Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load

International Journal of Mechanical Sciences 60 (2012) 59–71

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International Journal of Mechanical Sciences

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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: [email protected]

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

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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]

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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


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


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.


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


sinðknxÞXII

i ¼ I

1

oni

Z t

0ZniðsÞ sin oniðt�sÞ½ � ds, ð44Þ


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


sinðknxÞ An1 sinðOtÞþXII

i ¼ I

Bni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð49Þ



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


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.


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


ZnIIðtÞ ¼anI


where

MRn ¼M14lq

np, n¼ 1,3,5,. . .




i ¼ I

Bni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð56Þ


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


An1 sinðknxÞ, n¼ 1,3,5,. . ., ð58Þ


An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð59Þ




nII�anIo2nI

anII�anI,


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


ZnIIðtÞ ¼anI


where

MRn ¼ 2FM1 sinnp2

� �, n¼ 1,3,5,. . .




i ¼ I

Bni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð63Þ



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


An1 sinðknxÞ, n¼ 1,3,5,. . ., ð65Þ


An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð66Þ




nII�anIo2nI

anII�anI,


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


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


, 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


: 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
�J1J2
ðP1P2�Q1Q2Þ

, ð84Þ

o2nII ¼

J2P1þ J1P2�H2Q1�H1Q2þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðH2Q1þH1Q2�J2P1�J1P2Þ

2þ4ðH1H2�J1J2ÞðP1P2�Q1Q2Þ

q2ðJ1J2�H1H2Þ

, ð85Þ


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


XnðxÞXII

i ¼ I

SniðtÞ, ð89Þ


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


Z l

0f 2þ

1

C2s2

€f 2�C2

b2

C2s2C2

r2

f 002

" #sinðknxÞ dx,

ð96Þ

ZnIIðtÞ ¼2m1ðH2�J2anIÞ


Z l

0f 1þ

1

C2s1

€f 1�C2

b1

C2s1C2

r1

f 001

" #sinðknxÞ dx

þ2m2ðJ1�H1anIÞ


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


sinðknxÞXII

i ¼ I

1

oni

Z t

0ZniðsÞ sin oniðt�sÞ½ � ds, ð99Þ


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


Z l

01�

O2

C2s1

!qðxÞ

"

�C2

b1

C2s1C2

r1

q00ðxÞ

#sinðknxÞ dx, n¼ 1,3,5,. . ., ð102Þ

ZnIIðtÞ ¼2m1ðH2�J2anIÞ sinðOtÞ


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



i ¼ I

BnisinðonitÞ

" #, n¼ 1,3,5,. . .,

ð104Þ



i ¼ I

aniBni sinðonitÞ

" #, n¼ 1,3,5,. . .,

ð105Þ


where

An1 ¼MTðJ2anII�H2Þ

ðo2nI�O

2ÞðanI�anIIÞðH1H2�J1J2Þ

"

þðH2�J2anIÞ

ðo2nII�O


#, ð106Þ

An2 ¼MTanIðJ2anII�H2Þ

ðo2nI�O


" #

þanIIðH2�J2anIÞ

ðo2nII�O


#, ð107Þ

BnI ¼OðJ2anII�H2Þ

onIðo2nI�O

2ÞðanII�anIÞðH1H2�J1J2Þ

MT ,

BnII ¼OðH2�J2anIÞ

onIIðo2nII�O


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


An1 sinðknxÞ, n¼ 1,3,5,. . ., ð110Þ


An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð111Þ



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


O2

C2s1

!, n¼ 1,3,5,. . ., ð114Þ



sin ðknxÞ An1 sinðOtÞþXII

i ¼ I

Bni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð115Þ



i ¼ I

aniBni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð116Þ

where


ðo2nI�O


"

þðH2�J2anIÞ

ðo2nII�O


#, ð117Þ


ðo2nI�O


"


ðo2nII�O


#, ð118Þ


onIðo2nI�O


MT ,


onIIðo2nII�O


MT , ð119Þ

MT ¼4qm1ðJ2anII�H2Þ


O2

C2s1

!: ð120Þ



An1 sinðknxÞ, n¼ 1,3,5,. . ., ð121Þ


An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð122Þ



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.


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


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


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




i ¼ I

Bni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð126Þ


sinðknxÞ An2 sin ðOtÞþXII

i ¼ I

aniBni sinðonitÞ

" #,

n¼ 1,3,5,. . ., ð127Þ

where


ðo2nI�O


"

þðH2�J2anIÞ

ðo2nII�O


#, ð128Þ


ðo2nI�O


"


ðo2nII�O


#, ð129Þ


onIðo2nI�O


MT ,


onIIðo2nII�O


MT , ð130Þ

MT ¼2Fm1

l1�

O2

C2s1

!þ

C2b1

C2s1C2

r1

n2p2

l2

" #sin

np2

� �, i¼ 1,2: ð131Þ



An1 sinðknxÞ, n¼ 1,3,5,. . ., ð132Þ


An2 sinðknxÞ, n¼ 1,3,5,. . .,: ð133Þ



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


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.


s¼ F1=P for different mode number n.


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


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


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.

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Forced transverse vibration of Rayleigh and Timoshenko double-beam system with effect of compressive axial load