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Research Papers

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

[+] Author and Article Information
Yichao Guo

Department of Mechanical Engineering, Ohio State University, 201 West 19th Avenue, Columbus OH 43210

Robert G. Parker

State Key Lab for Mechanical Systems and Vibration, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, 200240, Chinaparker.242@osu.edu

There is a typo in Ref. 11. It is corrected in Eq. 14.

An asterisk $( ∗)$ in a superscript means any element, such as the planet train of a planet set, where the arbitrary element is dictated by the position of the asterisk.

J. Vib. Acoust 132(1), 011006 (Jan 11, 2010) (13 pages) doi:10.1115/1.4000461 History: Received March 13, 2008; Revised August 14, 2009; Published January 11, 2010; Online January 11, 2010

## 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.

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## Figures

Figure 1

The example system

Figure 2

Mode shape of vibration mode 4 (ω4=871 Hz) for the example system with nominal parameter values.

Figure 3

Modal (a) strain and (b) kinetic energy distributions associated with mode 4 (ω4=871 Hz).

Figure 4

(a) ω4 versus ε=(kgp32∗1−k¯gp32∗1)/k¯gp32∗1=(kgp42∗1−k¯gp42∗1)/k¯gp42∗1, and (b) ω4 versus τ=(Ip2∗1−I¯p2∗1)/I¯p2∗1. k¯gp32∗1 and kgp32∗1 are the nominal and perturbed values of the mesh stiffness between the sun gear in stage 2 (central gear 3) and planet 1 in any planet train of planet set 2. k¯gp42∗1 and kgp42∗1 are the nominal and perturbed values of the mesh stiffness between the ring gear in stage 2 (central gear 4) and planet 1 in any planet train of planet set 2. I¯p2∗1 and Ip2∗1 are the nominal and perturbed values of moment of inertia of planet 1 in any planet train of planet set 2.

Figure 5

(a) ω12 and ω13 versus ε1=(kp3211−k¯p3211)/k¯p3211, and (b) ω12 and ω13 versus ε2=(kp211−k¯p211)/k¯p211. k¯gp3211 and kgp3211 are the nominal and perturbed values of the mesh stiffness between the sun gear in stage 2 (central gear 3) and planet 1 in train 1 of planet set 2. k¯p211 and kp211 are the nominal and perturbed values of the bearing stiffness of planet 1 in train 1 of planet set 2.

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