Vibration of planetary gears having an elastic continuum ring gear

The primary goal of this work is to develop mathematical models for planetary gears having an elastic ring gear to conduct numerical and analytical studies to understand the modal properties, parametric instability, and nonlinear gear dynamic behaviors.;First, natural frequencies and vibration modes are determined as closed-form expressions for a ring having a circumferentially varying foundation of very general description through perturbation and Galerkin analyses. The simple eigensolution expressions explicitly show the parameter dependencies, lead to natural frequency splitting rules for degenerate unperturbed eigenvalues at both first and second orders of perturbation, and identify which nodal diameter Fourier components contaminate a given n nodal diameter base mode of the free ring. As an application and as the motivating problem for the study, the natural frequencies and vibration modes of a ring gear used in helicopter planetary gears with unequally spaced planets are investigated.;Second, the distinctive modal properties of equally spaced planetary gears with elastic ring gears are analytically studied through perturbation and a candidate mode method based on an elastic-discrete model. Two perturbations are used to obtain closed-form expressions of all the eigenfunctions. In the Discrete Planetary Perturbation (DPP), the unperturbed system is a discrete planetary gear with a rigid ring. In the Elastic Ring Perturbation (ERP), the unperturbed system is an elastic ring supported by the ring-planet mesh springs; the sun, planet and carrier motions are treated as small perturbations. All vibration modes are classified into rotational, translational, planet and purely ring modes. The well defined properties of each type of mode are analytically determined. All modal properties are verified numerically. Also the modal properties of planetary gears having diametrically opposed planets and an elastic ring gear are studied through the candidate mode method. Two types of modes are found: rotational and translational modes.;Fourth, the parametric instability of planetary gears having elastic continuum ring gears is analytically investigated based on the elastic-discrete model. By using the structured modal properties of planetary gears and the method of multiple scales, the instability boundaries are obtained as simple expressions in terms of mesh parameters. Instability existence rules for in-phase and sequentially phased planet meshes are also discovered. For in-phase planet meshes, instability existence depends only on the type of gear mesh deformation. For sequentially phased planet meshes, the number of teeth on the sun (or the ring) and the type of gear mesh deformation govern the instability existence. The instability boundaries are validated numerically.;Finally, the nonlinear dynamics of planetary gears having an elastic ring gear is studied. The frequency-response functions are presented for primary, subharmonic, and superharmonic resonances. The tooth separation phase between meshes depends on the mode type excited and the position of the planets. The resonances associated with a translational or planet mode are investigated. Parametric instability rule and mesh deflection phase expression are confirmed in the numerical simulation for both in-phase and sequentially phased planets. Response of planetary gears having commensurate natural frequencies is also studied.