Abstract This work investigates the modal properties and parametric instabilities of high-speed spur epicyclic and planetary gears with an elastically deformable ring. Coriolis (i.e., gyroscopic) and centripetal acceleration effects are modeled. The gyroscopic effects lead to complex-valued (i.e., traveling-wave) modes. These modes fall into three categories: rotational, translational, and planet modes. Each category has highly structured modal deflections. The modal properties are proved analytically by discretizing the elastic ring such that the system falls into the class of general cyclically symmetric systems with proven modal properties. Such a proof readily extends to helical planetary gears with any or all of the sun, carrier, and ring modeled as elastic bodies. The time-varying sun-planet and ring-planet mesh stiffnesses are the sources of parametric excitation. The conditions where combinations of the mesh frequency and mesh stiffness amplitudes cause parametric instabilities are determined in closed form with the method of multiple scales. The analytical results agree with numerical results from Floquet theory. Many potential parametric instabilities are suppressed. A rule to determine whether a given potential parametric instability is suppressed or not is given. The rule is closely linked to the modal properties noted above and the planet mesh phasing parameters (sun and ring tooth numbers and number of planets).

中文翻译：

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具有可变形环的旋转行星齿轮/行星齿轮的模态特性和参数激发振动

摘要 这项工作研究了具有弹性变形环的高速正行星齿轮和行星齿轮的模态特性和参数不稳定性。对科里奥利（即陀螺）和向心加速度效应进行建模。陀螺效应导致复值（即行波）模式。这些模式分为三类：旋转、平移和行星模式。每个类别都有高度结构化的模态偏转。通过对弹性环进行离散化来解析证明模态特性，使得系统属于具有已证明模态特性的一般循环对称系统。这种证明很容易扩展到螺旋行星齿轮，其中任何或所有太阳、支架和环都被建模为弹性体。随时间变化的太阳行星和环行星网格刚度是参数激励的来源。网格频率和网格刚度幅值的组合导致参数不稳定性的条件用多尺度方法以封闭形式确定。分析结果与 Floquet 理论的数值结果一致。许多潜在的参数不稳定性被抑制。给出了确定给定的潜在参数不稳定性是否被抑制的规则。该规则与上述模态属性和行星网格定相参数（太阳和环齿数以及行星数）密切相关。网格频率和网格刚度幅值的组合导致参数不稳定性的条件用多尺度方法以封闭形式确定。分析结果与 Floquet 理论的数值结果一致。许多潜在的参数不稳定性被抑制。给出了确定给定的潜在参数不稳定性是否被抑制的规则。该规则与上述模态属性和行星网格定相参数（太阳和环齿数以及行星数）密切相关。网格频率和网格刚度幅值的组合导致参数不稳定性的条件用多尺度方法以封闭形式确定。分析结果与 Floquet 理论的数值结果一致。许多潜在的参数不稳定性被抑制。给出了确定给定的潜在参数不稳定性是否被抑制的规则。该规则与上述模态属性和行星网格定相参数（太阳和环齿数以及行星数）密切相关。