1 Introduction

Axially moving beams are common dynamical models of many engineering applications, such as mechanical arms,automotive belts,and aerial cable tramways. In many cases,the occurrence of the nonlinear vibration in these continua is unwanted. Therefore,the transverse vibration of axially moving beams has been extensively investigated ^{[1, 2, 3]} .

Many studies reported over the recent years emphasized the vibration behaviors of the mov- ing systems with a single excitation. Pellicano and Vestroni ^[4] focused on the dynamic responses of a moving beam subjected to a transverse load in the super-critical speed range. Kiani et al. ^[5] extended the work on numerical parametric research on design parameters of multi-span viscoelastic shear deformable beams subjected to a moving mass. Ding and Chen ^[6] investigated the steady-state periodical response for an axially moving viscoelastic beam with a spatially uniform and temporally harmonic excitation via the approximate analysis. The dynamic re- sponses of an axially moving viscoelastic beam subjected to a randomly disordered periodic excitation were investigated by Liu et al. ^[7] . Little research has been found on vibrations of the moving beam with two-frequency excitations. Considering the nonlinear terms,Chakraborty and Mallik ^[8] pioneered the investigations on the effects of parametric excitation on a traveling beam with and without an external harmonic excitation.

The above papers showed the abundant application of the Euler-Bernoulli beam theory in the study of the nonlinear vibration of axially moving continua. Nevertheless,the Timoshenko beam theory should be taken into account in the case that shear deformation cannot be ne- glected. Based on the Timoshenko model,Tang et al. ^[9] studied the stability of the response amplitudes in the nonlinear vibration of axially moving beams under weak and strong external excitations. Ghayesh and Amabili ^[10] presented the analysis on the three-dimensional nonlinear planar dynamics of an axially moving Timoshenko beam by two numerical techniques. An and Su ^[11] used the generalized integral transform technique to find a semi-analytical numerical solution for dynamic response of an axially moving Timoshenko beam. Yan et al. ^[12] explored the steady-state periodic response and the chaos and bifurcation of an axially accelerating viscoelastic Timoshenko beam with a parametric excitation. However,there is no literature published on the nonlinear dynamic behaviors of an axially moving viscoelastic Timoshenko beam subjected to multi-frequency excitations,to the best knowledge of the authors.

To the nonlinear dynamics of axially moving systems,it is impossible to solve the governing equations exactly,thereby allowing the application of many numerical methods. Much research has been done on the transverse nonlinear vibration by the finite difference method ^[13] and the differential quadrature scheme ^[14] . The Galerkin truncation method is a common tool to obtain effective results in solving initial values and boundary values problems of engineering application. Parker and Lin ^[15] examined the parametric instability of axially moving media via the Galerkin discretization. Ding et al. ^[16] acquired the approximate solutions of axially moving viscoelastic beams based on 4-term Galerkin truncation.

Since Tang et al. ^[17] first founded the assumption that the axial tension of the moving continua is not accurate,the longitudinally varying axial tension due to the axial acceleration has been considered in the beam model. On the other hand,the support rigidity of axially moving beams may not be boundless in engineering circumstances. Therefore,the present investigation will incorporate the finite support rigidity into the vibration analysis of the axially accelerating viscoelastic beam.

Based on the prior studies,the present work involves the nonlinear dynamic behaviors of an axially moving viscoelastic Timoshenko beam under parametric excitation and external excitation synchronously. In view of the effect of the axially tension variation and the finite axial support rigidity on the nonlinear vibration in axially moving Timoshenko beams,the dynamical model is established by Newton’s second law. The Galerkin method is introduced to solve the governing equations associated with the fourth-order Runge-Kutta algorithm. On account of the numerical calculation,the paper studies the transverse vibration of the moving Timoshenko beam via bifurcation diagrams. Moreover,the effects of different relationships between parametric excited frequency and external harmonic excited frequency are explored. In addition,the nonlinear dynamic behaviors are identified by means of some standard indicators as the two excited frequencies are incommensurable.

2 Equation of motion

Based on the Timoshenko theory,the dynamical model of a uniform axially accelerating beam is established. As shown in Fig. 1,this axially moving viscoelastic beam travels between two simple supports with an axial time-dependent speed γ(t). The beam is under an axial varying tension P as well as a harmonic cosine force bcos(ω_ft) along the whole length L,in which b represents the excitation amplitude,and ω_f stands for the external excited frequency. E is Young’s modulus,G is the shearing modulus,ρ is the mass density,A is the cross-sectional area, and I is the area moment of inertia of the cross-section about the neutral axis of the beam. Since the effects of rotary inertia and shear deformation are considered,the vibration of the beam is described by two variables,namely,the transverse displacement u(x,t) and the slope ϕ(x,t) due to bending deformation of the beam,which are both dependent on the axial coordinate x and time t.

According to the Timoshenko beam theory,κ is denoted as the shape factor of the model. The viscoelastic material of the beam obeys the Kelvin model with the following constitutive relations ^[18] :

where σ is the axial stress,ε is the axial strain,α is the viscosity coefficient,τ is the shear stress,and υ is the shear strain. The comma preceding x or t denotes partial differentiation with respect to x or t.

The strain-displacement relationship can be described as follows:

Therefore,the axial stress can be written as The shear deformation is expressed as

Due to the extended Hamilton principle and Newton’s second law,the governing equations of the axially accelerating viscoelastic Timoshenko beam are derived as the following form ^{[14, 17]} :

where and the dot denotes differentiation with respect to t. The bending moment M(x,t) is given by

In the present paper,the tension variation along the longitudinal direction due to the ax- ial acceleration of the beam is considered. Therefore,the axial tension can be expressed as follows ^[17] :

where P₀ is the initial axial tension,and η is the axial support rigidity parameter,varying from 0 (infinite rigidity) to 1 (no rigidity). Substitution of (6) and (7) into (5) leads to

The boundary conditions that correspond to the simple supports at both ends are

The axial speed includes a constant mean value with a small simple harmonic fluctuation, i.e.,

where γ₀ ,γ₁ ,and ω_p account for the mean axial speed,the amplitude of the axial speed variation,and the frequency of the axial speed variation,respectively.

Employ the dimensionless variables and parameters as follows:

where the dimensionless parameter k₁ denotes the effect of the shear deformation,k₂ signifies the effect of the rotary inertia,k²_f stands for the bending stiffness of the beam,and k_N represents the nonlinear coefficient.

Substituting (11) into (8) yields the dimensionless governing equations of the Timoshenko beam as follows:

The simply supported boundary conditions at both ends are cast into the dimensionless forms as follows:

It should be noted that Chen and Ding ^[19] have examined the effect of the nonlinear terms on the dynamic responses of the moving beam. The authors found that these nonlinear terms can be neglected. Therefore,in the following calculations,these nonlinear terms are also ne-glected. 3 Galerkin truncation

For the nonlinear axially moving beams (see (5)),it is impossible to acquire the exact solution. The Galerkin truncation method ^[20] is a simple and efficient numerical technique to solve the integro-partial-differential equations. In the present study,the Galerkin method is applied to investigating the nonlinear vibration of the axially moving system. Considering the boundary conditions (13),the approximate solutions of (12) can be expanded as follows ^{[21, 22]} :

where q_n(t) and p_n(t) (n = 1,2,···,N) are sets of generalized coordinates for the transverse displacement and the slope,respectively.

Substituting (14) into (12) yields

It should be remarked that the weight function is chosen as a trial function for the Galerkin truncation method. Multiplying (15) by the weight functions and then integrating it under the interval from 0 to 1,the application of the Galerkin procedure leads to the second-order ordinary differential equations of the governing equations of the axially accelerating beam as follows:

where the value of M (M = m+n) is odd.

With a given N,the solutions of (16) can be obtained by the fourth-order Runge-Kutta method ^{[23, 24]} . Meanwhile,the initial conditions stay the same in the numerical calculation given as follows:

In the results presented in this paper,the nonlinear dynamics of the axially moving Timo- shenko beam is numerically investigated based on 4-term Galerkin truncation. For N=4,the transverse displacement and the slope of the center of the Timoshenko beam are expressed as

4 Nonlinear dynamic behaviors

In this section,the nonlinear dynamic behaviors of the axially accelerating Timoshenko beam are studied. The physical and geometrical properties of the Timoshenko beam and the corresponding dimensionless parameter values are listed in Table 1,which are selected as the fixed parameters in the following numerical examples.

In this case,the first natural frequency of the linear elastic system is numerically obtained as ω₁ = 6.8314. In the following numerical examples,the forcing frequency is set equal to the first natural frequency: ω_f = ω₁ = 6.8314. First,the parametric excited frequency is set as 3.4157.Let 2ω_p = ω_f = ω₁ in Section 4.1. In addition,the bifurcations of the Timoshenko beam are compared by changing the forcing amplitude b. In Section 4.2,the frequency of the axial speed variation is 3.0. Therefore,the characteristics of the transverse vibration of the axially moving beam are identified by five standard indicators,such as the time history,the amplitude spectrum,the phase portrait,the Poincar´ e map,and the sensitivity to initial conditions. In the following numerical calculations,the nonlinear dynamic response is obtained in the interval of [0,5000×2π/ω_p] dimensionless time units. By abandoning the first 96% of the time histories,the transient response of the motion has been moved away. As a result,only the last 200 steps are plotted in the following bifurcation diagrams. Furthermore,the transverse displacement and the slope with the corresponding velocity of the system at the maximum amplitude in the last 4% time histories at each step are assumed as the initial conditions of the next step.

4.1 2ω_p = ω_f = ω₁

Based on the numerical solutions obtained by the 4-term Galerkin method,Fig. 2 provides the investigation of the bifurcation behavior in the nonlinear vibration of the axially accelerating viscoelastic Timoshenko beam when the axial mean speed varies in the interval ^{[2, 5]}. The excitation amplitude is set as b= 0.2 while the amplitude of the axial speed variation is set as γ₁= 0.8. From Fig. 2,the importance of the mean speed in the motion can be obviously observed. In detail,Fig. 2 shows that the displacement and the slope of the center of the axially moving beam,including the corresponding velocity,increase simultaneously with the increasing axial mean speed. Furthermore,Fig. 2 illustrates that the periodic motion of the accelerating beam occurs at first. After a series of period-doubling bifurcation,the chaotic or quasiperiodic vibration appears when the axial mean speed reaches a certain large value. With the further increasing axial mean speed,the complicated vibration suddenly disappears while the periodic vibration appears again. In brief,the periodic motion and the chaotic or quasiperiodic motion of the axially moving Timoshenko beam alternately appear with the varying axial mean speed. Since the bifurcation diagrams of the displacement and the slope predict the same motion form as their corresponding velocity,only the bifurcation of the displacement and the slope are presented in the following numerical examples.

To study the effect of the excitation amplitude on the nonlinear vibration,Figs.3-5 provide the bifurcations of the transverse vibration of the accelerating Timoshenko beam. In Figs.3,4,5,respectively,the excitation amplitude is specified as b=0.2×3,b=0.2×5,and b=0.2×10. Compared with Fig. 2,the numerical results also indicate that the displacement and the slope increase with the growing value of the axial mean speed. Moreover,with the increasing mean speed,after a series of period-doubling bifurcation,the chaotic or quasiperiodic motion appears as well. However,the periodic motion area becomes larger under a larger forcing amplitude from the observation of Figs.2-5. The displacement and the slope of the moving beam also increase when the excitation amplitude becomes larger.

4.2 ω_p= 3,ω_f = ω₁

To study the nonlinear dynamics of the moving beam under an incommensurable relation- ship between the parametric frequency and the forcing excited frequency,Fig. 6 presents the dynamical characteristics of the transverse vibration of the Timoshenko beam with γ₀ = 3.9. In Fig. 6,five standard indicators,including the time history,the amplitude spectrum,the phase portrait,the Poincar´ e map,and the sensitivity to initial conditions,are used to identify the nonlinear dynamic behaviors of the complicated motion of the beam. From the observation of the time histories of the displacement and the slope in Figs.6(a) and 6(e),and the homologous amplitude spectra shown in Figs.6(b) and 6(f),it is found that the nonlinear vibration of the accelerating Timoshenko beam does not show chaotic characteristics. Moreover,Figs.6(c) and 6(g) give the phase-plane portraits of the beam. Meanwhile,Figs.6(d) and 6(h),respectively, show the Poincar´ e maps of the displacement and the slope,which consist of infinite points,and all the points are located on a smooth closed curve. As shown in Figs.6(c), 6(d),Figs.6(g) ,6(h), it is observed obviously that the transverse vibration of the moving beam shows aperiodic char- acteristics. Besides,the numerical results in Figs.6(i) and 6(j) reveal that the nonlinear dynamic response is not sensitive to the initial conditions,where the lines show the results of the initial amplitude D = 0.001 while the dots exhibit the results of D = 0.1. Therefore,Figs.6(i) and 6(g) also illustrate that there are non-chaotic characteristics in the motion of the axially moving Timoshenko beam. In a word,the observation of all the numerical results in Fig. 6 demonstrates that the transverse nonlinear vibration of the axially accelerating Timoshenko beam presents quasiperiodic characteristics when the relationship between the parametric frequency and the external excited frequency is incommensurable.

5 Conclusions

The nonlinear dynamic behaviors of an axially moving viscoelastic Timoshenko beam under parametric excitation and external excitation are studied for the first time. The 4-term Galerkin method combined with the Runge-Kutta time discretization is used to solve the nonlinear governing equation. The transverse nonlinear vibration is investigated when the frequencies of the two excitation are multiple and incommensurable relationships,respectively. Based on the numerical solutions,the bifurcation diagrams are exhibited to study the influ-ence of the axial mean speed. Furthermore,the effect of the external excitation amplitude is examined. Besides,five standard indicators,such as the amplitude spectrum,the phase por-trait,and the Poincar´ e map,are displayed to identify the nonlinear dynamic characteristics of the moving viscoelastic system. The following major conclusions are drawn from the present study:

(i) The 4-term Galerkin truncation reveals that the periodic motion and the chaotic or quasiperiodic motion in the nonlinear vibration exchange alternately with the increase of the axial mean speed.

(ii) The forcing excitation amplitude is a significant parameter for influencing on the non- linear dynamic response of the axially moving Timoshenko beam.

(iii) The nonlinear vibration of the accelerating Timoshenko beam shows quasiperiodic char- acteristics under an incommensurable relationship between the external excited frequency and the parametric excited frequency.