基于最大Lyapunov指数的行星齿轮传动系统混沌特性分析

为了分析行星齿轮系统的混沌特性,基于集中参数理论,考虑时变啮合刚度、齿隙和综合啮合误差等非线性因素,建立行星齿轮系统扭转振动模型.采用Runge-Kutta数值解法求解振动方程,利用分岔图和最大Lyapunov指数图分析系统随各种参数变化的分岔与混沌特性.数值仿真得出:随激励频率的增加,系统首先从周期运动进入阵发性混沌,再通过逆倍化分岔由混沌回到周期运动,之后再次通过跳跃激变和倍化分岔由周期运动进入混沌运动,最后通过逆倍化分岔稳定到1周期运动.随阻尼比的增加,系统通过逆倍化分岔由混沌运动进入周期运动.随综合啮合误差幅值、齿隙和刚度幅值分别增加的三种情况下,系统都是通过倍化分岔由周期运动进入混沌运动.随荷载的增加,系统通过跳跃激变和逆倍化分岔由混沌运动进入周期运动.以上分析结果可为行星齿轮系统参数设计提供理论依据.

Chaos Analysis of Planetary Gear Transmission System Based on Largest Lyapunov Exponent

In order to analyze the chaotic characteristics of the planetary gear system, based on the centralized parameter theory, a torsional vibration model of the planetary gear system is established, which considers nonlinear factors such as time-varying meshing stiffness, backlash and comprehensive meshing error. The Runge-Kutta numerical solution is used to solve the vibration equation. The bifurcation diagram and the largest Lyapunov exponent diagram are used to analyze the bifurcation and chaos characteristics of the system with various parameters. The numerical simulation results show that with the increase of the excitation frequency, the system first enters paroxysmal chaos from periodic motion, and then returns to periodic motion from chaos through inverse doubling bifurcation. The system again enters chaotic motion from periodic motion through jump shock and doubling bifurcation, and finally stabilizes to 1-periodic motion through inverse doubling bifurcation. With the increase of the damping ratio, the system moves from chaotic motion to periodic motion through inverse doubling bifurcation. In the three cases where the amplitude of the integrated meshing error, the backlash, and the stiffness increase respectively, the system moves from periodic motion to chaotic motion by doubling the bifurcation. With the increase of load, the system moves from chaotic motion to periodic motion through jump shock and inverse doubling bifurcation. The above analysis results can provide a theoretical basis for the parameter design of the planetary gear system.