Sensitivity of General Compound Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters

Abstract

This paper studies the sensitivity of general compound planetary gear natural frequencies and vibration modes to inertia and stiffness parameters. The model admits planetary gears having any combination of stepped-planet, meshed-planet, and multiple stage arrangements. Eigensensitivities in terms of eigenvalue and eigenvector derivatives are analytically derived for both tuned (i.e., cyclically symmetric) and mistuned systems. The results are expressed in compact closed-form formulas. The well-defined modal properties of general compound planetary gears simplify the expressions of eigenvalue sensitivities to ones that are proportional to modal strain/kinetic energies. Inspection of the modal strain/kinetic energy distribution plots provides an effective way to quantitatively and qualitatively determine the parameters that have the largest impact on a certain mode. For parameter perturbations that preserve the system symmetry, the structured modal properties imply that the modes of the same type are independent of the same group of system parameters. Parameter mistuning, with a few exceptions, splits a degenerate natural frequency of the unperturbed system into two frequencies; one frequency keeps its original value and retains its well-defined modal properties, while the other frequency changes and its associated mode lose its structured modal properties.