多自由度齿轮系统非线性动力学分析

Nonlinear Dynamics Analysis of a Multi-DOF Gear System

建立三自由度多间隙齿轮系统耦合振动模型,综合考虑时变啮合刚度、齿侧间隙和轴承纵向响应等非线性特性因素的影响,并采用变步长4～5 阶Runge-Kutta 法对系统状态方程进行数值求解.构建系统的Poincaré 截面,得到系统的位移-时间映像图,通过分析位移-时间映像图,发现系统在支承间隙较小而支承刚度较大时更加稳定;根据分析位移-时间映像图的结论,选择合理的参数,画出系统随频率变化的分岔图,结合相图和Poincaré 映射图分析系统的非线性动力学特性,发现系统在不同激励频率下会发生Hopf分岔、倍化分岔和混沌等现象.

A three-DOF multi-gap coupled vibration model of a gear system was established considering the nonlinear properties such as time-varying meshing stiffness, backlash and bearing longitudinal response factors. Then, the system state equation was solved by using 4-5 order Runge-Kutta integration method with variable step size. The bifurcation diagrams, phase portraits and Poincaré maps which describe the system vibration displacement with the change of excitation frequency under different supporting damping and stiffness were obtained. According to the analysis of the displacement-time map of the system, it was concluded that the system is more stable with larger supporting damping and large stiffness. The bifurcation diagram of the system against frequency variation was plotted based on the reasonable selection of the parameters. The Hopf bifurcation, torus tumble bifurcation and chaos phenomena of the system under different excitation frequencies were discussed based on the phase diagrams and Poincaré maps.