Non-linear dynamic behaviour of compound planetary gear trains: Model formulation and semi-analytical solution

Abstract

A discrete, non-linear, time-varying, torsional dynamic model of a multi-stage planetary train that is formed by any number of simple planetary stages is proposed in this study. Each planetary stage has a distinct fundamental mesh frequency and any number of planets spaced in any angular positions. The model allows the analysis of the gear train in all possible power flow configurations suitable for various gear drive ratios. It includes periodic variation of gear mesh stiffnesses as well as clearance (backlash) non-linearities that allow tooth separations. Equations of motion for the general case are formulated and solved semi-analytically using a hybrid harmonic balance method (HBM) in conjugate with inverse Fourier transform. Relative mesh displacements along lines of action of individual gear pairs were used as the continuation parameters to pass singular points and ill-conditioned equations in their proximity. At the end, a case study of a two-stage planetary train is used to demonstrate the effectiveness of the model and solution methods. The HBM solutions are compared to those obtained by a direct numerical integration method to assess their accuracy.

A nonlinear dynamic model for analysis of the combined influences of nonlinear internal excitations on the load sharing behavior of a compound planetary gear set

“…Al-Shyyab et al. 20 employed the purely rotational model of compound planetary gears with the vibration modes analysis to research the structured vibration properties. In 2007, a rotational-translational model for general compound planetary gear set was developed by Kiracofe and Parker 21 to characterize modal properties.…”

Dynamic Characteristics of Encased Differential Gear Train with Journal Bearing

“…Lin and Parker [10] used a multiscale method to investigate the influence of meshing stiffness waves on the stability of the planetary gear train, and they derived the analytical formula of the relationship between meshing stiffness variation and system dynamic stability. Al-Shyyab et al [11] constructed a torsional dynamics model of a discrete two-stage planetary transmission system, and they studied the frequency response of the system under stiffness excitation. Considering time-varying meshing stiffness, Xiang et al [12] developed a torsional nonlinear dynamic model of multistage planetary gear trains and studied the effects of bifurcation parameters such as excitation frequency, backlash, and damping.…”

Influence of transmission errors to load sharing behaviour in Ravigneaux planetary gear sets

“…Ligata [12] designed the gear transmission system experiment to investigate the influence of the number of planetary gears and the amplitude of errors on the load sharing characteristics of gear sets. Ai-shyyab [13] used a hybrid harmonic balance method (HBM) to semi-analytically solve the torsional dynamic model of a multi-stage gear sets and provided a case study to verify its accuracy.…”