关键词: Lyapunov指数, 齿轮, 非线性动力学, 关联维数

Abstract: 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 Poincaré 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.

Key words: Correlation dimension, Gear, Lyapunov exponent, Nonlinear dynamics