1 Introduction

Transverse vibration problems of axially moving beams received a lot attention due to its association with various applications,such as band saw blades,textile machines,power transmission belts,high-speed magnetic tapes,and fluid conveying pipes.

Many analytical methods and numerical approaches have been used to study the vibration behavior and dynamic characteristics of axially moving beams. Pakdemirli et al.^[1] investigated the non-linear transverse vibrations of an axially moving Euler-Bernoulli beam,where the multiple-scale method was employed to the motion equation for finding approximate solutions. Yang and Chen^[2] analyzed the stability of an axially moving viscoelastic beam with simply supported ends and with pulsating speed,where the Galerkin method and the averaging approach were used to investigate the instability interval of the subharmonic resonance and the combination of resonances for the axially moving beam. Lee and Oh^[3] proposed a spectral element model for the transverse response of an axially moving viscoelastic beam,and conducted numerical studies to analyze the viscoelasticity effect on the dynamic characteristics and stability of the moving beams. Liu and Deng^[4] introduced a subspace-based identification algorithm to obtain the pseudo-modal parameters of an axially moving cantilever beam, and checked the capabilities of the algorithm by comparing the results with the ones observed from the experimental study. By means of Newton’s second law,Chen and Yang^[5] defined the governing equation of nonlinear vibration of axially moving beams,obtained the integropartial- differential formulation through replacing the axial tension by its average over the full beam length,and evaluated the natural frequencies using the multiple-scale method. Jakšić^[6] presented a numerical algorithm for the solution of natural frequencies of axially moving beams, the validity of which was proved by comparing the calculated results with the analytical solutions of its limiting cases. Ponomareva and van Horssen^[7] studied the transverse vibration of an axially moving beam by adopting a combined model that is a string model at lower frequencies and a stretched beam model at higher frequencies. Based on the Hamilton principle and the finite element method,Chang et al.^[8] derived the governing equations of an axially moving Rayleigh beam,the natural frequencies and the transverse response of which were obtained from the eigenvalue problem and by using the time integration method,respectively. Huang et al.^[9] examined the nonlinear vibration behavior of an axially moving beam with periodic excitation. The incremental harmonic balance method and the multivariable Floquet theory were employed to evaluate the dynamic behavior and the stability of the system,respectively. Wang^[10] investigated the weakly forced vibration of an axially accelerating beam defined by the standard linear solid model,and used the multiple-scale method to analyze the steady-state response. Marynowski^[11] investigated the dynamics of the axially moving sandwich beams with elastic faces and a viscoelastic core,where the two-parameter Kelvin-Voigt rheological model was employed to define the material characteristics of the core.

In order to take into account the shear deformation and rotational inertia effects for nonslender beams,there are also comprehensive studies based on the Timoshenko theory. Lee et al.^[12] presented a spectral element model for the axially moving Timoshenko beam,and analyzed the effects of the axial speed and the axial tension on the vibration behaviour and stability of the moving beam. To investigate the vibration response of the railway track due to a moving train,Cojocaru et al.^[13] modeled the train as a Timoshenko beam with distributed stiffness moving at a constant velocity,where the derived eighth-order system of ODEs was solved by the mathematical software Maple. Using the Galerkin method,Yang et al.^[14] discretized the governing equation for transverse vibration of a simply-supported axially accelerating Timoshenko beam,and analyzed the instability conditions of the system by the averaging method. Tang et al.^[15] investigated vibration characteristics and critical speed of axially moving Timoshenko beams,the results of which were compared to the ones for Rayleigh beams,shear beams and Euler-Bernoulli beams. Tang et al.^[16] extended their previous work by performing the parametric resonance analysis of axially moving Timoshenko beams with variable velocity, where the multiple-scale method was applied to study the steady-state behavior. Subsequently, Tang et al.^[17] studied the nonlinear vibration of simply-supported axially moving Timoshenko beams under external excitations,and adopted the multiple-scale method to consider superharmonic resonance and subharmonic resonance. Moreover,the results of nonlinear transverse vibration of an axially moving Timoshenko beam with two free edges were reported by Li et al.^[18]. To avoid the approximate boundary conditions in previous studies,Tang et al.^[19] derived the governing equations for axially accelerating viscoelastic Timoshenko beams with the accurate boundary conditions by means of the generalized Hamilton principle. Ghayesh and Balar^[20] obtained two dynamic models of axially moving Timoshenko beams including one regarding only the transverse displacement and the other regarding both longitudinal and transverse displacements,and employed the multiple-scale method to obtain the natural frequencies and steady-state behaviour of the system. By considering the longitudinal,transverse, and rotational displacements,Ghayesh and Amabili^[21] investigated the three-dimensional nonlinear dynamic phenomenon of axially moving Timoshenko beams,and solved the reduced set of nonlinear ODEs using the direct time integration method and pseudo-arclength continuation technique.

Recently,a semi-analytical numerical approach,known as the generalized integral transform technique (GITT),has been well developed in the understanding of heat transfer and fluid flow phenomenons^[22,23,24]. The feature of automatic global error control in this method makes it quite suitable for benchmarking. The GITT has been applied in analyzing the bending problem of orthotropic plates^[25],solving the dynamic response of axially moving strings^[26],axially moving beams^[27],fluid-conveying pipes^[28],and predicting the vortex-induced vibration of long flexible cylinders^[29],which demonstrated high accuracy,excellent convergence behavior,and good longtime numerical stability of the approach.

To our best knowledge,the dynamic response of axially moving Timoshenko beams has not been studied using the GITT. Due to the lack of study in this area,the GITT approach is used to investigate the dynamic behavior of axially moving Timoshenko beams subjected to the following two types of boundary conditions,respectively: (i) clamped-clamped and (ii) simplysupported. The paper is organized as follows. Section 2 defines the problem. In Section 3,the semi-analytical numerical solution is given by performing integral transform. The numerical results of proposed method,such as the transverse deflections and the angles of rotation are presented in Section 4. A parameter study is then performed to investigate the effects of the axial speed,the axial force and the amplitude of external distributed force on the dynamic behavior of axially moving Timoshenko beams. Finally,the paper ends in Section 5 with conclusions and perspectives. 2 Mathematical formulation

Consider an axially moving Timoshenko beam with constant velocity c in the x direction, as illustrated in Fig. 1,with length L,density ρ,cross-sectional area A,moment of inertia I,shear coefficient κ,modulus of elasticity E,axial tension N_x,and shear modulus G. The transverse displacement and the angle of rotation due to bending are indicated by w(x,t) and θ(x,t),respectively. The governing equations of transverse motion of an axially moving uniform Timoshenko beam can be derived as follows^[12]:

where q(x,t) represents an exciting load as a function of spatial coordinate x and time t,and is taken to be q0 cos(Ωt) (not depend on the spatial variable x for simplicity) in this study. Ω is the frequency of the distributed harmonic force.

Two types of boundary conditions are considered: (i) clamped-clamped,

and (ii) simply-supported,

By substituting dimensionless variables and parameters,

into (1)-(4),the dimensionless governing equation can be obtained as follows (dropping the superposed asterisks for simplicity): with the dimensionless clamped-clamped boundary conditions and the dimensionless simply-supported boundary conditions respectively. The initial conditions are defined in the dimensionless form as follows: 3 Integral transform solution

On the basis of the principle of the GITT,the additional eigenvalue problem should be selected for the dimensionless governing equations (6) and (7) with the homogeneous boundary conditions (8) or (9). For the clamped-clamped boundary conditions,the spatial coordinate ‘x’ is eliminated through integral transformation. The related eigenvalue problems are adopted for the transverse deflection as follows:

with the following boundary conditions and for the angle of rotation as follows: with the following boundary conditions respectively,where X_i,μ_i,and Y_i,φ_i are the eigenfunctions and eigenvalues of problems (11) and (12),satisfying the following orthogonality properties:

Problems (13) and (14) are readily solved analytically to yield

and the eigenvalues become

The norms,or normalization integrals,are written as

For the simply-supported boundary conditions,the related eigenvalue problem for the transverse deflection is the same as the one above mentioned,while the eigenvalue problem for the angle of rotation is defined as follows:

with the following boundary conditions where Z_i and ψ_i are the eigenfunctions and eigenvalues of problems (21),satisfying the following orthogonality property:

Problem (22) is readily solved analytically to yield

and the eigenvalues become

The norms,or normalization integrals,are written as

The eigenvalue problems (11),(12),and (21) allow to establish the following integral transform pair for the transverse deflection and the angle of rotation:

where _i(ξ) and _i(ξ) are the normalized eigenfunctions Now,to carry out the integral transform,the dimensionless equations (6) and (7) are multiplied by the operator (ξ)dξ and (ξ)dξ,respectively,where the inverse formulas (26b) and (26d) are applied. After some derivation,the following set of ODEs can be obtained: where the coefficients are given from the following integrals: and λ_j equals to φ_j and ψ_j for the clamped-clamped and simply-supported boundary conditions, respectively.

Similarly,initial conditions (10) are also integral transformed through the operator,generating

For computational purposes,the expansions for the transverse deflection and the angle of rotation are truncated to finite orders NW and NA,respectively. (27) and (28) in the truncated series are subsequently calculated by the NDSolve routine of Mathematica^[30]. Once the transformed potential,_i and _j ,the numerically evaluated,the inversion formulas (26b) and (26d) are then used to yield the explicit analytical expressions for the dimensionless transverse deflection w(ξ,τ) and the angle of rotation θ(ξ,τ). 4 Results and discussion

We now present the convergence behavior of numerical results for the dimensionless transverse deflection w(ξ,τ) and the angle of rotation θ(ξ,τ) of an axially moving Timoshenko beam calculated by the GITT approach. For the case examined,the following dimensionless parameters are taken in (6) and (7):

The solution of the systems (27) and (28) is obtained with N_W ≤ 160 and N_A ≤ 60 to analyze the convergence behavior.

The dimensionless transverse displacement w(ξ,τ) and the angle of rotation θ(ξ,τ) at different positions,ξ = 0.1,0.3,0.5,0.7,and 0.9,of an axially moving Timoshenko beam with clamped-clamped boundary conditions are presented in Tables 1 and 2,respectively. The convergence behavior of the GITT solution is studied for increasing truncation orders N_W = 40, 80,120,160 and N_A = 40,60 at τ = 5,20,50,respectively. For the transverse deflection of clamped-clamped axially moving Timoshenko beams,convergence is basically achieved at some points with truncation orders (N_W ≤ 40) and (N_A ≤ 40). For the convergence achieved at most points,more terms (e.g.,N_W = 80 and N_A = 40) are required. For the angle of rotation of the moving beam with clamped-clamped boundary conditions,the convergence is also achieved essentially at most points with truncation orders N_W = 80 and N_A = 40. The results at τ = 50 show that the good convergence behavior of the GITT solution does not change with time,considered as a long-time stability proof. The convergence behaviors of the transverse deflection and the angle of rotation of axially moving beams with simply-supported boundary conditions are given in Tables 3 and 4. Figures 2 and 3 present the GITT results with different truncation orders NW for time history of the transverse deflection at τ ∈ [15, 20] and τ ∈ [35, 40] of axially moving Timoshenko beams with clamped-clamped and simply-supported boundary conditions, respectively. The good convergence behavior of our method is clearly exhibited.

The dynamic response is obtained at different moving speed. Figures 4 and 5 show the effect of the axial speed (v = 0.1,0.5,0.7,and 1.0) on the transverse deflection at the central point of the axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions,respectively,while α = 1.0,β = 1.0,γ = 1.0,and η = 1.0. It can be seen that the amplitudes of vibration do not vary monotonously with the axial speed. When v = 0.7,the maximum absolute value of the transverse deflection is 0.265 60,much higher than the results obtained with v = 0.1,0.5,and 1.0. For the vibration with v = 0.7,the amplitude builds up and then dies down in a regular pattern,which is clearly the occurrence of the beating phenomenon. Figures 6 and 7 show the effect of the axial force (α = 1.0,1.5,1.8,and 2.0) on the transverse deflection at the central point of the axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions,respectively,while β = 1.0,γ = 1.0,η = 1.0,and v = 0.8. It can be seen that for both cases,the maximum absolute amplitude value of the system decreases with the axial force. Finally,the effect of the amplitude of external distributed force (η = 1.0,1.5,1.8,and 2.0) on the dimensionless transverse deflection at the central point of an axially moving Timoshenko beam is examined,where α = 1.0,β = 1.0,γ = 1.0,and v = 0.8 are employed,as shown in Figs. 8 and 9. It can be clearly seen that the maximum absolute amplitude value of the system increases proportionally with increasing amplitude of external distributed force for both clamp-clamped and simply-supported cases.

5 Conclusions

The GITT has shown in this study to be a practical method for the analysis of vibration behavior of axially moving Timoshenko beams,providing a semi-analytical numerical solution for the transverse deflection and the angle of rotation. The solutions for both clamped-clamped and simply-supported cases converge essentially at truncation orders (N_W ≤ 80 and N_A ≤ 40). Good long-time stability is also shown. The parametric study indicates that,the maximum absolute amplitude values of the system do not vary monotonously with the axial speed,and the beating phenomenon can occur at a certain value of the axial speed (e.g.,v = 0.7). Besides,the maximum absolute amplitude value of the system decreases with the axial force and increases proportionally with the amplitude of external distributed force. The proposed approach can be employed to predict the dynamic response of axially moving viscoelastic Timoshenko beams for future investigation.