Session: VIB-08-02: Nonlinear Systems & Phenomena II

Paper Number: 97593

97593 - Analytical Solutions for Dynamics of Lumped-Parameter Planetary Gears Having Degenerate Modes 

Vibration modes of planetary gears can be categorized into rotational modes with distinct natural frequencies, translational modes with degenerate natural frequencies of multiplicity two, and planet modes with natural frequency multiplicity depending on the total number of planets. Resonances and parametric instabilities of these modes or natural frequencies, under periodic tooth mesh excitation, lead to large-amplitude vibrations that damage the planetary gear system. This work presents analytical solutions for resonances and parametric instabilities of all natural frequencies with a focus on the degenerate translational and planet mode natural frequencies. A resonance of a degenerate natural frequency involves multiple modes. Analytically solving a multi-mode resonance problem is algebraically prohibitive. A resonance suppression rule from the literature gives that only one mode associated with a degenerate translational or planet mode natural frequency is responsible for the resonance of this natural frequency. This reduces a multi-mode resonance problem to a single-mode one. Analytically solving the single-mode resonance problem yields a closed-form amplitude-frequency relation that applies for resonances of all natural frequencies. The analytical solution allows for fast computation of the response curve near a resonance (including its peak amplitude) without running any long-term dynamic simulations, and the accuracy is sufficiently high. A parametric instability suppression rule governs the occurrence or suppression of a parametric instability between any two natural frequencies (which can be the same), regardless of their multiplicity. For a parametric instability that is not suppressed according to the instability suppression rule, closed-form instability boundaries that bound the range of mesh frequencies where this parametric instability occurs are given with and without damping. Counter-intuitively, damping can destabilize a parametric instability by enlarging the instability bandwidth.

Presenting Author: Chenxin Wang University of Utah

Presenting Author Biography: 2014 - 2019, Ph.D. in Mechanical Engineering, Virginia Tech <br/>2010 - 2014, B.S. in Mechanical Engineering, Tianjin University

Authors:

Chenxin Wang University of Utah
Robert Parker University of Utah