Abstract: Based on gearing theory and Lagrange equation, the torsional dynamic model for multi-stages planetary gears train is developed by lumped-parameter method. Trapezoid waves are used to express the time-varying mesh stiffness, and the mesh phasing relations among planetary gears are analyzed. For the 3-stages planetary gears of shield tunnelling machine, the torsional vibration frequencies are calculated at the different mesh position of sun and planet gears. According to the characteristics of the gears mesh errors, the harmonic waves composited with shaft frequencies and mesh frequencies are presented to describe the transmission errors of the planetary gears. Considering the time-varying mesh stiffness with different mesh phases of planet gears, the inner excitations of 3-stages planetary gears are analyzed systematically. The dynamic responses of the 3-stages planetary gears are simulated under the inner and external excitation got from a statistical load, and the spectrum properties of gears vibration are also analyzed. The dynamic response of system components are calculated under different speeds for the reducer, and the relationship between gears vibration and the rotational speed of the reducer is studied, which provides the base for choosing the suitable operating condition of driving system of the shield tunnelling machine

Key words: Mesh phase, Multi-stage planetary gears, Shield tunnelling machine, Time-varying stiffness, Vibration response