Dynamic modeling and mesh phasing-based spectral analysis of quasi-static deformations of spinning planetary gears with a deformable ring

Abstract

Abstract With a focus on high-speed, lightweight planetary (or epicyclic) gears, this work develops an analytical dynamic model with the sun, carrier, and planets modeled as rigid bodies coupled to an elastic continuum ring having bending, extensional, and shear deformations. The model allows for simultaneous rotation of the sun, carrier, and ring, or for any one of them to be stationary. Coriolis and centripetal accelerations generated by ring and carrier rotation are included. The coupled ordinary (sun, carrier, and planets) and partial (elastic ring) differential equations are cast into a standard gyroscopic system form that is solved using Galerkin discretization. Steady deformations and measured spectral content of the ring deflections are examined with a quasi-static model reduced from the dynamic one. The steady deformations calculated from the analytical model compare well to those from a finite element/contact mechanics model. Ring extensional effects meaningfully influence steady deformations, especially as a result of high-speed rotation of the compliant ring gear, but ring shear effects are small. The spectra of ring deflections measured by sensors fixed to the rotating ring, space-fixed ground, and the rotating carrier are much different, even though the sensors are simultaneously directed at the same ring. Planet mesh phasing significantly affects the measured spectral content. Simple rules are derived to explain the spectra for all three sensor locations for in-phase and out-of-phase systems.