Single-gap Nonlinear Dynamics Of The Gear System

This dissertation mainly studies the nonlinear characters of a spur gear pair with piecewise linear clearance. Three problems are solved in this paper., which are: to solve the nonlinear dynamic gear pair model, to calculate the Lyapunov exponents and to find the fractal dimensions of the attractors of the system.Using the characters of the piecewise system this paper derives a precise analytical solution. On this basis, the applicability of the numerical methods is investigated and the calculation shows that the four order variable step size Gill numerical integration method is better than Runge-Kutta method in numerical stability. Additionally, parametric studies are performed to understand the effect of system parameters such as damping ratio, excitation frequency and the ratio of alternate force to mean force etc. on the nonlinear dynamic behaviors.For the nonlinear model of a gear system with backlash between the gear pair, the system's Jacobi matrix does not always exist. Under this condition, a new method for calculate the Lyapunov exponent is developed. The validity is verified by using the system's phase plane plots and the Poincare map plots. Farther more the conclusion that the maximum Lyapunov Exponent should not be less than the twice of the damping ratio is obtained. The plots of the maximum Lyapunov exponent with respect to the damping ratio and the excitation frequency are given respectively.With reference to the fractal dimension of the system, the correlation dimension and the Lyapunov dimension are studied. The calculation samples with periodic and chaotic response are illuminated respectively.