# Vibration of Rotating and Revolving Planet Rings with Discrete and Partially Distributed Stiffness
## 1 Introduction  
Vibration analysis of rotation rings in different structures, such as gears, rotors, bearings, etc., is of great interest for complex dynamic characteristics. This study was motivated by planetary gear systems, where planet gears often have a thin rim [1], especially in aerospace applications, where weight is an important consideration. Planet gears do not only rotate around their axis supported by bearings but revolve around the center of planetary gear systems meshed with central gears. Meanwhile, they are supported by both discrete stiffnesses of mesh pairs and partially distributed stiffnesses of support bearings. These planet gears may undergo significant elastic continuum vibration during operation, for their compliance, combined movement, and complex elastic supports. Compliance of gears has received attention in some references. [PERSON] and [PERSON] [2] found elastic vibration in aerospace gears experimentally. [PERSON] et al. [3] investigated the flexible gear vibration, using FEM method. Elastic deformations are also found in planetary gears by experiments [4]. [PERSON] and [PERSON] [5] studied the vibration modes of compliant gears by multibody dynamic formulation. [PERSON] et al. [6] investigated the effect of flexibility of gears on the dynamic behavior of a planetary gear set. Their study shows that dynamic gear tooth bending stress values were found to be significantly higher for all of the gears for a flexible internal gear as compared to the rigid internal gear. Vibrations of stationary or rotating rings with different types of supports have been investigated by many researchers. [PERSON] and [PERSON] [7] studied the in-plane and out-plane vibration of rotating rings on an elastic foundation. Free vibrations of rings on rigid radial supports were investigated by [PERSON] and [PERSON] [8], where the ring was separated into several parts with supports located at the ends of each segment. [PERSON] [9] studied the free vibration of a thick ring on multiple radial supports by the Galerkin method. [PERSON] and [PERSON] [10] investigated rotating rings, where discrete stiffnesses are attached to and rotate with the ring. [PERSON] et al. [11] investigated the vibration of non-axisymmetric rings with lumped masses and stiffnesses added. [PERSON] and [PERSON] [12] studied the cylindrical shell with ring support. [PERSON] and [PERSON] [13] studied the steady state response of an elastic ring subjected to a uniformly moving load by the method of images. The results show that, for small velocities of the load, the ring pattern is almost perfectly symmetric with respect to the loading point. The pattern becomes slightly asymmetric due to the viscosity of the springs if the load velocity is small. [PERSON] and [PERSON] [14] investigated the free non-linear vibration of a rotating thin ring with in-plane and out-plane motions. [PERSON] and [PERSON] [15] used two observer systems, that is, Lagrangian and Eulerian systems, to derive rotating ring models. [PERSON] and [PERSON] [16] studied the vibration of rings on a general elastic foundation. There are fewer studies on the vibration of rotating rings with discrete supports. Afterward, they [17] studied the modal properties of equally spaced planetary gears with elastic ring gears. The results show that all eigenfunctions fall into four mode types, rotational, translational, planet, and purely ring modes. [PERSON] and [PERSON] [18] mathematically proved modal properties of planetary gears having diametrically opposed planets and an elastic ring gear. Two types of modes were found, that is, rotational and translational modes. [PERSON] and [PERSON] [19; 20] studied the parametric instability of rotating rings connected to moving discrete springs with time-varying stiffness. They analytically studied the effects of different system parameters on the instability boundaries and well-defined properties for the occurrence or suppression of instabilities are obtained as simple relations in the system parameters.