单级齿轮非线性系统吸引子的数值特性研究

为了定量地判断吸引子的特性,在建立间隙函数呈分段线性时单级齿轮系统的量纲一化的动力学方程的基础上,考虑到系统在传统意义下的Jacobi矩阵并不是处处存在的,故直接从Lyapunov指数(LE)的定义出发,给出了计算系统最大Lyapunov指数的方法;基于稳态数值响应阐明了计算系统吸引子关联分维数的方法;通过与系统相图及Poincare截面图进行比较,验证了计算Lyapunov指数及关联维数的方法的正确性;在此基础上,分别对阻尼比、齿轮综合误差以及齿侧间隙等参数对系统动力学特性的影响进行了分析,分别计算了单独改变系统阻尼比、齿轮综合误差和齿侧间隙时,系统振动的分岔图、最大Lyapunov指数图以及系统的关联维数,得到了系统振动特性随这些参数变化时的变化规律。

RESEARCH ON NUMERICAL CHARACTERS OF THE ATTRACTORS IN A NONLINEAR GEAR SYSTEM

A three-degree freedom nonlinear dynamics model of a gear pair system is established. Since the system's Jacobi matrix does not always exist in traditional understanding, then a numerical method for calculating the greatest Lyapunov exponent is presented directly based on the definition of Lyapunov exponent. As the system's chaotic attractors often have fractal dimension, then the method of how to calculate the system's correlation dimension is illuminated and the system's correlation dimension is calculated. By comparing the results with the system phase plot and the Poincare map, the validities of the methods to calculate the greatest Lyapunov exponent and the correlation dimension are proved. On this basis, the system's dynamic characters are analyzed by changing the damping ratio, composite error and the backlash in the system. The system's bifurcation plots, the greatest Lyapunov exponent plots and the correlation dimension are given when the parameter is changing respectively. Then the changing laws of the system's numerical characters can be obtained.