The Homotopy Analysis for the Nonlinear Dynamics of Planetary Gear Trains

In this paper, the homotopy analysis method (HAM) is proposed to study the nonlinear oscillators of planetary gear trains, in which the periodically time-varying mesh stiffness and gear backlash are included through a nonlinear displacement function. In contrast to the traditional perturbation methods, the HAM does not require a small parameter in the equation under study, and then can be applied to both of the weakly and strongly nonlinear problems. In this article, firstly the closed-form approximations for the dynamic response of planetary gear trains are obtained by HAM. The analytical solutions give insight into the nonlinear dynamics and the impact of system parameters on dynamic response. The accuracy of HAM solutions is evaluated by numerical integration simulations. Results indicate that with large tooth separation times, the amplitude-frequency curves obtained by HAM agree better with the results obtained by NI than those obtained by the MMS.