] H Xiao M Sheng Z Liu and ZWei ldquoThe study on free vibra-tion of elastically restrained beams carrying various types ofattachments with arbitrary spatial distributionsrdquo Shock andVibration vol 20 no 3 pp 369ndash383 2013

[5] H Su and J R Banerjee ldquoExact natural frequencies of structuresconsisting of two-part beam-mass systemsrdquo Structural Engi-neering and Mechanics vol 19 no 5 pp 551ndash566 2005

[6] K Saeedi and R B Bhat ldquoClustered natural frequencies inmulti-span beams with constrained characteristic functionsrdquoShock and Vibration vol 18 no 5 pp 697ndash707 2011

[7] Y Yesilce ldquoFree vibrations of a Reddy-Bickford multi-spanbeam carryingmultiple spring-mass systemsrdquo Shock and Vibra-tion vol 18 no 5 pp 709ndash726 2011

[8] Y Chen ldquoOn the vibration of a beam or rods carrying a con-centrated massrdquo Journal of Applied Mechanics vol 30 no 2 pp310ndash311 1963

[9] S Naguleswaran ldquoTransverse vibrations of an Euler-Bernoulliuniform beam carrying two particles in-spanrdquo InternationalJournal of Mechanical Sciences vol 43 no 12 pp 2737ndash27522001

[10] J-SWu andT-L Lin ldquoFree vibration analysis of a uniform can-tilever beamwith pointmasses by an analytical-and-numerical-combinedmethodrdquo Journal of Sound and Vibration vol 136 no2 pp 201ndash213 1990

[11] M Gurgoze ldquoOn the alternative formulations of the frequencyequation of a Bernoulli-Euler beam to which several spring-mass systems are attached in-spanrdquo Journal of Sound andVibration vol 217 no 3 pp 585ndash595 1998

[12] P D Cha ldquoNatural frequencies of a linear elastica carrying anynumber of sprung massesrdquo Journal of Sound and Vibration vol247 no 1 pp 185ndash194 2001

[13] O R Barry J Zu J W Zu and D C D Oguamanam ldquoFreevibration analysis of a beam under axial load carrying a mass-spring-mass systemrdquo in Proceedings of the ASME InternationalDesign Engineering Technical Conferences amp Computers andInformation in Engineering Conference (IDETCCIE rsquo12) pp791ndash796 Chicago Ill USA August 2012

[14] M N Hamdan and B A Jubran ldquoFree and forced vibrationsof a restrained cantilever beam carrying a concentrated massrdquoJournal of Engineering Science vol 3 no 1 pp 71ndash83 1991

[15] F S Samani and F Pellicano ldquoVibration reduction on beamssubjected to moving loads using linear and nonlinear dynamicabsorbersrdquo Journal of Sound and Vibration vol 325 no 4-5 pp742ndash754 2009

[16] J-S Wu and D-W Chen ldquoDynamic analysis of a uniform can-tilever beam carrying a number of elastically mounted pointmasses with dampersrdquo Journal of Sound and Vibration vol 229no 3 pp 549ndash578 2000

[17] J-SWu andC-G Huang ldquoFree and forced vibrations of a Tim-oshenko beam with any number of translational and rotationalsprings and lumped massesrdquo Communications in NumericalMethods in Engineering vol 11 no 9 pp 743ndash756 1995

[18] B Zhang and S Shepard ldquoDynamic response of supportedbeams with intermediate supports under moving loadsrdquo Shockand Vibration vol 19 no 6 pp 1403ndash1413 2012

[19] T-P Chang and C-Y Chang ldquoVibration analysis of beams witha two degree-of-freedom spring-mass systemrdquo InternationalJournal of Solids and Structures vol 35 no 5-6 pp 383ndash4011998

[20] K H Low ldquoComments on ldquonon-linear vibrations of a beam-mass system under different boundary conditionsrdquo (with

authorsrsquo reply)rdquo Journal of Sound and Vibration vol 207 no 2pp 284ndash287 1997

[21] E Ozkaya andM Pakdemirli ldquoNon-linear vibrations of a beam-mass system with both ends clampedrdquo Journal of Sound andVibration vol 221 no 3 pp 491ndash503 1999

[22] M H Ghayesh F Alijani and M A Darabi ldquoAn analyticalsolution for nonlinear dynamics of a viscoelastic beam-heavymass systemrdquo Journal ofMechanical Science and Technology vol25 no 8 pp 1915ndash1923 2011

[23] M H Ghayesh S Kazemirad and M A Darabi ldquoA generalsolution procedure for vibrations of systems with cubic non-linearities and nonlineartime-dependent internal boundaryconditionsrdquo Journal of Sound and Vibration vol 330 no 22 pp5382ndash5400 2011

[24] C L Kirk and S M Wiedemann ldquoNatural frequencies andmode shapes of a free-free beamwith large endmassesrdquo Journalof Sound and Vibration vol 254 no 5 pp 939ndash949 2002

[25] P A Hassanpour E Esmailzadeh W L Cleghorn and J KMills ldquoGeneralized orthogonality condition for beams withintermediate lumped masses subjected to axial forcerdquo Journalof Vibration and Control vol 16 no 5 pp 665ndash683 2010

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

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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