时变张力作用下轴向运动梁的分岔与混沌  

BIFURCATION AND CHAOS OF AXIALLY MOVING BEAMS UNDER TIME-VARYING TENSION

作　　者：陈玲 唐有绮 Chen Ling;Tang Youqi(School of Mechanical Engineering, Shanghai Institute of Technology, Shanghai 201418, China)

机构地区：[1]上海应用技术大学机械工程学院

出　　处：《力学学报》2019年第4期1180-1188,共9页Chinese Journal of Theoretical and Applied Mechanics

基　　金：国家自然科学基金资助项目(11672186)

摘　　要：轴向运动结构的横向参激振动一直是非线性动力学领域的研究热点之一.目前研究较多的是轴向速度摄动的动力学模型,参数激励由速度的简谐波动产生.但在工程应用中,存在轴向张力波动的运动结构较为广泛,而针对轴向张力摄动的模型研究较少.本文研究了时变张力作用下轴向变速运动黏弹性梁的分岔与混沌.考虑随着时间周期性变化的轴向张力,计入线性黏性阻尼,采用Kelvin模型的黏弹性本构关系,给出了梁横向非线性振动的积分--偏微分控制方程.首先应用四阶Galerkin截断方法将控制方程离散化,然后采用四阶Runge-Kutta方法计算系统的数值解,进而确定其动力学行为.基于梁中点的横向位移和速度的数值结果,仿真了梁沿平均轴速、张力摄动幅值、张力摄动频率以及黏弹性系数变化的倍周期分岔与混沌运动,并且通过计算系统的最大李雅普诺夫指数来识别其混沌行为.结果表明:较小的平均轴速有助于梁的周期运动,梁在临界速度附近容易发生倍周期分岔与混沌行为.随着张力摄动幅值的增大,梁的振动幅值的混沌区间不断增大.较小的黏弹性系数和张力摄动频率更容易使梁发生混沌运动.最后,给出时程图、频谱图、相图以及Poincaré映射图来确定梁的混沌运动.The transverse parametric vibration of the axially moving structure is always one of the hot topics in the field of nonlinear dynamics. At present, most of the studies are considering the time-varying speed of dynamic model. The parametric excitation comes from harmonic fluctuations of the axial speed. However, the fluctuation of the axial tension in an axially moving structure is more extensive in the engineering application. There are few researches considering the time-varying tension. The bifurcation and the chaotic behavior of axially accelerating viscoelastic beams under time-varying tension are studied in this paper. A nonlinear integropartia-differential governing equation of the moving beam is established. The linear viscous damping and the Kelvin model in the viscoelastic constitution relation are introduced. The axial tension is assumed as a harmonic variation with time. The fourth-order Galerkin truncation is employed to discretize the governing equation. The dynamic behavior of axially accelerating viscoelastic beams is determined by applying the fourth-order Runge-Kutta algorithm. The influences of material's viscoelastic coefficients, the mean axial speeds, the axial tension fluctuation amplitudes, and the axial tension fluctuation frequencies on the bifurcation diagrams are demonstrated by some numerical results of the displacement and velocity at the midpoint of the beam. The maximum Lyapunov exponent diagram of the system is used to identify the period motion and chaos motion. The results show that the smaller mean axial speed leads to the periodic motion. The period-doubling bifurcation and chaotic behavior are easy to occur near the critical speed. The larger axial tension fluctuation amplitude results in the larger chaos interval. The less viscoelastic coefficient and axial tension fluctuation frequencies lead to the chaotic behavior of the axially moving beam. Furthermore, chaos motions are confirmed using different factors, such as the time history, the fast Fourier transforms, the phase-pl

关 键 词：轴向运动梁 时变张力 GALERKIN法 RUNGE-KUTTA法 分岔 混沌 

分 类 号：O323[理学—一般力学与力学基础]