Planetary gears are widely used in mechanical transmission systems, but the vibration and noise affect their reliability and life. In this paper, the torsional dynamic model of an encased differential planetary gear with coaxial contrarotating outputs considering the time-varying meshing stiffness, damping, and phase difference of all gear pairs is established. By solving the equations of the derived system, three types of natural frequencies with different multiplicities of the system are obtained. The multiscale method is used to study the parametric vibration stability caused by the time-varying meshing stiffness, and the results are verified by numerical simulation. The dynamic characteristics of elastic meshing force are analyzed from time domain and frequency domain. The variation of the dynamic load factor of each gear pair with input speed and the relationship between its peak position and the natural frequency of the derived system are discussed. The results show that there is an unequal coupling phenomenon of meshing frequency between the meshing forces of different planetary sets. In the absence of external excitation, the meshing stiffness parameters not only excite the main resonance response of the system but also cause superharmonic resonance, subharmonic resonance, and combined resonance.

中文翻译：

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同轴旋转包裹式差动轮系的啮合刚度参数振动。

行星齿轮广泛用于机械传动系统，但振动和噪声会影响其可靠性和寿命。在本文中，考虑了所有齿轮副的时变啮合刚度，阻尼和相位差，建立了带有同轴反向旋转输出的被包裹的差动行星齿轮的扭转动力学模型。通过求解导出系统的方程，可以得到三种具有不同系统多重性的固有频率。采用多尺度方法研究了随时间变化的啮合刚度引起的参数振动稳定性，并通过数值模拟验证了结果。从时域和频域分析了弹性啮合力的动态特性。讨论了每个齿轮副的动态负载系数随输入速度的变化以及其峰值位置与导出系统的固有频率之间的关系。结果表明，不同行星组的啮合力之间存在不均等的啮合频率耦合现象。在没有外部激励的情况下，啮合刚度参数不仅会激发系统的主共振响应，还会引起超谐共振，次谐波共振和组合共振。