以行星轮系柔性内齿圈为研究对象，为计入拉伸、剪切和弯曲变形等因素，使数学模型更好与工程实际相匹配，根据其结构及边界特点采用平面梁单元建立弹性边界柔性直齿内齿圈的振动分析模型，并对其固有特性进行了研究。基于叠加原理和力向一点平移定理，采用组合激振法等效模拟了沿啮合线作用的啮合激励，并从频域角度揭示了啮合激励对弹性边界柔性内齿圈振动响应的影响。研究结果表明，弹性边界柔性内齿圈振型可归结为三类：平移振动模式（内齿圈各点具有相同的运动），弯曲振动模式（内齿圈发生弯曲变形的振动）和扩展振动模式（内齿圈各点仅产生径向振动）；啮合激励对扩展振动模式影响较小，而对弯曲振动模式影响显著；对直齿内齿圈而言，弯曲振动模式是引起振动与噪声的主要原因；当扩展振型（波数m=0）发生共振时，内齿圈呈等幅振动；当波数m不为零的振型共振时，内齿圈呈m个波形。
The flexible planetary ring gear is taken as the research object. In order to formulate a more accurate mathematical model in compliance with common engineering practices, the tension, transverse shear, and bending deformation, of the ring gear are considered. According to the structure and boundary, vibration analysis model for flexible spur ring gear with elastic foundation is established by using the plane beam element, and the inherent characteristics are investigated. Based on the superposition principle and force to a point translation theorem, the combined excitation apporach is appled to simulate the meshing excitation acting along the line of action. Effects of meshing excitation on the vibration response of flexible ring gear is revealed from the frequency domain. The results indicate that, the vibration type of flexible ring gear with elastic foundation can be divided into three categories:Translational vibration mode (each point of the ring gear has the same motion), flexural vibration mode (the ring gear occurs vibration of bending deformation), and extensible vibration mode (each point of the ring gear has only radial vibration). The meshing excitation has a slight impact on extensible vibration mode, while it influences the flexural vibration mode apparently. For the spur ring gear, flexural vibration mode is the main cause of vibration and noise. When extensible vibration mode (wave number m=0) occurs resonance, each point of the ring gear has equal vibration amplitude. When the ring gear occurs resonance and the wave number m is not zero, the ring gear has m waveform.
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