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Bifurcation and chaos analysis of multistage planetary gear train

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Abstract

A nonlinear time-varying dynamic model for a multistage planetary gear train, considering time-varying meshing stiffness, nonlinear error excitation, and piece-wise backlash nonlinearities, is formulated. Varying dynamic motions are obtained by solving the dimensionless equations of motion in general coordinates by using the varying-step Gill numerical integration method. The influences of damping coefficient, excitation frequency, and backlash on bifurcation and chaos properties of the system are analyzed through dynamic bifurcation diagram, time history, phase trajectory, Poincaré map, and power spectrum. It shows that the multi-stage planetary gear train system has various inner nonlinear dynamic behaviors because of the coupling of gear backlash and time-varying meshing stiffness. As the damping coefficient increases, the dynamic behavior of the system transits to an increasingly stable periodic motion, which demonstrates that a higher damping coefficient can suppress a nonperiodic motion and thereby improve its dynamic response. The motion state of the system changes into chaos in different ways of period doubling bifurcation, and Hopf bifurcation.

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Abbreviations

b :

backlash

c :

damping

e :

error of transmission

F :

force

g :

nonlinear displacement function

I :

rotary inertia

K,k :

stiffness

M,m :

mass

N :

number of planet

r :

radius

t :

time

T :

torque

u :

displacement

x :

relative displacement

Z :

number of tooth

α :

pressure angle

θ :

angular displacement

ξ :

damping coefficient

τ :

dimensionless time

φ :

phase angle

Ω,ω :

frequency

b:

base circle

c:

carrier

in:

input

out:

output

p:

planet gear

r:

ring gear

s:

sun gear

(n):

number of stage

T:

matrix transpose

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (No. 51105280) and the Fundamental Research Funds for the Central Universities of China (No. 2012208020206).

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Authors and Affiliations

  1. School of Power and Mechanical Engineering, Wuhan University, Wuhan, 430072, China

    Sheng Li, Qingming Wu & Zhiqiang Zhang

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  1. Sheng Li
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  2. Qingming Wu
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  3. Zhiqiang Zhang
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Correspondence to Zhiqiang Zhang.

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Li, S., Wu, Q. & Zhang, Z. Bifurcation and chaos analysis of multistage planetary gear train. Nonlinear Dyn 75, 217–233 (2014). https://doi.org/10.1007/s11071-013-1060-z

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