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Research Papers

Analytical Solution for the Nonlinear Dynamics of Planetary Gears

[+] Author and Article Information
Cheon-Jae Bahk

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

Robert G. Parker1

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

1

Corresponding author.

J. Comput. Nonlinear Dynam 6(2), 021007 (Oct 22, 2010) (15 pages) doi:10.1115/1.4002392 History: Received October 30, 2009; Revised May 16, 2010; Published October 22, 2010; Online October 22, 2010

Planetary gears are parametrically excited by the time-varying mesh stiffness that fluctuates as the number of gear tooth pairs in contact changes during gear rotation. At resonance, the resulting vibration causes tooth separation leading to nonlinear effects such as jump phenomena and subharmonic resonance. This work examines the nonlinear dynamics of planetary gears by numerical and analytical methods over the meaningful mesh frequency ranges. Concise, closed-form approximations for the dynamic response are obtained by perturbation analysis. The analytical solutions give insight into the nonlinear dynamics and the impact of system parameters on dynamic response. Correlation between the amplitude of response and external torque demonstrates that tooth separation occurs even under large torque. The harmonic balance method with arclength continuation confirms the perturbation solutions. The accuracy of the analytical and harmonic balance solutions is evaluated by parallel finite element and numerical integration simulations.

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References

Figures

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Figure 1

Rotational lumped parameter model of planetary gear system

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Figure 2

Vibration modes of planetary gear rotational model for parameters in Table 1

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Figure 3

Steady state response of (a) RMS (mean removed) sun rotational deflection and (b) sun mean rotation. Both increasing and decreasing speed sweeps are performed. (—) NI; (…) FE analysis.

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Figure 4

Waterfall spectra of sun and planet rotational deflection for decreasing speed by ((a) and (c)) finite element and ((b) and (d)) numerical integration.

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Figure 5

Tooth separation function and mesh deflection. (a) In-phase and (b) out-of-phase (by π) sun-planet and ring-planet mesh deflections.

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Figure 6

RMS (mean removed) planet rotational deflection from numerical integration for (a) the first and (b) the second distinct primary resonance for full and limited harmonics of the mesh stiffness variations. (—) Full harmonics of stiffness, (- - -) the first and second harmonics of stiffness, (⋯) The first harmonic of stiffness.

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Figure 7

Waterfall spectra of planet rotational deflection for (a) only the first, (b) up to the second, and (c) full harmonics of the mesh stiffness variations by numerical integration

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Figure 8

RMS (mean removed) sun and planet rotational deflection for primary resonance of ((a) and (c)) the first and ((b) and (d)) the second distinct mode by NI, HB, and MMS. Unstable solutions are shown as dashed lines.

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Figure 9

RMS (mean removed) planet rotational deflection for subharmonic resonance of the second distinct mode. The dashed line is the unstable HB solution.

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Figure 10

Waterfall spectra of planet rotational deflection for subharmonic resonance by finite element for decreasing speed

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Figure 11

rms (mean removed) sun rotational deflection for subharmonic resonance of the second distinct mode by MMS and NI. Both increasing and decreasing speed are performed by numerical integration.

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Figure 12

Sun-planet tooth separation time and points denoting 5% and 10% deviation between perturbation and numerical integration solutions. (▽) subharmonic and (●) primary for the second distinct mode.

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Figure 13

RMS (mean removed) sun rotational deflection for (a) second harmonic excitation and (b) superharmonic resonance by NI, HB, and MMS

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Figure 14

Sun response time history for different resonances at mesh frequency 2200 Hz. (⋯) Second harmonic excitation only, (- - -) superharmonic resonance (only excitation at mesh frequency), sum of second harmonic excitation, and superharmonic resonance, and (—) resonance excited by first and second harmonics of mesh stiffness.

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