Hierarchical multi-objective optimization of vibration and dynamic load in a two-stage planetary gear transmission based on phase tuning theory

A multi-dimensional hierarchical design methodology is proposed to reduce dynamic vibration in a two-stage Planetary Gear Transmission (PGT) system. A transversal–torsional coupled nonlinear dynamic model is established using the Lagrange method. Response-based sensitivity analysis is applied to quantify the influence of mass and stiffness parameters of each component. On the basis of phase-tuning theory, the effect of gear tooth numbers on forced vibration response and resonance behaviour is analysed, and a phase-tuning-oriented tooth-number design for both stages is derived. A hierarchical multi-objective optimization framework is then constructed, which sequentially optimizes coupling stiffness, vibration-related mass and support stiffness parameters, and gear tooth numbers, under strength and assembly constraints. Simulation and experimental validation on a two-stage PGT test bench demonstrate that the proposed design strategy reduces the mean root-mean-square values of translational and torsional vibration displacement by 33.75% and 12.51%, respectively, decreases the mean dynamic load coefficient by 55.79%, and lowers low-order resonance peak amplitudes by 19.35%. These results verify that the hierarchical optimization method can effectively improve vibration performance and dynamic load distribution in high-speed, heavy-load multi-stage planetary drives.