The overall transmission error and phase difference between the internal and external gear meshing pairs in planetary gear trains (PGTs) play crucial roles in vibration reduction design. However, these factors have received limited attention in existing studies concerning the dynamics and tooth profile modification of PGTs. In this paper, a shifted spur planetary gear system is employed as a case study. The formulation for calculating the phase difference of both internal and external meshing pairs is derived. Furthermore, a new time-varying meshing stiffness model incorporating the phase difference is proposed based on Tooth Contact Analysis (TCA) and Loaded Tooth Contact Analysis (LTCA). The calculation method for the overall transmission error considering the phase difference is provided. An optimization method for tooth profile modification is developed to minimize the fluctuation amplitude of the overall transmission error, and applied to a shifted spur planetary gear system. Utilizing the time-varying meshing stiffness and phase differences of both internal and external meshing pairs, a dynamic model of the planetary transmission system is established. Dynamic simulations based on this model confirm the effectiveness of the proposed profile modification strategy, demonstrating a significant reduction in the vibration acceleration of the sun gear. In summary, optimizing the tooth profile to suppress fluctuations in overall transmission error enhances the vibration reduction performance of planetary transmission systems.
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