# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 1 Introduction
[PERSON] and [PERSON] [21] mathematically proved the structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears. The results show that planetary gears have the only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds, and flutter instabilities occur at extremely high speeds. Then, they [22] studied the vibration of high-speed rotating rings coupled to space-fixed stiffnesses. [PERSON] et al. [23] examined the parametric vibration of an elastic dual-ring structure by an analytical model including stationary supports and rotating loads. The results imply that there exist four types of excitations and that vibration can occur in a coupled, uncoupled, principal, or combination manner. [PERSON] et al. [24] investigated the in-plane free vibration of high-speed rotating thin ring with an elastic foundation. [PERSON] and [PERSON] [25] investigated eigenvalue sensitivity to model parameters and veering in gyroscopic systems by a perturbation approach. The results are shown to be accurate over a wide range of rotation speeds. A veering parameter is defined to analyze veering in high-speed planetary gears. [PERSON] et al. [26] investigated the vibration of rotating elastic rings that are excited by an arbitrary number of space-fixed discrete stiffnesses with periodically fluctuating stiffnesses. The results show that the bandwidths of the instability regions correlate with the fractional strain energy stored in the discrete stiffnesses. For rings with multiple discrete stiffnesses, the phase differences between them can eliminate large amplitude response under certain conditions. [PERSON] et al. [27] developed a new high-order model for in-plane vibrations of rotating rings. The study shows that the new high-order model is superior to the existing ring models in predicting the dynamic behavior of either stationary or rotating rings, and the model is applicable to both plane strain and plane stress configurations. In this paper, the through-thickness variation of the radial stress and geometrical nonlinearity were considered. As mentioned above, there has been no study on the vibration of planet rings, which have both rotating and revolving movements, with discrete stiffnesses and partially distributed stiffnesses. In this study, the vibration properties of high-speed rotating and revolving planet rings with discrete and partially distributed stiffnesses were investigated. The governing equations of motion for the radial and tangential deformations of planet rings were derived based on a rotating-basis system. This model considered the support zone variation and revolving movement of planet rings. Matrix differential operators were used to express the governing equations. The vibration properties of planet rings with and without stiffnesses were investigated by using the Galerkin method. The influence of parameters on vibration characteristics was studied. The forced responses excited by rotating and revolving movement were also investigated.