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An analytical technique, namely the homotopy analysis method (HAM), is applied to solve periodic solutions for nonlinear oscillations with parametric excitation in a gear system, in which the periodically time-varying mesh stiffness and gear backlash by using a nonlinear displacement function are included. In this paper, periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations. Thus the frequency-response curves are obtained by using HAM. Unlike perturbation methods, HAM does not depend on any small physical parameters at all; besides, it is valid for both weakly and strongly nonlinear problems. Moreover, different from all other analytic techniques, the HAM provides us a simple way to adjust and control the convergence region of the series solution by means of an auxiliary parameter  ħ .

Study on Nonlinear Oscillations of Gear Systems with Parametric Excitation Solved by Homotopy Analysis Method