Parametric Instability in a Gear Train System Due to Stiffness Variation

The excitation from mesh stiffness variation in a tunnel gear driving system can cause excessive noise and vibrations. Since the stiffness variation may induce parametric instability, the system could be damaged on a permanent-basis. Therefore, the study of parametric instability in such system is of paramount importance. In this work, a rigid-elastic model is developed using the energy method, where the ring gear is treated as a rotating thin ring having radial and tangential deflections, whereas the pinions are assumed to be rigid bodies having translational motion relative to the radial directions of the ring gear as well as rotational motions around their centers. All gear meshes are modeled as interactions caused by time-varying springs, and the supports of the pinions are modeled as linear springs in the radial direction relative to the ring gear. The modeling leads to a set of partial-ordinary linear differential equations with time-varying coefficients. For an N planet system, the discretization process yields 2N+2 ordinary differential equations. Stability boundaries are determined using Floque’t theory for a wide range of parameter values. Specifically, the effects of mesh stiffness on the parametric instability are examined. The results show that the instability behaviors are closely related to the basic parameters when considering the time-varying excitation. This could be a serious consideration in the preliminary design of such systems.