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International Journal of Mechanical Sciences 60 (2012) 59–71
Contents lists available at SciVerse ScienceDirect
International Journal of Mechanical Sciences
0020-74
http://d
n Corr
Nis, A. M
fax: þ3
E-m
journal homepage: www.elsevier.com/locate/ijmecsci
Forced transverse vibration of Rayleigh and Timoshenko double-beamsystem with effect of compressive axial load
Vladimir Stojanovic a,b,n, Predrag Kozic a
a Department of Mechanical Engineering, University of Nis, A. Medvedeva 14, 18000 Nis, Serbiab DEMec/IDMEC, Faculdade de Engenharia, Universidade do Porto, Porto, Portugal
a r t i c l e i n f o
Article history:
Received 23 May 2011
Received in revised form
9 April 2012
Accepted 27 April 2012Available online 10 May 2012
Keywords:
Forced vibration
Timoshenko double beam
Rayleigh double beam
Winkler elastic layer
Critical buckling force
03/$ - see front matter & 2012 Elsevier Ltd. A
x.doi.org/10.1016/j.ijmecsci.2012.04.009
esponding author at: Department of Mechani
edvedeva 14, 18000 Nis, Serbia. Tel.: þ381
81 18 588 244.
ail address: stojanovic.s.vladimir@gmail.com
a b s t r a c t
Forced vibration and buckling of a Rayleigh and Timoshenko double-beam system continuously joined
by a Winkler elastic layer under compressive axial loading are considered in this paper. Based on the
Timoshenko beam theory, deflections of the beams are shown. The general solutions of forced
vibrations of beams subjected to arbitrarily distributed continuous loads are found. The analytical
solution of forced vibration with associated amplitude ratios is determined. The dynamic responses of
the system caused by arbitrarily distributed continuous loads are obtained. Vibrations caused by the
harmonic exciting forces are discussed, and conditions of resonance and dynamic vibration absorption
are formulated. Thus the beam-type dynamic absorber can be used to suppress the excessive vibrations
of corresponding beam systems. The effects of compressive axial load on the forced vibrations of the
Rayleigh and Timoshenko double-beam system are discussed for three cases of particular excitation
loadings. Numerical results of the present method are verified by comparing with those available in the
literature.
& 2012 Elsevier Ltd. All rights reserved.
1. Introduction
Structural members made of two parallel simply supportedbeams continuously joined by a linear, elastic layer of Winklertype are increasingly used in aeronautical, mechanical and civilengineering applications. Three important structural perfor-mances of these complex continuous systems are: weight reduc-tion, strength and stiffness increase, and vibration absorption. Theone-dimensional continuous elements such as strings and beamsare often used to resist tension, compression or bending in manymodern engineering structures. Being a simple model of a one-dimensional continuous system, a beam has been a subject ofgreat scientific interest. As a matter of fact, the phenomenon oftransverse vibration and buckling problems of such systems is ofpractical interest and has a wide application in engineeringpractice. In the past few decades, such structures have beenextensively covered by many investigators. Most of them havebeen done within the scope of the classical Bernoulli-Euler beamtheory to investigate the vibration and buckling behavior of thebeam on elastic foundations. This theory leads to a significantover prediction of the natural frequencies and buckling stresses ofdeep beams due to the neglect of the effects of transverse shear
ll rights reserved.
cal Engineering, University of
18 500 666;
(V. Stojanovic).
deformation, depth change, and rotary inertia. For deep beamswith small length-to-depth ratio and/or beams in which highermodes may appear, the Timoshenko beam theory, which takesinto account the effects of shear deformation and rotary inertia, isapplied in the analysis by Matsunaga [1–3]. Higher-order shear-deformable theories have been developed for beams with rectan-gular cross-sections that account for the strain distributionthrough the depth to satisfy the stress-free boundary conditionson the upper and lower surfaces without the need for shearcorrection coefficient. In retaining the parabolic distribution ofthe transverse shear strain, a shear deformation theory forrectangular beams that accounts for the shear free boundaryconditions on the lateral surfaces of the beam is proposed inLevinson [4,5]. However, it has been shown that the behavior offoundation materials in engineering practice cannot be repre-sented by this foundation model which consists of independentlinear elastic springs. In order to find a physically close andmathematically simple foundation model, Pasternak proposed aso-called two-parameter foundation model with shear interac-tion. In Wang, Stephens and De Rosa [6,7], a study of the naturalvibrations of a Timoshenko beam on a Pasternak-type foundationis presented. Frequency equations are derived for beams withdifferent end restraints. A specific example is given to show theeffects of rotary inertia, shear deformation, and foundation con-stants on the natural frequencies of the beam. Li-Qun et al. [8]study dynamic stability of an axially accelerating viscoelasticbeam undergoing parametric resonance. Oniszczuk [9–11]
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–7160
analyzes free and forced transverse vibrations of an elasticallyconnected complex simply supported double-beam system. Thefree and forced transverse vibration and buckling of a double-beam system under compressive axial loading are investigated inthe paper by Zhang et al. [12,13]. Explicit expressions are derivedfor the natural frequencies and the associated amplitude ratios ofthe two beams, and the analytical solutions of the critical bucklingload are obtained. The influences of the compressive axial loadingon the responses of the double-beam system are discussed. It isshown that the critical buckling load of the system is related to theaxial compression ratio of the two beams and the Winkler elasticlayer, and the properties of forced transverse vibration of thesystem greatly depend on the axial compressions. Stojanovic et al.[14] analyze the free transverse vibration and buckling of adouble-beam system with effect of rotary inertia and shear undercompressive axial loading. Explicit expressions are derived for thenatural and shear frequencies and the associated amplitude ratiosof the two beams, and the analytical solutions of the criticalbuckling load are obtained. The influences of the compressive axialloading on the responses of the double-beam system are dis-cussed. It is shown that the critical buckling load of the system isrelated to the axial compression ratio of the two Timoshenkobeams and the Winkler elastic layer. Mey et al. [15] analyze thefree and forced vibration of axially loaded cracked Timoshenkobeams. Li et al. [16] analyze elastically connected Euler double-beam systems by using the spectral finite element method. As anextension of the work of Zhang et al. [12,13], the forced vibrationand buckling of a double-beam system for a Rayleigh andTimoshenko double-beam system under axial loading are studiedin the present paper. Both beams have the same length. It is alsosupposed that the buckling can only occur in the plane where thedouble-beam system lies. The explicit expressions are derived fornatural frequencies and associated amplitude ratio of the twobeams, and the analytical solutions of the transverse vibration andthe critical buckling are obtained. Since the beams have largecross-sectional dimensions in comparison to their lengths, theTimoshenko theory, which considers the effects of rotary inertiaand shear, gives a better approximation to the true behavior of thebeams. The two higher shear frequencies which are associatedwith a shear vibration are of much lesser technical interest.
2. Structural model and formulation of the problem
The basic differential equations of motion for the analysis willbe deduced by considering the Timoshenko-beam of length l,Fig. 1a, subjected to axial compressive force F, and to a distributedlateral loads of intensity q1 and q2 which vary with the distance x
along the beam. This will be applied on the basis of the following
Fig. 1. (a) Timoshenko-beam subjected to an axial compressive force F and to a distri
element of length dx.
assumptions: (a) the behavior of the beam material is linearelastic; (b) the cross-section is rigid and constant throughout thelength of the beam and has one plane of symmetry; (c) sheardeformations of the cross-section of the beam are taken intoaccount while elastic axial deformations are ignored; (d) theequations are derived bearing in mind the geometric axialdeformations; (e) the axial forces F acting on the ends of thebeam are not changed with time. An element of length dx
between two cross-sections taken normal to the deflected axisto the beam is shown in Fig. 1b.
Since the slope of the beam is small, the normal forces actingon the sides of the element can be taken to be equal to the axialcompressive force F. The shearing force FT is related to thefollowing relationship
FT ¼ kGA@w
@x�c
� �, ð1Þ
where w¼w(x,t) is the displacement of a cross-section in y
direction, qw/qx is the global rotation of the cross-section, c isthe bending rotation, G is the shear modulus, A is the area of thebeam cross section, and k is the shear factor. Analogously, therelationship between bending moments M and bending anglesc¼c(x,t) is given by
M¼�EI@c@x
, ð2Þ
where E is the Young modulus and I is the second moment of thearea of the cross-section. Finally, forces and moments of inertiaare given by
f in ¼�rA@2w
@t2, Jin ¼�rI
@2c@t2
, ð3Þ
respectively, where r is the mass density. The forces acting on adifferential layered-beam element are shown in Fig. 1b. Thedynamic-force equilibrium conditions of these forces are givenby the following equations
rA@2w
@t2�kGA
@2w
@x2�@c@x
!þF
@2w
@x2�q1ðxÞþq2ðxÞ ¼ 0, ð4aÞ
rI@2c@t2�EI
@2c@x2�kGA
@w
@x�c
� �¼ 0: ð4bÞ
Fig. 2 shows the structural model of a layered-beam systemcomposed of two parallel beams of uniform properties axiallyloaded with a flexible a Winkler elastic layer in-between. Thebeams are subjected to axial compressions F1 and F2 that arepositive in compression and arbitrarily distributed transversecontinuous loads f1 and f2 that are positive when they actdownward. Assumption is that the two beams have the same
buted lateral loads of intensity q1 and q2; (b) Deflected differential layered-beam
Fig. 2. Double-beam complex system.
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–71 61
effective material constants. The transverse displacements of thebeams and bending rotations are wi ¼wiðx,tÞ,ci ¼ciðx,tÞ, i¼ 1,2respectively. If we apply the above mentioned procedure to adifferential element of each beam, the following set of coupleddifferential equations will be obtained:
GA1k@c1
@x�@2w1
@x2
!þrA1
@2w1
@t2þF1
@2w1
@x2þKðw1�w2Þ ¼ f 1ðx,tÞ,
ð5aÞ
EI1@2c1
@x2þGA1k
@w1
@x�c1
� ��rI1
@2c1
@t2¼ 0, ð5bÞ
GA2k@c2
@x�@2w2
@x2
!þrA2
@2w2
@t2þF2
@2w2
@x2þKðw2�w1Þ ¼ f 2ðx,tÞ,
ð6aÞ
EI2@2c2
@x2þGA2k
@w2
@x�c2
� ��rI2
@2c2
@t2¼ 0, ð6bÞ
where K is the stiffness modulus of a Winkler elastic layer. Keet al. [17] obtained adapted coupled differential equations as Eqs.(5) and (6) which are adapted to the nonlocal Timoshenko theoryin analysis the free nonlinear vibration of embedded double-walled carbon nanotubes. Also, if we reduce the number of beamsin two in multi-Timoshenko beam model for analysis of multi-walled carbon nanotubes proposed by Wang et al [18], Eqs.(5) and (6) can be obtained with included external excitation.Eliminating c1 from Eqs. (5a) and (5b) and c2 from Eqs. (6a) and(6b), one can obtain the following two fourth-order partialdifferential equations
EI1 1�F1
GA1k
� �@4w1
@x4þrA1 1þ
KI1
GA21k
!@2w1
@t2�rKI1
GA1k
@2w2
@t2
þF1 1�KEI1
F1GA1k
� �@2w1
@x2þ
KEI1
GA1k
@2w2
@x2
�rI1 1þE
kG�
F1
GA1k
� �@4w1
@x2@t2þr2I1
Gk
@4w1
@t4
þKðw1�w2Þ ¼ f 1ðx,tÞþrI1
GA1k
@2f 1ðx,tÞ
@t2�
EI1
GA1k
@2f 1ðx,tÞ
@x2, ð7Þ
EI2 1�F2
GA2k
� �@4w2
@x4þrA2 1þ
KI2
GA22k
!@2w2
@t2�rKI2
GA2k
@2w1
@t2
þF2 1�KEI2
F2GA2k
� �@2w2
@x2þ
KEI2
GA2k
@2w1
@x2
�rI2 1þE
kG�
F2
GA2k
� �@4w2
@x2@t2þr2I2
Gk
@4w2
@t4
�Kðw1�w2Þ ¼ f 2ðx,tÞþrI2
GA2k
@2f 2ðx,tÞ
@t2�
EI2
GA2k
@2f 2ðx,tÞ
@x2: ð8Þ
Eqs. (7) and (8) can be reduced to fourth-order partialdifferential equations for forced vibration of the Timoshenkodouble-beam model
C2b1 1�
m1F1
C2s1C2
r1
!@4w1
@x4þ 1þ
H1
C2s1
!@2w1
@t2�
H1
C2s1
@2w2
@t2
þ m1F1�C2
b1H1
C2s1C2
r1
!@2w1
@x2þ
C2b1H1
C2s1C2
r1
@2w2
@x2
� C2r1þ
C2b1
C2s1C2
r1
�m1F1
C2s1
!@4w1
@x2@t2þ
1
C2s1
@4w1
@t4
þH1ðw1�w2Þ ¼m1 f 1ðx,tÞþ1
C2s1
@2f 1ðx,tÞ
@t2�
C2b1
C2s1C2
r1
@2f 1ðx,tÞ
@x2
!,
ð9Þ
C2b2 1�
m2F2
C2s2C2
r2
!@4w2
@x4þ 1þ
H2
C2s2
!@2w2
@t2�
H2
C2s2
@2w1
@t2
þ m2F2�C2
b2H2
C2s2C2
r2
!@2w2
@x2þ
C2b2H2
C2s2C2
r2
@2w1
@x2
� C2r2þ
C2b2
C2s2C2
r2
�m2F2
C2s2
!@4w2
@x2@t2þ
1
C2s2
@4w2
@t4þH2ðw2�w1Þ
¼m2 f 2ðx,tÞþ1
C2s2
@2f 2ðx,tÞ
@t2�
C2b2
C2s2C2
r2
@2f 2ðx,tÞ
@x2
!, ð10Þ
where
m1 ¼1
rA1, m2 ¼
1
rA2, H1 ¼
K
rA1, H2 ¼
K
rA2:
The coefficients
Cbi ¼
ffiffiffiffiffiffiffiffiEIi
rAi
s, Csi ¼
ffiffiffiffiffiffiffiffiffiffiGAik
rIi
s, Cri ¼
ffiffiffiffiffiIi
Ai
s, i¼ 1,2:
related to bending stiffness, shear stiffness and rotational effects,respectively, are now introduced. The shear beam model, theRayleigh beam model and the simple Euler beam model can beobtained from the Timoshenko beam model by setting Cri to zero(that is, ignoring the rotational effect), Csi to infinity (ignoring theshear effect) and setting both Cri to zero and Csi to infinity,respectively. If we ignore only the shear effect (Csi-N), we canobtain fourth-order partial differential equations for forced vibra-tion of the Rayleigh double-beam model
C2b1
@4w1
@x4þ@2w1
@t2þm1F1
@2w1
@x2�C2
r1
@4w1
@x2@t2þH1ðw1�w2Þ ¼m1f 1ðx,tÞ,
ð11Þ
C2b2
@4w2
@x4þ@2w2
@t2þm2F2
@2w2
@x2�C2
r2
@4w2
@x2@t2þH2ðw2�w1Þ ¼m2f 2ðx,tÞ:
ð12Þ
The initial conditions in general form and boundary conditionsfor simply supported beams of the same length l are assumed asfollows
wiðx,0Þ ¼wi0ðxÞ, _wiðx,0Þ ¼ vi0ðxÞ, ð13Þ
wiðx,0Þ ¼w00i ð0,tÞ ¼wiðl,0Þ ¼w00i ðl,tÞ ¼ 0, i¼ 1,2: ð14Þ
When the effect of both shear deformation and rotary inertia isignored, Eqs. (9) and (10) can be reduced to the general equationsfor forced transverse vibrations given by Zhang et al. [13].
3. Solution of equations
In order to solve the non-homogeneous partial differentialeqs. (9) and (10) representing forced vibrations of a double
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–7162
Timoshenko beam system and the non-homogeneous partialdifferential eqs. (11) and (12) representing forced vibrations of adouble Rayleigh beam system, the natural frequencies and thecorresponding mode shapes of the system should be obtained bysolving the undamped free vibration with appropriate boundaryconditions. Assuming time harmonic motion and using separationof variables, the solutions to Eqs. (9), (10), (11) and (12) with thegoverning boundary conditions (14) can be written in the form
wiðx,tÞ ¼X1n ¼ 1
XnðxÞTniðtÞ, i¼ 1,2, ð15Þ
where Tin(t) is the unknown time function, and Xn(x) is the knownmode shape function for simply supported single beam, which isdefined as
XnðxÞ ¼ sinðknxÞ, kn ¼ np=l, n¼ 1,2,3,���: ð16Þ
Introducing the general solutions (15) into Eqs. (9) and (10),one gets the following ordinary differential equations for theTimoshenko double-beam model
X1n ¼ 1
1
C2s1
d4Tn1
dt4þ 1þC2
r1k2nþ
C2b1k2
n
C2s1C2
r1
þ1
C2s1
ðH1�F1Z1Þ
" #d2Tn1
dt2
(
�H1
C2s1
d2Tn2
dt2þ C2
b1k4nþðH1�F1Z1Þ 1þ
C2b1k2
n
C2s1C2
r1
!" #Tn1
�H1 1þC2
b1k2n
C2s1C2
r1
!Tn2
)Xn ¼m1 f 1þ
1
C2s1
€f 1�C2
b1
C2s1C2
r1
f 001
" #, ð17Þ
X1n ¼ 1
1
C2s2
d4Tn2
dt4þ 1þC2
r2k2nþ
C2b2k2
n
C2s2C2
r2
þ1
C2s2
ðH2�F2Z2Þ
" #d2Tn2
dt2
(
�H2
C2s2
d2Tn1
dt2þ C2
b2k4nþðH2�F2Z2Þ 1þ
C2b2k2
n
C2s2C2
r2
!" #T2n
�H2 1þC2
b2k2n
C2s2C2
r2
!Tn1
)Xn ¼m2 f 2þ
1
C2s2
€f 2�C2
b2
C2s2C2
r2
f 2’’
" #, ð18Þ
where
Z1 ¼k2
n
rA1, Z2 ¼
k2n
rA2:
Introducing the general solutions (13) into Eqs. (9) and (10),one gets the following ordinary differential equations for theRayleigh double-beam model
X1n ¼ 1
J1d2Tn1
dt2þðN1�F1Z1ÞTn1�H1Tn2
" #Xn ¼m1f 1, ð19Þ
X1n ¼ 1
J2d2Tn2
dt2þðN2�F2Z2ÞTn2�H2Tn1
" #Xn ¼m2f 2, ð20Þ
where
J1 ¼ 1þC2r1k2
n , J2 ¼ 1þC2r2k2
n, N1 ¼ C2b1k4
nþH1, N2 ¼ C2b2k4
nþH2:
4. Forced vibration of a Rayleigh double-beam system
The natural frequencies and the corresponding mode shapes ofthe system should be obtained by solving the undamped freevibration with appropriate boundary conditions of ordinarydifferential equations for the unknown time functions
J1d2Tn1
dt2þðN1�F1Z1ÞTn1�H1Tn2 ¼ 0, ð21Þ
J2d2Tn2
dt2þðN2�F2Z2ÞTn2�H2Tn1 ¼ 0: ð22Þ
The solutions of Eqs. (21) and (22) can be assumed to have thefollowing forms
Tn1 ¼ Cnejont , Tn2 ¼Dnejont , j¼ffiffiffiffiffiffiffi�1p
: ð23Þ
Where on denotes the natural frequency of the system. Substitut-ing Eq. (23) into Eqs. (21) and (22) results in the following systemof homogeneous algebraic equations for the unknown constantsCn, Dn
ðN1�F1Z1�J1o2nÞCn�H1Dn ¼ 0, ð24Þ
ðN2�F2Z2�J2o2nÞDn�H2Cn ¼ 0: ð25Þ
When the determinant of the coefficients in Eqs. (24) and (25)vanishes, non-trivial solutions for the constants Cn and Dn can beobtained, which yields the following frequency (characteristic)equation:
J1J2o4n�ðN1J2þN2J1�F1Z1J2�F2Z2J1Þo2
nþðN1�F1Z1ÞðN2�F2Z2Þ�H1H2 ¼ 0:
ð26Þ
It can be observed that the discriminant of this biquadraticalgebraic equation is positive
D¼ ½J1ðN1�F1Z1Þ�J2ðN2�F2Z2Þ�2þ4J1J2H1H240,
then from the characteristic Eq. (26), we obtain
o2nI ¼
J2ðN1�F1Z1Þþ J1ðN2�F2Z2Þ
2J1J2
�1
2J1J2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½J2ðN1�F1Z1Þ�J1ðN2�F2Z2Þ�
2þ4J1J2H1H2
q, ð27Þ
o2nII ¼
J2ðN1�F1Z1Þþ J1ðN2�F2Z2Þ
2J1J2
þ1
2J1J2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½J2ðN1�F1Z1Þ�J1ðN2�F2Z2Þ�
2þ4J1J2H1H2
q, ð28Þ
where onI is the lower natural frequency of the system, and onII isthe higher natural frequency. For each of the natural frequencies,the associated amplitude ratio of vibration modes of the twobeams is given by
a�1ni ¼
Cn
Dn¼
H1
N1�F1Z1�J1o2n
¼N2�F2Z2�J2o2
n
H2: ð29Þ
Introducing onI and onII into Eq. (27), respectively, we have
a�1nI ¼
Cn
Dn¼
H1
N1�F1Z1�J1o2nI
¼N2�F2Z2�J2o2
nI
H2, ð30Þ
a�1nII ¼
Cn
Dn¼
H1
N1�F1Z1�J1o2nII
¼N2�F2Z2�J2o2
nII
H2: ð31Þ
The forced vibrations of beams subjected to arbitrarily dis-tributed continuous loads can be determined applying the classi-cal modal expansion method [15]. Following the above analysisfor the undamped free transverse vibration, particular solutions ofnon-homogeneous differential Eqs. (11) and (12) representingforced vibrations of a Rayleigh double model can be assumed inthe following
w1ðx,tÞ ¼X1n ¼ 1
XnðxÞXII
i ¼ I
SniðtÞ, ð32Þ
w2ðx,tÞ ¼X1n ¼ 1
XnðxÞXII
i ¼ I
aniSniðtÞ, ð33Þ
where SniðtÞði¼ I,IIÞ is the unknown time function correspondingto the natural frequency oni. Introduction of Eqs. (32) and (33)
Fig. 3. Double-beam complex system subjected to harmonic distributed contin-
uous load.
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–71 63
into Eqs. (11) and (12) results in
X1n ¼ 1
XnðxÞXII
i ¼ I
½ J1€SniþðN1�F1Z1�H1aniÞSni� ¼m1f 1, ð34Þ
X1n ¼ 1
XnðxÞXII
i ¼ I
½ J2€SniþðN2�F2Z2�H2a�1
ni ÞSni�ani ¼m2f 2: ð35Þ
By multiplying the relations (34) and (35) by the eigenfunctionXm, then integrating them with respect to x from 0 to l and usingorthogonality conditionZ l
0XnXm dx¼
Z l
0sinðknxÞ sinðkmxÞdx¼ cdnm, ð36Þ
c¼
Z l
0X2
n dx¼
Z l
0½sinðknxÞ�2 dx¼
l
2,
where dnm is the Kronecker delta function, we have
XII
i ¼ I
J1€SniþðN1�F1Z1�H1aniÞSni
h i¼ 2
m1
l
Z l
0Xnf 1 dx, ð37Þ
XII
i ¼ I
J2€SniþðN2�F2Z2�H2a�1
ni ÞSni
h iani ¼ 2
m2
l
Z l
0Xnf 2 dx: ð38Þ
From Eqs. (30), (31), (37) and (38) after some algebra weobtain
XII
i ¼ I
€Sniþo2nISni
h i¼ 2
m1
J1l
Z l
0Xnf 1 dx, ð39Þ
XII
i ¼ I
€Sniþo2nISni
h iani ¼ 2
m2
J2l
Z l
0Xnf 2 dx: ð40Þ
From Eqs. (39) and (40), we obtain
€Sniþo2nISni ¼ ZniðtÞ, i¼ I,II, ð41Þ
where
ZnIðtÞ ¼2
anII�anI
Z l
0M1anIIf 1ðx,tÞ�M2f 2ðx,tÞ� �
sinðknxÞdx, ð42Þ
ZnIIðtÞ ¼2
anI�anII
Z l
0M1anIf 1ðx,tÞ�M2f 2ðx,tÞ� �
sinðknxÞ dx, ð43Þ
where
M1 ¼m1
J1l, M2 ¼
m2
J2l:
By combining Eqs. (32), (33) and (41), the forced vibrations ofan elastically connected Rayleigh double-beam system can bedescribed by
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞXII
i ¼ I
1
oni
Z t
0ZniðsÞ sin oniðt�sÞ½ � ds, ð44Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞXII
i ¼ I
ani
oni
Z t
0ZniðsÞ sin oniðt�sÞ½ � ds: ð45Þ
Now these general solutions (44) and (45) are used to find thevibrations of the two coupled Rayleigh beams.
Case 1. Stationary harmonic loads. For simplicity of furtheranalysis, it is assumed that only one of the two beams is subjectedto an arbitrarily distributed harmonic load acting on the entirelength of the beam. An arbitrarily distributed harmonic load actson the first Rayleigh beam. The second Rayleigh beam is unloaded(Fig. 3). The exciting loading of a Rayleigh double-beam system is
f 1ðx,tÞ ¼ qðxÞ sinðOtÞ, f 2ðx,tÞ ¼ 0 ð46Þ
where q(x) is the arbitrary function of spatial coordinate x and Ois the frequency of the load. Substituting Eq. (46) into Eqs. (42)and (43), we obtain
ZnIðtÞ ¼anII
ðanII�anIÞMRn sinðOtÞ, n¼ 1,3,5,. . ., ð47Þ
ZnIIðtÞ ¼anI
ðanI�anIIÞMRn sinðOtÞ, n¼ 1,3,5,. . ., ð48Þ
where
MRn ¼ 2M1
Z l
0qðxÞ sinðknxÞ dx,n¼ 1,3,5,. . .
Introduction of Eqs. (47) and (48) into Eqs. (44) and (45) gives
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An1 sinðOtÞþXII
i ¼ I
Bni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð49Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An2 sinðOtÞþXII
i ¼ I
aniBni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð50Þ
where
An1 ¼MRn
anII�anI
anII
o2nI�O
2�
anI
o2nII�O
2
" #,
An2 ¼MRn
anII�anI
anIanII
o2nI�O
2�
anIanII
o2nII�O
2
" #,
BnI ¼MRnanII
anII�anI
OonIðO
2�o2
nIÞ
" #,
BnII ¼MRnanI
anII�anI
OonIIðo2
nII�O2Þ
" #:
Ignoring the free response, the forced vibrations of the Ray-leigh double-beam system can be obtained by
w1ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An1 sinðknxÞ, n¼ 1,3,5,. . ., ð51Þ
w2ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð52Þ
The following fundamental conditions of resonance anddynamic vibration absorption have practical significance:
aÞ Resonance : O¼oni, n¼ 1,3,5,. . .,
bÞ Dynamic vibration absorption : O2¼anIIo2
nII�anIo2nI
anII�anI,
An1 ¼ 0, An2 ¼MRnanI�anII
o2nI�o2
nII
, n¼ 1,3,5,. . .
Case 2. Uniformly distributed harmonic load. The harmonic uni-formly distributed continuous load acts on the first Rayleigh
Fig. 4. Double-beam complex system subjected to harmonic uniform distributed
continuous load.
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–7164
beam. The second Rayleigh beam is unloaded (Fig. 4). The excitingloading of a Rayleigh double-beam system is
f 1ðx,tÞ ¼ q sinðOtÞ, f 2ðx,tÞ ¼ 0 ð53Þ
whereq and O are the amplitude and exciting frequency of theload, respectively. Substituting Eq. (53) into Eqs. (42) and (43), weobtain
ZnIðtÞ ¼anII
ðanII�anIÞMRn sinðOtÞ, n¼ 1,3,5,. . ., ð54Þ
ZnIIðtÞ ¼anI
ðanI�anIIÞMRn sinðOtÞ, n¼ 1,3,5,. . ., ð55Þ
where
MRn ¼M14lq
np, n¼ 1,3,5,. . .
Introduction of Eqs. (54) and (55) into Eqs. (44) and (45) gives
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An1 sinðOtÞþXII
i ¼ I
Bni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð56Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An2 inðOtÞþXII
i ¼ I
aniBni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð57Þ
where
An1 ¼MRn
anII�anI
anII
o2nI�O
2�
anI
o2nII�O
2
" #,
An2 ¼MRn
anII�anI
anIanII
o2nI�O
2�
anIanII
o2nII�O
2
" #,
BnI ¼MRnanII
anII�anI
OonIðO
2�o2
nIÞ
" #,
BnII ¼MRnanI
anII�anI
OonIIðo2
nII�O2Þ
" #:
Ignoring the free response, the forced vibrations of the Ray-leigh double-beam system can be obtained by
w1ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An1 sinðknxÞ, n¼ 1,3,5,. . ., ð58Þ
w2ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð59Þ
The following fundamental conditions of resonance anddynamic vibration absorption have practical significance:
aÞ Resonance : O¼oni, n¼ 1,3,5,. . .,
bÞ Dynamic vibration absorption : O2¼anIIo2
nII�anIo2nI
anII�anI,
An1 ¼ 0, An2 ¼MRnanI�anII
o2nI�o2
nII
, n¼ 1,3,5,. . .:
Case 3. Harmonic concentrated force. The first Rayleigh beam issubjected to the harmonic concentrated force applied at themidspan of the beam (Fig. 4). The exciting loading of a Rayleighdouble-beam system is
f 1ðx,tÞ ¼ F sinðOtÞdðx�0:5lÞ, f 2ðx,tÞ ¼ 0, ð60Þ
where F and O are the amplitude and frequency of the excitingharmonic force, respectively, and d(x) is the Dirac delta function.Substituting Eq. (60) into Eqs. (42) and (43), we obtain
ZnIðtÞ ¼anII
ðanII�anIÞMRn sinðOtÞ, n¼ 1,3,5,. . ., ð61Þ
ZnIIðtÞ ¼anI
ðanI�anIIÞMRn sinðOtÞ, n¼ 1,3,5,. . ., ð62Þ
where
MRn ¼ 2FM1 sinnp2
� �, n¼ 1,3,5,. . .
Introduction of Eqs. (61) and (62) into Eqs. (44) and (45) gives
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An1 sinðOtÞþXII
i ¼ I
Bni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð63Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An2 sinðOtÞþXII
i ¼ I
aniBni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð64Þ
where
An1 ¼MRn
anII�anI
anII
o2nI�O
2�
anI
o2nII�O
2
" #,
An2 ¼MRn
anII�anI
anIanII
o2nI�O
2�
anIanII
o2nII�O
2
" #,
BnI ¼MRnanII
anII�anI
OonIðO
2�o2
nIÞ
" #,
BnII ¼MRnanI
anII�anI
OonIIðo2
nII�O2Þ
" #:
The forced vibrations of the Rayleigh double-beam system canbe obtained by
w1ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An1 sinðknxÞ, n¼ 1,3,5,. . ., ð65Þ
w2ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð66Þ
The following fundamental conditions of resonance anddynamic vibration absorption have practical significance:
aÞ Resonance : O¼oni, n¼ 1,3,5,. . .,
bÞ Dynamic vibration absorption : O2¼anIIo2
nII�anIo2nI
anII�anI,
An1 ¼ 0, An2 ¼MRnanI�anII
o2nI�o2
nII
, n¼ 1,3,5,. . .:
5. Forced vibration of a Timoshenko double-beam system
Now, when we consider the influence of rotary inertia andshear, we have a structural model composed of two parallelTimoshenko beams continuously joined with a flexible Winkler
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–71 65
elastic layer in-between. Solving the undamped free vibrationgives four frequencies, two shear frequencies which are asso-ciated with a shear vibration and two frequencies associated witha transverse vibration with appropriate boundary conditions ofordinary differential equations for the unknown time functions
d4Tn1
dt4þ J1
d2Tn1
dt2�H1
d2Tn2
dt2þP1Tn1�Q1Tn2 ¼ 0, ð67Þ
d4Tn2
dt4þ J2
d2Tn2
dt2�H2
d2Tn1
dt2þP2Tn2�Q2Tn1 ¼ 0, ð68Þ
where
R1 ¼ 1þC2
b1k2n
C2s1C2
r1
, R2 ¼ 1þC2
b2k2n
C2s2C2
r2
, Q1 ¼H1C2s1R1, Q2 ¼H2C2
s2R2,
J1 ¼ C2s1 R1þC2
r1k2nþ
1
C2s1
ðH1�F1Z1Þ
" #,
J2 ¼ C2s2 R2þC2
r2k2nþ
1
C2s2
ðH2�F2Z2Þ
" #,
P1 ¼ C2s1½C
2b1k4
nþðH1�F1Z1ÞR1�, P2 ¼ C2s2½C
2b2k4
nþðH2�F2Z2ÞR2�:,
In order to find natural frequencies of the structural model, thesolution of Eqs. (67) and (68) could be expressed as
Tn1 ¼ Cnejont , Tn2 ¼Dnejont , j¼ffiffiffiffiffiffiffi�1p
, ð69Þ
where on denotes the natural frequencies of the system. Bysubstituting of Eq. (69) into Eqs. (67) and (68), the equivalentalgebraic eigenvalue equations are obtained
ðo4n�J1o2
nþP1ÞCn�ðQ1�H1o2nÞDn ¼ 0, ð70Þ
�ðQ2�H2o2nÞCnþðo4
n�J2o2nþP2ÞDn ¼ 0, ð71Þ
and the Eqs. (70) and (71) have non-trivial solutions when thedeterminant of the coefficient matrix of the Cn and Dn vanishes.Setting the determinant equal to zero yields
o8n�ðJ1þ J2Þo6
nþðP1þP2þ J1J2�H1H2Þo4n
�ðJ1P2þ J2P1�H1Q2�H2Q1Þo2nþP1P2�Q1Q2 ¼ 0: ð72Þ
Therefore the fourth order polynomial equation for the roots lmust be solved, where l¼o2
n is substituted into Eq. (72) toexplicitly reduce it to a fourth order polynomial equation. Thesolutions can be found in close form as follows. Eq. (72) can berewritten as
l4þa1l
3þa2l
2þa3lþa4 ¼ 0, ð73Þ
where
a1 ¼�ðJ1þ J2Þ, a2 ¼ P1þP2þ J1J2�H1H2,
a3 ¼�ðJ1P2þ J2P1�H1Q2�H2Q1Þ, a4 ¼ P1P2�Q1Q2:
The fourth order Eq. (73) can be factorized as
ðl2þp1lþq1Þðl
2þp2lþq2Þ ¼ 0, ð74Þ
where
p1
p2
( )¼
1
2a17
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia1�4a2þ4w1
ph i,
q1
q2
( )¼
1
2w17
a1w1�2a3ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2
1�4a2þ4w1
q264
375,
o2nI ¼
J2P1þ J1P2�H2Q1�H1Q2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH2Q1þH1Q2�J2P1�J1P2Þ
2þ4 H1H2
q2ðJ1J2�H1H2Þ
and w1 is one of the roots of the following cubic equation:
w3�a2w2þða1a3�4a4Þwþð4a2a4�a23�a2
1a4Þ ¼ 0: ð75Þ
Then the four roots of Eq. (73) can be written as
l1
l2
( )¼�
p1
27
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
1
4�q1
s,
l3
l4
( )¼�
p2
27
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffip2
2
4�q2
s: ð76Þ
The three roots of Eq. (75) can be written as
w1 ¼ a2=3þ2ffiffiffiffiffiffiffiffiffi�Q
pcosðy=3Þ,
w2 ¼ a2=3þ2ffiffiffiffiffiffiffiffi�Q
pcos½ðyþ2pÞ=3�,
w3 ¼ a2=3þ2ffiffiffiffiffiffiffiffi�Q
pcos½ðyþ4pÞ=3�, ð77Þ
where
y¼ cos�1ðS=
ffiffiffiffiffiffiffiffiffiffi�Q3
qÞ,Q ¼�
1
9ða2
2�3a1a3þ12a4Þ,
S¼1
54ð2a3
2�9a1a2a3þ27a23þ27a2
1a4�72a2a4Þ:
Then from the characteristic Eq. (73), we obtain two naturalfrequencies which are associated with a transverse vibrationonI ¼
ffiffiffiffiffil1
p,onII ¼
ffiffiffiffiffil3
p, and two much higher natural shear fre-
quencies which are associated with a shear vibrationonIðsÞ ¼
ffiffiffiffiffil2
p,onIIðsÞ ¼
ffiffiffiffiffil4
p. The analytical expressions for the free
natural and shear frequencies of the double-beam complexsystem with the influence of rotary inertia and shear are deter-mined and given in Ref. [14]. Shear effect makes two frequenciesonI and onII which are associated with a transverse vibrationlower. The two higher shear frequencies onI(s) and onII(s) whichare associated with a shear vibration are of much lesser technicalinterest. Ordinary differential equations under the influence ofshear and rotary inertia for the unknown time functions can bewritten as
J1d2T1n
dt2�H1
d2T2n
dt2þP1T1n�Q1T2n ¼ 0, ð78Þ
J2d2T2n
dt2�H2
d2T1n
dt2þP2T2n�Q2T1n ¼ 0: ð79Þ
In order to find natural frequencies associated with a trans-verse vibration of the structural model, the solution of Eqs. (78)and (79) could be expressed as
Tn1 ¼ Cnejont , Tn2 ¼Dnejont , j¼ffiffiffiffiffiffiffi�1p
: ð80Þ
By substituting Eq. (80) into Eqs. (78) and (79), we obtain
ð�J1o2nþP1ÞCn�ðQ1�H1o2
nÞDn ¼ 0, ð81Þ
�ðQ2�H2o2nÞCnþð�J2o2
nþP2ÞDn ¼ 0, ð82Þ
When the determinant of the coefficients in Eqs. (81) and (82)vanishes, non-trivial solutions for the constants Cn and Dn can beobtained, which yield the following frequency equation:
ðJ1J2�H1H2Þo4nþðH1Q2þH2Q1�J1P2�J2P1Þo2
nþP1P2�Q1Q2 ¼ 0:
ð83Þ
It can be observed that the discriminant of this biquadraticalgebraic equation is positive
D¼ ðH2Q1þH1Q2�J2P1�J1P2Þ2þ4ðH1H2�J1J2ÞðP1P2�Q1Q2Þ40,
Then from the characteristic Eq. (83), we obtain
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
�J1J2ðP1P2�Q1Q2Þ
, ð84Þ
o2nII ¼
J2P1þ J1P2�H2Q1�H1Q2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH2Q1þH1Q2�J2P1�J1P2Þ
2þ4ðH1H2�J1J2ÞðP1P2�Q1Q2Þ
q2ðJ1J2�H1H2Þ
, ð85Þ
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–7166
Where onI is the lower natural frequency of the system, and onII
is the higher natural frequency associated with a transversevibration of a Timoshenko double-beam model. For each of thenatural frequencies, the associated amplitude ratio of vibrationmodes of the two beams is given by
a�1ni ¼
Cn
Dn¼
Q1�H1o2n
P1�J1o2n
¼P2�J2o2
n
Q2�H2o2n
: ð86Þ
Introducing onI and onII into Eq. (86), respectively, we have
a�1nI ¼
Cn
Dn¼
Q1�H1o2nI
P1�J1o2nI
¼P2�J2o2
nI
Q2�H2o2nI
, ð87Þ
a�1nII ¼
Cn
Dn¼
Q1�H1o2nII
P1�J1o2nII
¼P2�J2o2
nII
Q2�H2o2nII
: ð88Þ
Following analysis for the undamped free transverse vibration,particular solutions of non-homogeneous differential Eqs. (9) and(10) and negligible shear frequencies onI(s) and onII(s) which aredescribed by ordinary differential eqs. (78) and (79), representingforced vibrations of a Timoshenko double model and can beassumed in the following
w1ðx,tÞ ¼X1n ¼ 1
XnðxÞXII
i ¼ I
SniðtÞ, ð89Þ
w2ðx,tÞ ¼X1n ¼ 1
XnðxÞXII
i ¼ I
aniSniðtÞ, ð90Þ
where SniðtÞði¼ I,IIÞ is the unknown time function correspondingto the natural frequency oni. Introduction of Eqs. (89) and (90)into Eqs. (9) and (10), and negligible shear frequencies onI(s) andonII(s) results in
X1n ¼ 1
XnðxÞXII
i ¼ I
ðJ1�H1aniÞ€SniþðP1�Q1aniÞSni
h i¼m1 f 1þ
1
C2s1
€f 1�C2
b1
C2s1C2
r1
f 001
" #,
ð91Þ
X1n ¼ 1
XnðxÞXII
i ¼ I
ðJ2�H2a�1ni Þ
€SniþðP2�Q2a�1ni ÞSni
h iani ¼m2 f 2þ
1
C2s2
€f 2�C2
b2
C2s2C2
r2
f 002
" #:
ð92Þ
By multiplying the relations (69) and (70) by the eigenfunctionXm, then integrating them with respect to x from 0 to l and usingorthogonality condition (36), we have
XII
i ¼ I
ðJ1�H1aniÞ€SniþðP1�Q1aniÞSni
h i¼ 2
m1
l
Z l
0Xn f 1þ
1
C2s1
€f 1�C2
b1
C2s1C2
r1
f 001
" #dx,
ð93Þ
XII
i ¼ I
ðJ2�H2a�1ni Þ
€SniþðP2�Q2a�1ni ÞSni
h iani ¼ 2
m2
l
Z l
0Xn f 2þ
1
C2s2
€f 2�C2
b2
C2s2C2
r2
f 002
" #dx:
ð94Þ
From Eqs. (87), (88), (93) and (94), after some algebra, weobtain
€Sniþo2niSni ¼ ZniðtÞ, i¼ I,II, ð95Þ
where
ZnIðtÞ ¼2m1ðJ2anII�H2Þ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
0f 1þ
1
C2s1
€f 1�C2
b1
C2s1C2
r1
f 1’’
" #sinðknxÞ dx
þ2m2ðH1anII�J1Þ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
0f 2þ
1
C2s2
€f 2�C2
b2
C2s2C2
r2
f 002
" #sinðknxÞ dx,
ð96Þ
ZnIIðtÞ ¼2m1ðH2�J2anIÞ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
0f 1þ
1
C2s1
€f 1�C2
b1
C2s1C2
r1
f 001
" #sinðknxÞ dx
þ2m2ðJ1�H1anIÞ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
0f 2þ
1
C2s2
€f 2�C2
b2
C2s2C2
r2
f 002
" #sinðknxÞ dx:
ð97Þ
From Eq. (95) we have
SniðtÞ ¼1
oni
Z t
0ZniðsÞ sin oniðt�sÞ½ � ds, i¼ I,II: ð98Þ
By combining Eqs. (89), (90) and (98), the forced vibrations ofan elastically connected Timoshenko double-beam system can bedescribed by
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞXII
i ¼ I
1
oni
Z t
0ZniðsÞ sin oniðt�sÞ½ � ds, ð99Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞXII
i ¼ I
ani
oni
Z t
0ZniðsÞ sin oniðt�sÞ½ � ds: ð100Þ
Now these general solutions (99) and (100) are used to find thevibrations of the two coupled Timoshenko beams for certainexciting loadings.
Case 1. Stationary harmonic loads. An arbitrarily distributedharmonic load acts on the first Timoshenko beam. The secondTimoshenko beam is unloaded (Fig. 3). The exciting loading of aTimoshenko double-beam system is
f 1ðx,tÞ ¼ qðxÞ sinðOtÞ, f 2ðx,tÞ ¼ 0 ð101Þ
where q(x) is the arbitrary function of spatial coordinate x and Ois the frequency of the load. Substituting Eq. (101) into Eqs. (96)and (97), we obtain
ZnIðtÞ ¼2m1ðJ2anII�H2Þ sinðOtÞ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
01�
O2
C2s1
!qðxÞ
"
�C2
b1
C2s1C2
r1
q00ðxÞ
#sinðknxÞ dx, n¼ 1,3,5,. . ., ð102Þ
ZnIIðtÞ ¼2m1ðH2�J2anIÞ sinðOtÞ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
01�
O2
C2s1
!qðxÞ�
C2b1
C2s1C2
r1
q00ðxÞ
" #
sinðknxÞ dx, n¼ 1,3,5,. . ., ð103Þ
Introduction of Eqs. (100) and (101) into Eqs. (97) and (98)gives
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An1 sinðOtÞþXII
i ¼ I
BnisinðonitÞ
" #, n¼ 1,3,5,. . .,
ð104Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An2 sinðOtÞþXII
i ¼ I
aniBni sinðonitÞ
" #, n¼ 1,3,5,. . .,
ð105Þ
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–71 67
where
An1 ¼MTðJ2anII�H2Þ
ðo2nI�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
"
þðH2�J2anIÞ
ðo2nII�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
#, ð106Þ
An2 ¼MTanIðJ2anII�H2Þ
ðo2nI�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
" #
þanIIðH2�J2anIÞ
ðo2nII�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
#, ð107Þ
BnI ¼OðJ2anII�H2Þ
onIðo2nI�O
2ÞðanII�anIÞðH1H2�J1J2Þ
MT ,
BnII ¼OðH2�J2anIÞ
onIIðo2nII�O
2ÞðanII�anIÞðH1H2�J1J2Þ
MT , ð108Þ
MT ¼2m1
l
Z l
01�
O2
C2s1
!qðxÞ�
C2b1
C2s1C2
r1
q00ðxÞ
" #sinðknxÞ dx, i¼ 1,2:
ð109Þ
Ignoring the free response, the forced vibrations of theTimoshenko double-beam system can be obtained by
w1ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An1 sinðknxÞ, n¼ 1,3,5,. . ., ð110Þ
w2ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð111Þ
The following fundamental conditions of resonance anddynamic vibration absorption have practical significance:
aÞ Resonance : O¼oni, n¼ 1,3,5,. . .,
bÞ Dynamic vibration absorption : O2
¼o2
nIðJ2anI�H2Þþo2nIIðH2�J2anIÞ
J2ðanI�anIIÞ,
An1 ¼ 0, An2 ¼MTnJ2ðanII�anIÞ
ðH1H2�J1J2Þðo2nI�o2
nIIÞ, n¼ 1,3,5,. . .
Case 2. Uniformly distributed harmonic load. The harmonicuniformly distributed continuous load acts on the firstTimoshenko beam. The second beam is unloaded (Fig. 4). Theexciting loading of a Timoshenko double-beam system is
f 1ðx,tÞ ¼ q sinðOtÞ, f 2ðx,tÞ ¼ 0 ð112Þ
whereq and O are the amplitude and exciting frequency of theload, respectively. Substituting Eq. (110) into Eqs. (96) and (97),we obtain
ZnIðtÞ ¼4qm1ðJ2anII�H2Þ sinðOtÞ
npðanI�anIIÞðH1H2�J1J2Þ1�
O2
C2s1
!, n¼ 1,3,5,. . ., ð113Þ
ZnIIðtÞ ¼4qm1ðH2�J2anIÞsinðOtÞ
npðanI�anIIÞðH1H2�J1J2Þ1�
O2
C2s1
!, n¼ 1,3,5,. . ., ð114Þ
Introduction of Eqs. (113) and (114) into Eqs. (99) and (100)gives
w1ðx,tÞ ¼X1n ¼ 1
sin ðknxÞ An1 sinðOtÞþXII
i ¼ I
Bni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð115Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An2 sinðOtÞþXII
i ¼ I
aniBni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð116Þ
where
An1 ¼MTðJ2anII�H2Þ
ðo2nI�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
"
þðH2�J2anIÞ
ðo2nII�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
#, ð117Þ
An2 ¼MTanIðJ2anII�H2Þ
ðo2nI�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
"
þanIIðH2�J2anIÞ
ðo2nII�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
#, ð118Þ
BnI ¼OðJ2anII�H2Þ
onIðo2nI�O
2ÞðanII�anIÞðH1H2�J1J2Þ
MT ,
BnII ¼OðH2�J2anIÞ
onIIðo2nII�O
2ÞðanII�anIÞðH1H2�J1J2Þ
MT , ð119Þ
MT ¼4qm1ðJ2anII�H2Þ
npðanI�anIIÞðH1H2�J1J2Þ1�
O2
C2s1
!: ð120Þ
Ignoring the free response, the forced vibrations of theTimoshenko double-beam system can be obtained by
w1ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An1 sinðknxÞ, n¼ 1,3,5,. . ., ð121Þ
w2ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð122Þ
The following fundamental conditions of resonance anddynamic vibration absorption have practical significance:
aÞ Resonance : O¼oni, n¼ 1,3,5,. . .,
bÞ Dynamic vibration absorption :
O2¼o2
nIðJ2anI�H2Þþo2nIIðH2�J2anIÞ
J2ðanI�anIIÞ,
An1 ¼ 0, An2 ¼MTJ2ðanII�anIÞ
ðH1H2�J1J2Þðo2nI�o2
nIIÞ, n¼ 1,3,5,. . .
Case 3. Harmonic concentrated force. For the case when theexcitation force is not changed with time, effect of the axial forceon deflection of the single beam is determined on pages 3–6 ofTimoshenko and Gere [19]. The effect of the axial load on double-beam subjected to the concentrated static force on the first beamcan be also determined if we apply the mentioned procedure fromRef. [19]. In our case the first Timoshenko beam is subjected tothe harmonic concentrated force applied at the midspan of thebeam (Fig. 5). The exciting loading of a Timoshenko double-beamsystem is
f 1ðx,tÞ ¼ F sinðOtÞdðx�0:5lÞ, f 2ðx,tÞ ¼ 0, ð123Þ
Fig. 5. Double-beam complex system subjected to harmonic concentrated load.
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–7168
where F and O are the amplitude and frequency of the excitingharmonic force, respectively, and d(x) is the Dirac delta function.Substituting Eq. (123) into Eqs. (96) and (97), we obtain
ZnIðtÞ ¼2Fm1ðJ2anII�H2Þ sinðOtÞ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
0dðx�0:5lÞ 1�
O2
C2s1
!(
�C2
b1
C2s1C2
r1
@2½dðx�0:5lÞ�
@x2
)sinðknxÞ dx, n¼ 1,3,5,. . .,
ZnIIðtÞ ¼2Fm1ðH2�J2anIÞ sinðOtÞ
lðanI�anIIÞðH1H2�J1J2Þ
Z l
0dðx�0:5lÞ 1�
O2
C2s1
!(
�C2
b1
C2s1C2
r1
@2½dðx�0:5lÞ�
@x2
)sinðknxÞ dx, n¼ 1,3,5,. . .,
and after integrating it can be described by
ZnIðtÞ ¼2Fm1ðJ2anII�H2Þ sinðOtÞ
lðanI�anIIÞðH1H2�J1J2Þ1�
O2
C2s1
!þ
C2b1
C2s1C2
r1
n2p2
l2
" #
sinnp2
� �, n¼ 1,3,5,. . ., ð124Þ
ZnIIðtÞ ¼2Fm1ðH2�J2anIÞ sinðOtÞ
lðanI�anIIÞðH1H2�J1J2Þ1�
O2
C2s1
!þ
C2b1
C2s1C2
r1
n2p2
l2
" #
sinnp2
� �, n¼ 1,3,5,. . ., ð125Þ
Introduction of Eqs. (124) and (125) into Eqs. (99) and (100)gives
w1ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An1 sinðOtÞþXII
i ¼ I
Bni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð126Þ
w2ðx,tÞ ¼X1n ¼ 1
sinðknxÞ An2 sin ðOtÞþXII
i ¼ I
aniBni sinðonitÞ
" #,
n¼ 1,3,5,. . ., ð127Þ
where
An1 ¼MTðJ2anII�H2Þ
ðo2nI�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
"
þðH2�J2anIÞ
ðo2nII�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
#, ð128Þ
An2 ¼MTanIðJ2anII�H2Þ
ðo2nI�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
"
þanIIðH2�J2anIÞ
ðo2nII�O
2ÞðanI�anIIÞðH1H2�J1J2Þ
#, ð129Þ
BnI ¼OðJ2anII�H2Þ
onIðo2nI�O
2ÞðanII�anIÞðH1H2�J1J2Þ
MT ,
BnII ¼OðH2�J2anIÞ
onIIðo2nII�O
2ÞðanII�anIÞðH1H2�J1J2Þ
MT , ð130Þ
MT ¼2Fm1
l1�
O2
C2s1
!þ
C2b1
C2s1C2
r1
n2p2
l2
" #sin
np2
� �, i¼ 1,2: ð131Þ
Ignoring the free response, the forced vibrations of theTimoshenko double-beam system can be obtained by
w1ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An1 sinðknxÞ, n¼ 1,3,5,. . ., ð132Þ
w2ðx,tÞ ¼ sinðOtÞX1n ¼ 1
An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð133Þ
The following fundamental conditions of resonance anddynamic vibration absorption have practical significance:
aÞ Resonance : O¼oni, n¼ 1,3,5,. . .,
bÞ Dynamic vibration absorption : O2¼o2
nIðH2�J2anIIÞþo2nIIðJ2anI�H2Þ
J2ðanI�anIIÞ,
An1 ¼ 0, An2 ¼MTnJ2ðanI�anIIÞ
2
o2nII�o2
nI
, n¼ 1,3,5,. . .
6. Numerical experiment and discussion
Most recently, Oniszczuk [9–11] and Zhang et al. [12,13]analysed a structural model of a layered-beam system composedof two parallel Euler beams of uniform properties axially loadedwith a flexible Winkler elastic layer in-between ðH1 ¼H2,Z1 ¼ Z2Þ.For simplicity, it is assumed that both beams are geometricallyand physically identical. Without the loss of generality, weassume
F2 ¼ zF1, 0rzr1, ð134Þ
E¼ 1� 1010 Nm�2, G¼ 0:417� 1010 Nm�2, k¼ 5=6,
K ¼ 2� 105 Nm�2, r¼ 2� 103 kgm�3,
l¼ 10 m, A¼ 5� 10�2 m2, I¼ 4� 10�4 m4: ð135Þ
If the axial compressions vanish, we have for the Rayleighdouble-beam
ðo0nIÞ
2¼
J2N1þ J1N2
2J1J2
�1
2J1J2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ2N1�J1N2Þ
2þ4J1J2H1H2
q, ð136Þ
ðo0nIIÞ
2¼
J2N1þ J1N2
2J1J2
þ1
2J1J2
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðJ2N1�J1N2Þ
2þ4J1J2H1H2
q, ð137Þ
ða0nIÞ�1¼
H1
N1�J1ðo0nIÞ
2¼
N2�J2ðo0nIÞ
2
H2, ð138Þ
ða0nIIÞ�1¼
H1
N1�J1ðo0nIIÞ
2¼
N2�J2ðo0nIIÞ
2
H2, ð139Þ
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–71 69
and for the Timoshenko double-beam
Fig. 6. Relationship between ratio j1 ¼ An1=A0n1 and dimensionles parameter
s¼ F1=P for different axial compression ratio z.
Fig. 7. Relationship between ratio j2 ¼ An2=A0n2 and dimensionless parameter
s¼ F1=P for different axial compression ratio z.
ðo0nIÞ
2¼
J02P0
1þ J01P0
2�H2Q1�H1Q2�
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH2Q1þH1Q2�J0
2P01�J0
1P02Þ
2þ4ðH1H2�J0
1J02ÞðP
01P0
2�Q1Q2Þ
q2ðJ0
1J02�H1H2Þ
, ð140Þ
ðo0nIIÞ
2¼
J02P0
1þ J01P0
2�H2Q1�H1Q2þ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH2Q1þH1Q2�J0
2P01�J0
1P02Þ
2þ4ðH1H2�J0
1J02ÞðP
01P0
2�Q1Q2Þ
q2ðJ0
1J02�H1H2Þ
, ð141Þ
ða0nIÞ�1¼
Q1�H1ðo0nIÞ
2
P01�J0
1o2nI
¼P0
2�J02ðo0
nIÞ2
Q2�H2o2nI
, ð142Þ
ða0nIIÞ�1¼
Q1�H1ðo0nIIÞ
2
P01�J0
1ðo0nIIÞ
2¼
P02�J0
2ðo0nIIÞ
2
Q2�H2ðo0nIIÞ
2, ð143Þ
where
J01 ¼ C2
s1 R1þC2r1k2
nþH1
C2s1
" #, J0
2 ¼ C2s2 R2þC2
r2k2nþ
H2
C2s2
" #,
P01 ¼ C2
s1½C2b1k4
nþH1R1�, P02 ¼ C2
s2½C2b2k4
nþH2R2�:
To determine the effect of compressive axial load on thesteady-state vibration amplitudes of the system, the results undercompressive axial load and those without axial load are com-pared. Introducing the related
j1 ¼An1
A0n1
, j2 ¼An2
A0n2
, ð144Þ
where A0n1 and A0
n2 are the steady-state vibration amplitudes ofthe two beams without axial compression. Using non-dimen-sional ratio
s¼F1
P, Euler and Rayleigh beam-P¼ PER ¼
EIp2
l2,
Timoshenko beam-P¼ PT ¼ðEIp2=l2Þ
1þðEIp2=GAkl2Þ, ð145Þ
where PER is known as the Euler load for Euler and Rayleigh beam,which is the smallest load at which the single beam ceases to bein stable equilibrium and PT is known as the Euler load forTimoshenko beam, as shown in Ref. [19], which is the smallestload at which the single beam ceases to be in stable equilibrium.For the case of uniformly distributed harmonic load, the steady-state vibration amplitudes for the Rayleigh and Timoshenkobeams ðAn1, An2, A0
n1, A0n2Þ can be determined.
With the vibration mode number n¼3 and the excitingfrequency O¼0.6onII, the effects of compressive axial load onthe steady-state vibration amplitudes An1 and An2 of the twoRayleigh and Timoshenko beams represented by the ratios j1 andj2 are compared with results for Euler beams, and shown inFigs. 6 and 7, respectively. Solid lines represent the case withoutthe rotary inertia and shear - Euler beam, Zhang et al. [13], dashed
lines represent the case with the influence of rotary inertia -Rayleigh beam, present study and dot dashed lines represent thecase with the influence of rotary inertia and shear - Timoshenkobeam, present study. As can be seen, the ratios j1 and j2 increasewith the increase of the axial compression, which implies that themagnitudes of the steady-state vibration amplitudes An1 and An2
become larger when the axial compression increases. Dot dashedlines which refer to the Timoshenko beam, show that the ratiosj1 and j2 increase faster than the ratios j1 and j2 for the Eulerbeam with increasing the dimensionless parameter s, Ref. [13].The difference becomes larger with the increasing axial compres-sion ratio z. Influence of rotary inertia is smaller then influence of
the shear with rotary inertia and the difference between Euler andRayleigh beam can not be observed on Figs. 6 and 7. Because of
that, calculated numerical differences in chosen points are givenin Table 1.
Table 1 shows also the effects of compressive axial load ondifference between the steady-state vibration amplitudes ratiosj1 and j2 of the Euler, Rayleigh and Timoshenko beam fordifferent values of parameter z. It can be observed that the effectof compressive axial load on the magnitude of An1 is almostindependent of the axial compression ratio z of the two beamswhereas it is significantly dependent on the magnitude of An2.This difference is unobservable in Figs. 6 and 7. The decrease ofthe axial compression ratio z causes an evident reduction of themagnitude of An2 with higher difference for the Rayleigh andTimoshenko beam. Numerical values of the ratios j1 and j2 inTable 1 which refer to the Rayleigh beam show that the differencewith ratios j1 and j2 for the Euler beam increases with increas-ing the dimensionless parameter s.
Fig. 8. Relationship between ratio j1 ¼ An1=A0n1 and dimensionless parameter
s¼ F1=P for different mode number n.
Fig. 9. Relationship between ratio j2 ¼ An2=A0n2 and dimensionless parameter
s¼ F1=P for different mode number n.
Table 2Effects of compressive axial load on the steady-state vibration amplitudes ratios
j1 and j2.
s
0 0.2 0.4 0.6 0.8 1
j1(n¼3)
Euler 1 1.03514 1.07216 1.11125 1.15261 1.19647
Rayleigh 1 1.03530 1.07250 1.11171 1.15333 1.19742
Timoshenko 1 1.03630 1.07462 1.11514 1.15810 1.20377
j1(n¼5)
Euler 1 1.01257 1.02539 1.03847 1.05181 1.06544
Rayleigh 1 1.01272 1.01272 1.03841 1.05241 1.06620
Timoshenko 1 1.01372 1.01372 1.04205 1.05670 1.07168
j1(n¼3)
Euler 1 1.00640 1.01287 1.01940 1.02601 1.03268
Rayleigh 1 1.00655 1.01316 1.01984 1.02659 1.03342
Timoshenko 1 1.00757 1.01523 1.02298 1.03083 1.03878
j2(n¼3)
Euler 1 1.05277 1.10898 1.16898 1.23316 1.30196
Rayleigh 1 1.05301 1.10948 1.16978 1.23428 1.30345
Timoshenko 1 1.05451 1.11269 1.17493 1.24166 1.31338
j1(n¼5)
Euler 1 1.01890 1.03824 1.05806 1.07837 1.09919
Rayleigh 1 1.01912 1.01912 1.05873 1.07928 1.10036
Timoshenko 1 1.02063 1.02063 1.06351 1.08582 1.10875
j1(n¼3)
Euler 1 1.00961 1.01934 1.02920 1.03918 1.04928
Rayleigh 1 1.00983 1.01978 1.02986 1.04006 1.05041
Timoshenko 1 1.01137 1.02290 1.03460 1.04648 1.05854
Table 1Effects of compressive axial load on the steady-state vibration amplitudes ratios
j1 and j2.
s
0 0.2 0.4 0.6 0.8 1
j1(¼0.1)
Euler 1 1.03511 1.07101 1.11200 1.15440 1.19990
Rayleigh 1 1.03527 1.07268 1.11254 1.15516 1.20091
Timoshenko 1 1.03626 1.08225 1.11593 1.16001 1.20743
j1(¼0.5)
Euler 1 1.03514 1.07216 1.11125 1.15261 1.19647
Rayleigh 1 1.0353 1.07250 1.11177 1.15333 1.19742
Timoshenko 1 1.0363 1.07462 1.11514 1.1581 1.20377
j1(¼0.9)
Euler 1 1.03519 1.07204 1.11070 1.15130 1.19398
Rayleigh 1 1.03535 1.07238 1.11121 1.15200 1.19488
Timoshenko 1 1.03635 1.07450 1.11457 1.15672 1.20112
j2(¼0.1)
Euler 1 1.0386 1.07809 1.12320 1.16989 1.22003
Rayleigh 1 1.03877 1.07993 1.12379 1.17073 1.22114
Timoshenko 1 1.03986 1.08225 1.12752 1.17606 1.22831
j2(¼0.5)
Euler 1 1.05277 1.10898 1.16898 1.23316 1.30196
Rayleigh 1 1.05301 1.10948 1.16978 1.23428 1.30345
Timoshenko 1 1.05451 1.11269 1.17493 1.24166 1.31338
j2(¼0.9)
Euler 1 1.06727 1.13980 1.21810 1.30276 1.39442
Rayleigh 1 1.06757 1.14045 1.21913 1.30422 1.39636
Timoshenko 1 1.06950 1.14462 1.22593 1.31407 1.40976
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–7170
On the other hand, with the axial compression ratio z¼0.5 andthe exciting frequency O¼ 0:6onII , the effects of compressiveaxial load on the steady-state vibration amplitudes An1 and An2
are shown in Figs. 8 and 9, respectively. It can be seen that, withthe same axial compression, the ratios j1 and j2 diminish withthe increasing vibration mode number n, which implies that themagnitudes of the steady-state vibration amplitudes An1 and An2
get smaller when the vibration mode number n becomes largerwhere the differences for Timoshenko beams are higher. Thedifference between Euler and Rayleigh beam also can not beobserved on Figs. 8 and 9 and because of that, calculatednumerical differences in chosen points are given in Table 2.
Table 2 shows the effects of compressive axial load ondifference between the steady-state vibration amplitudes ratiosj1 and j2 of the Euler, Rayleigh and Timoshenko beam fordifferent mode number n. It can be observed that the differences
between ratios j1 and j2 of the Rayleigh and Euler beam increasewith increasing the dimensionless parameter s what is alsounobservable in Figs. 8 and 9. Differences between ratios j1
and j2 of the Timoshenko beam and ratios j1 and j2 of the Eulerbeam are greater and they increase with increasing the dimen-sionless parameter s but not significant with increasing the modenumber.
7. Conclusions
Based on the Timoshenko beam theory, the properties offorced transverse vibration and buckling of an elastically con-nected simply supported double-beam with the influence ofrotary inertia - Rayleigh and with the influence of rotary inertiaand shear - Timoshenko beam, under compressive axial loading
V. Stojanovic, P. Kozic / International Journal of Mechanical Sciences 60 (2012) 59–71 71
are studied. The dynamic response of the system causedby arbitrarily distributed continuous loads is obtained. Theeffects of compressive axial load on the forced vibrations of thedouble-beam system are discussed for three cases of particularexcitation loadings. The magnitudes of the steady-state vibrationamplitudes of the two beams are dependent on the influence ofrotary inertia and shear and axial compression. It is concludedthat they become larger when the axial compression increasesand more larger when the effect of rotary inertia and shear aretaken into consideration. The effect of rotary inertia and shearincrease the ratios j1 and j2 of magnitudes of the steady-statevibration amplitudes of the two beams with the increase of theaxial compression ratio z. The effect of compressive axial load onthe magnitudes of the steady-state vibration amplitudes An1 andAn2 is related to the vibration mode number n. Irrespective of thesame axial compression, the magnitudes of the steady-statevibration amplitudes An1 and An2 diminish with the and j2, andhigher values of the steady-state vibration amplitudes An1 and An2
than the Euler beam theory. Analytical forms found can be used inthe optimal design of a new type of a dynamic vibration absorber.Thus the Rayleigh and Timoshenko beam-type dynamic absorbercan be applied to suppress the excessive vibrations of correspond-ing beam systems. The beam-type dynamic damper is an acceptedconcept for a continuous dynamic vibration absorber (CDVA).
Acknowledgments
The research is supported bythe Ministry of Science andEnvironment Protection of the Republic of Serbia, grant No. ON174011.
References
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