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Research Papers

Parametric Instability of Planetary Gears Having Elastic Continuum Ring Gears

[+] Author and Article Information
Robert G. Parker1

 Ohio State University, Distinguished Professor Chair and Executive Dean, University of Michigan-Shanghai Jiao Tong University Joint Institute, Shanghai Jiao Tong University, Shanghai, China 200240parker.242@osu.edu

Xionghua Wu

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

1

Corresponding author.

J. Vib. Acoust 134(4), 041011 (May 29, 2012) (11 pages) doi:10.1115/1.4005836 History: Received June 26, 2011; Revised November 14, 2011; Published May 29, 2012; Online May 29, 2012

The parametric instability of planetary gears having elastic continuum ring gears is analytically investigated based on a hybrid continuous-discrete model. Mesh stiffness variations of the sun-planet and ring-planet meshes caused by the changing number of teeth in contact are the source of parametric instability. The natural frequencies of the time invariant system are either distinct or degenerate with multiplicity two, which indicates three types of combination instabilities: distinct-distinct, distinct-degenerate, and degenerate-degenerate instabilities. 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.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

Grahic Jump Location
Figure 1

Elastic-discrete model of a planetary gear and corresponding system coordinates. The distributed springs around the ring circumference are not shown.

Grahic Jump Location
Figure 2

Mesh stiffness variation for the nth ring-planet mesh. cr is the contact ratio (less than 2), and ρr is the trapezoidal wave slope coefficient.

Grahic Jump Location
Figure 3

Instability regions for a planetary gear with in-phase meshes as γsn=0, γsr=12, cs=1.4, cr=1.6, ɛ=μ, ρs=ρr=0, and other parameters in Table 1. —, analytical solution; ***, numerical solution.

Grahic Jump Location
Figure 4

Instability regions for a planetary gear with sequentially phased meshes as γsn=[0,14,12,34], γsr=12, cs=1.4, cr=1.6, ɛ=μ, ρs=ρr=0, and other parameters in Table 1. —, analytical solution; ***, numerical solution.

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