Dynamics of High-Speed Planetary Gears with a Deformable Ring
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This work investigates steady deformations, measured spectra of quasi-static ring deformations, natural frequencies, vibration modes, parametric instabilities, and nonlinear dynamics of high-speed planetary gears with an elastically deformable ring gear and equally-spaced planets. An analytical dynamic model is developed with rigid sun, carrier, and planets coupled to an elastic continuum ring. Coriolis and centripetal acceleration effects resulting from carrier and ring gear rotation are included. Steady deformations and measured spectra of the ring deflections are examined with a quasi-static model reduced from the dynamic one. The steady deformations calculated from the analytical model agree well with those from a finite element/contact mechanics (FE/CM) model. The spectra of ring deflections measured by sensors fixed to the rotating ring, space-fixed ground, and the rotating carrier are much different. Planet mesh phasing significantly affects the measured spectra. Simple rules are derived to explain the spectra for all three sensor locations for in-phase and out-of-phase systems. A floating central member eliminates spectral content near certain mesh frequency harmonics for out-of-phase systems. Natural frequencies and vibration modes are calculated from the analytical dynamic model, and they compare well with those from a FE/CM model. Planetary gears have structured modal properties due to cyclic symmetry, but these modal properties are different for spinning systems with gyroscopic effects and stationary systems without gyroscopic effects. Vibration modes for stationary systems are real-valued standing wave modes, while those for spinning systems are complex-valued traveling wave modes. Stationary planetary gears have exactly four types of modes: rotational, translational, planet, and purely ring modes. Each type has distinctive modal properties. Planet modes may not exist or have one or more subtypes depending on the number of planets. Rotational, translational, and planet modes persist with gyroscopic effects included, but purely ring modes evolve into rotational or one subtype of planet modes. Translational and certain subtypes of planet modes are degenerate with multiplicity two for stationary systems. These modes split into two different subtypes of translational or planet modes when gyroscopic effects are included. Parametric instabilities of planetary gears are examined with the analytical dynamic model subject to time-varying mesh stiffness excitations. With the method of multiple scales, closed-form expressions for the instability boundaries are derived and verified with numerical results from Floquet theory. An instability suppression rule is identified with the modal structure of spinning planetary gears with gyroscopic effects. Each mode is associated with a phase index such that the gear mesh deflections between different planets have unique phase relations. The suppression rule depends on only the modal phase index and planet mesh phasing parameters (gear tooth numbers and the number of planets). Numerical integration of the analytical model with time-varying mesh stiffnesses and tooth separation nonlinearity gives dynamic responses, and they compare well with those from a FE/CM model. Closed-form solutions for primary, subharmonic, superharmonic, and second harmonic resonances are derived with a perturbation analysis. These analytical results agree well with the results from numerical integration. The analytical solutions show suppression of certain resonances as a result of planet mesh phasing. The tooth separation conditions are analytically determined. The influence of the gyroscopic effects on dynamic response is examined numerically and analytically.
- Doctoral Dissertations