This work aims to advance the understanding of nonlinear dynamics of planetary gears and the influence of the key system parameters on dynamic response. Analytical solutions of nonlinear dynamic model are mainly used to conduct investigations on interesting nonlinear dynamic behaviors.
An analytical lumped-parameter model, which is parametrically excited by time-varying mesh stiffness and includes tooth separation, shows nonlinear dynamic response. The accuracy of the model is correlated against a benchmark finite element analysis over broad mesh frequency ranges.
The nonlinear dynamic model is analytically solved by perturbation analysis. Concise, closed-form expressions of planetary gear dynamic response are obtained for various resonances. The analytical solution is validated by numerical integration and the harmonic balance method. The rapid calculation of dynamic response with acceptable accuracy demonstrates that the analytical solutions are effective for performing parametric studies.
The explicit inclusion of key system parameters in the analytical solution shows the impact of the system parameters on planetary gear nonlinear vibration.
Mesh stiffness discontinuity from tooth contact loss is considered for the analytical solution that gives nonlinear vibrations. Correlation between the external torque and vibration amplitude proves that tooth contact loss can occur even under large torque. Resonances at multiple harmonics of the mesh frequency are distinguished by different excitation sources. Nonlinear subharmonic resonance characterized by response jump phenomena on both sides of the mesh frequency range where resonance occurs is examined.
The impact of system parameters on planetary gear vibrations is investigated by using a generalized planetary gear model including bearing stiffness and relative mesh phase. Use of the well-defined modal properties and closed-form expressions of resonant response confirm the existing mesh phasing rules to suppress selected vibration modes of primary resonance. Extended suppression rules for super- and subharmonic resonances are proposed. In addition to the suppression conditions, the analytical solution discovers the dependence of dynamic response on the system parameters and vibration modes, which provides practical guidance for finding optimal design parameters for vibration reduction.
A nonlinear analytical tooth profile modification model for planetary gear dynamic response is developed. Perturbation analysis gives a closed-form expression for the frequency response relation including the fundamental tooth profile modification parameters of the modification amount and length. Different effects of tooth profile modification on static transmission error and dynamic response are compared in terms of the modification amount. Strong influence of system parameters on the dynamic effect of tooth profile modification is discovered.