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Research Papers

Dynamic Analysis of Planetary Gears With Bearing Clearance

[+] Author and Article Information
Yi Guo2

Department of Mechanical Engineering,  Ohio State University, Columbus, OH 43210guo.83@osu. edu

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, Chinaparker.242@osu.edu

1

Address all correspondence to this author.

2

Currently at National Renewable Energy Laboratory.

J. Comput. Nonlinear Dynam 7(4), 041002 (Jun 13, 2012) (15 pages) doi:10.1115/1.4005929 History: Received June 15, 2011; Accepted December 19, 2011; Published June 13, 2012; Online June 13, 2012

This study investigates the dynamics of planetary gears where nonlinearity is induced by bearing clearance. Lumped-parameter and finite element models with bearing clearance, tooth separation, and gear mesh stiffness variation are developed. The harmonic balance method with arc length continuation is applied to the lumped-parameter model to obtain the dynamic response. Solution stability is analyzed using Floquet theory. Rich nonlinear behavior is exhibited, consisting of nonlinear jumps, a hardening effect induced by the transition from no bearing contact to contact, and softening induced by tooth separation. Bearings of the central members (sun, carrier, and ring) impact against the bearing races near resonances, which leads to coexisting solutions in wide speed ranges, grazing bifurcation, and chaos. Secondary Hopf and period-doubling bifurcations are the routes to chaos. Input torque can suppress some of the nonlinear effects caused by bearing clearance.

Copyright © 2012 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Translational-rotational, lumped-parameter model of a planetary gear

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Figure 2

Stiffness model of the bearings with clearance in [34]

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Figure 3

Time-varying mesh stiffness of the sun-planet (—) and ring-planet (——) meshes of the planetary gear calculated by finite element analysis

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Figure 4

The relative error of the sun bearing deformation (ds-ds,0)/ds-ds,0ds,0ds,0 in the x-direction for a varying number of bearing spring/gap pairs between races. The nominal torque 1130 Nm is applied, and the sun bearing clearance is 5 μm.

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Figure 5

Flow chart of the iteration solver in the harmonic balance method

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Figure 6

(a) Smoothing functions for the bearing reaction force with clearance and (b) smoothing functions for the tooth load with backlash

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Figure 7

(a) The rms (root mean square, mean removed) carrier xc for varying speeds with carrier clearance ρc  = 0, 1, 3, 5, 7, 15, 30, and infinity. Unstable solutions are denoted by ——. (b) Percent of bearing contact in a mesh cycle for varying speeds with carrier clearance ρc  = 0, 1, 3, 5, 7, 15, and 30. Results are calculated by the harmonic balance method. The modal damping ratio of this mode is equal 2.5%.

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Figure 8

The rms (root mean square, mean removed) carrier xc for varying speeds with ρc  = 5 clearance in the carrier bearing. Results are compared among finite element analysis (□), numerical integration (○), and harmonic balance (—). Unstable branches are denoted by (——).

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Figure 9

The rms (root mean square, mean removed) sun ys for varying speeds with ρs  = 2.63 clearance in the sun bearing. Results are compared among finite element analysis (□), numerical integration (○), and harmonic balance (—). Unstable branches are denoted by (——).

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Figure 10

(a) The rms (mean removed) planet 1 rotational displacement u1 for varying speeds with ρc  = 5. The dynamic response is compared among the finite element analysis (□), numerical integration (○), and harmonic balance (—). Unstable branches are denoted by (——). (b) Dynamic tooth loads at the (upper) sun-planet and (lower) ring-planet meshes when the mesh frequency equals 1620 Hz. The response is at the upper branch of the resonance in (a). Tooth loads at the 1st, 2nd, 3rd, and 4th meshes are denoted by —, ——, …,—·—, respectively.

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Figure 11

The rms (mean removed) value sun Xs with ρs  = 2.63 in the sun bearing (a) within the speed range of 1000 to 2000 Hz and (b) within the speed range 1100 to 1450 Hz. Results are compared among finite element analysis (□), numerical integration (○), and harmonic balance (—). Unstable branches are denoted by (——).

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Figure 12

The rms (mean removed) planet 1 rotational displacement u1 for varying speeds (a) without clearance, and (b) with ρp  = 1.5 clearance in the planet 1 bearing. Results are compared among finite element analysis (——), numerical integration (…), and harmonic balance (—).

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Figure 13

(a) The rms (mean removed) planet 1 rotational displacement u1 for varying speeds when ρp  = 0, 1.5, and 3.0 in the planet 1 bearing. Unstable branches are denoted by (——). (b) Planet bearing forces of planets 1 (—), 2 (——), 3 (…), and 4 (—·—) when the mesh frequency equals 1760 Hz and ρp  = 3.0 in the planet 1 bearing. System damping ratios are 1.25, 1.25, 1.003, 1.68, 1.68, 3.37, and 2.46%, which are increased to eliminate chaos. The occurrences of tooth separation (TS) and bearing contact loss are labeled.

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Figure 14

(a) Poincare map of the velocity of planet 1 rotation without clearance computed by numerical integration. (b) Real and imaginary parts of Floquet multipliers without bearing clearance from mesh frequency 1692 Hz to 1775 Hz. The multipliers are symmetric about the real axis. The response is at the upper branch of the resonance in Fig. 1.

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Figure 15

(a) Poincare map of the velocity of planet 1 in rotation with ρp  = 1.5 in the planet 1 bearing computed by numerical integration. (b) Real and imaginary parts of Floquet multipliers of the planetary gear with ρp  = 1.5 in the planet 1 bearing from mesh frequency 1684 Hz to 1762 Hz. The multipliers are symmetric about the real axis. The response is at the upper branch of the resonance in Fig. 1.

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Figure 16

Sensitivity of the static deflection measures an , n = s, c, p of the sun (——), carrier (…), and planets (—) to the input torque

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Figure 17

(a) The rms (mean removed) carrier yc for varying speeds with Δc  = 5 μm when the input torque varies from 1130 (nominal torque) to 5650 Nm. Unstable branches are denoted by (——) and (b) percent of bearing contact in a mesh cycle for changing speeds under the conditions of (a). Results are calculated by the harmonic balance method.

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