Modal properties and parametrically excited vibrations of spinning epicyclic/planetary gears with a deformable ring

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).