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.

References

References
1.
Oswald
,
F.
,
Zaretsky
,
E.
, and
Poplawski
,
J.
, 2012, “
Effects of Internal Clearance on Load Distribution and Life of Radially-Loaded Ball and Roller Bearings
,” STLE Tribol. Trans., Preprint.
2.
Kahraman
,
A.
, and
Singh
,
R.
, 1991, “
Nonlinear Dynamics of a Geared Rotor-Bearing System With Multiple Clearances
,”
J. Sound Vib.
,
144
, pp.
469
506
.
3.
Gurkan
,
N. E.
, and
Ozguven
,
H.
, 2007, “
Interactions Between Backlash and Bearing Clearance Nonlinearity in Geared Flexible Rotors
,”
ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, DETC2007–34101.
4.
Tiwari
,
M.
,
Gupta
,
K.
, and
Prakash
,
O.
, 2000, “
Effect of Radial Internal Clearance of a Ball Bearing on the Dynamics of a Balanced Horizontal Rotro
,”
J. Sound Vib.
,
238
(
5
), pp.
723
756
.
5.
Bai
,
C.-Q.
,
Xu
,
Q.-Y.
,
Zhang
,
X.-L.
, and
Qing
,
Y.
, 2006, “
Nonlinear Stability of Balanced Rotor Due to Effect of Ball Bearing Internal Clearance
,”
Appl. Math. Mech.
,
27
(
2
), pp.
175
186
.
6.
Kim
,
Y. B.
, and
Noah
,
S. T.
, 1990, “
Bifurcation Analysis for a Modified Jeffcott Rotor With Bearing Clearances
,”
Nonlinear Dyn.
,
1
, pp.
221
241
.
7.
Foale
,
S.
, and
Bishop
,
S. R.
, 1994, “
Bifurcations in Impact Oscillators
,”
Nonlinear Dyn.
,
6
, pp.
285
299
.
8.
Hinrichs
,
N.
,
Oestreich
,
M.
, and
Popp
,
K.
, 1997, “
Dynamics of Oscillators With Impact and Friction
,”
Chaos, Solitons Fractals
,
8
(
4
), pp.
535
558
.
9.
Nordmark
,
A. B.
, 1991, “
Non-Periodic Motion Caused by Grazing Incidence in an Impact Oscillator
,”
J. Sound Vib.
,
145
(
2
), pp.
279
297
.
10.
Peterka
,
F.
, 1996, “
Bifurcations and Transition Phenomena in an Impact Oscillator
,”
Chaos, Solitons Fractals
,
7
(
10
), pp.
1635
1647
.
11.
Nordmark
,
A. B.
, 2001, “
Existence of Periodic Orbits in Grazing Bifurcations of Impacting Mechanical Oscillators
,”
Nonlinearity
,
14
, p.
15171542
.
12.
Luo
,
A. C. J.
, 2010, “
Grazing and Chaos in a Periodically Forced, Piecewise Linear System
,”
ASME J. Vibr. Acoust.
,
128
, pp.
28
34
.
13.
Chin
,
W.
,
Ott
,
E.
,
Nusse
,
H. E.
, and
Grebogi
,
C.
, 1994, “
Grazing Bifurcation in Impact Oscillators
,”
Phys. Rev. E
,
50
(
6
), pp.
4427
4444
.
14.
Halse
,
C. K.
,
Wilson
,
R. E.
,
Bernardo
,
M. D.
, and
Homer
,
M. E.
, 2007, “
Coexisting Solutions and Bifurcations in Mechanical Oscillators With Backlash
,”
J. Sound Vib.
,
305
, pp.
854
885
.
15.
Ellermann
,
K.
, 2009, “
The Motion of Floating Systems: Nonlinear Dynamic in Periodic and Random Waves
,”
ASME J. Offshore Mech. Arct. Eng.
,
131
, pp.
041104
-1–041104-
7
.
16.
Blazejczyk-Okolewska
,
B.
, and
Kapitaniak
,
T.
, 1998, “
Co-Existing Attractors of Impact Oscillator
,”
Chaos, Solitons Fractals
,
9
(
8
), pp.
328
332
.
17.
Kahraman
,
A.
, and
Blankenship
,
G. W.
, 1997, “
Experiments on Nonlinear Dynamic Behavior of an Oscillator With Clearance and Periodically Time-Varying Parameters
,”
ASME J. Appl. Mech.
,
64
, pp.
217
226
.
18.
Parker
,
R. G.
,
Vijayakar
,
S. M.
, and
Imajo
,
T.
, 2000, “
Non-Linear Dynamic Response of a Spur Gear Pair: Modelling and Experimental Comparisons
,”
J. Sound Vib.
,
237
(
3
), pp.
435
455
.
19.
Seaman
,
R.
,
Johnson
,
C.
, and
Hamiltion
,
R.
, 1984, “
Component Inertial Effects on Transmission Design
,”
SAE Tech. Pap. Ser.
,
841686
, pp.
6.990
6.1008
.
20.
Botman
,
M.
, 1976, “
Epicyclic Gear Vibrations
,”
ASME J. Eng. Ind.
,
97
, pp.
811
815
.
21.
Lin
,
J.
, and
Parker
,
R. G.
, 2002, “
Planetary Gear Parametric Instability Caused by Mesh Stiffness Variation
,”
J. Sound Vib.
,
249
(
1
), pp.
129
145
.
22.
Parker
,
R. G.
,
Agashe
,
V.
, and
Vijayakar
,
S. M.
, 2000, “
Dynamic Response of a Planetary Gear System Using a Finite Element/Contact Mechanics Model
,”
ASME J. Mech. Des.
,
122
(
3
), pp.
304
310
.
23.
Sun
,
T.
, and
Hu
,
H.
, 2003, “
Nonlinear Dynamics of a Planetary Gear System With Multiple Clearances
,”
Mech. Mach. Theory
,
38
(
12
), pp.
1371
1390
.
24.
Al-shyyab
,
A.
, and
Kahraman
,
A.
, 2007, “
A Non-Linear Dynamic Model for Planetary Gear Sets
,”
Proc. Inst. Mech. Eng. Part K: J. Multi-Body Dyn.
,
221
, pp.
567
576
.
25.
Ambarisha
,
V. K.,
and
Parker
,
R. G.
, 2007, “
Nonlinear Dynamics of Planetary Gears Using Analytical and Finite Element Models
,”
J. Sound Vib.
,
302
, pp.
577
595
.
26.
Bahk
,
C.-J.
, and
Parker
,
R.
, 2011, “
Analytical Solution for the Nonlinear Dynamics of Planetary Gears
,”
ASME J. Comput. Nonlinear Dyn.
,
6
(
2
), p.
021007
.
27.
Zhu
,
F.
, and
Parker
,
R.
, 2005, “
Non-Linear Dynamics of a One-Way Clutch in Belt-Pulley Systems
,”
J. Sound Vib.
,
279
, pp.
285
308
.
28.
Blankenship
,
G. W.,
and
Kahraman
,
A.
, 1995, “
Steady State Forced Response of a Mechanical Oscillator With Combined Parametric Excitation and Clearance Type Non-Linearity
,”
J. Sound Vib.
,
185
(
5
), pp.
743
765
.
29.
Al-shyyab
,
A.
, and
Kahraman
,
A.
, 2005, “
Non-Linear Dynamic Analysis of a Multi-Mesh Gear Train Using Multi-Term Harmonic Balance Method: Subharmonic Motions
,”
J. Sound Vib.
,
279
(
1–2
), pp.
417
451
.
30.
Nayfeh
,
A.
, and
Balachandran
,
B.
, 1995,
Applied Nonlinear Dynamics
,
John Wiley and Sons
,
New York
.
31.
Raghothama
,
A.
, and
Narayanan
,
S.
, 1999, “
Bifurcation and Chaos in Geared Rotor Bearing System by Incremental Harmonic Balance Method
,”
J. Sound Vib.
,
226
(
3
), pp.
469
492
.
32.
Blair
,
K. B.,
Krousgrill
,
C. M.,
and
Farris
,
T. N.
, 1997, “
Harmonic Balance and Continuation Techniques in the Dynamic Analysis of Duffing’s Equation
,”
J. Sound Vib.
,
202
(
5
), p.
717731
.
33.
Seydel
,
R.
, 1994,
Practical Bifurcation and Stability Analysis: From Equilibrium to Chaos
,
Springer
,
New York, USA
.
34.
Lin
,
J.
, and
Parker
,
R. G.
, 1999, “
Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration
,”
ASME J. Vibr. Acoust.
,
121
(
3
), pp.
319
321
.
35.
Wu
,
X.
, and
Parker
,
R. G.
, 2012, “
Parametric Instability of Planetary Gears with Elastic Continuum Ring Gears
,” ASME J. Vibr. Acoust.
36.
Guo
,
Y.
, and
Parker
,
R. G.
, 2010, “
Dynamic Modeling and Analysis of a Spur Planetary Gear Involving Tooth Wedging and Bearing Clearance Nonlinearity
,”
Eur. J. Mech. A/Solids
,
29
, pp.
1022
1033
.
37.
Vijayakar
,
S. M.
, 2005, Calyx User’s Manual, http://ansol.com.
38.
Vijayakar
,
S. M.
, 1991, “
A Combined Surface Integral and Finite-Element Solution for a 3-Dimensional Contact Problem
,”
Int. J. Numer. Methods Eng.
,
31
(
3
), pp.
525
545
.
39.
Vijayakar
,
S. M.
, 1991, “
A Combined Surface Integral and Finite Element Solution for a Three-Dimensional Contact Problem
,”
Int. J. Numer. Methods Eng.
,
31
, pp.
524
546
.
40.
Kahraman
,
A.
, and
Vijayakar
,
S. M.
, 2001, “
Effect of Internal Gear Flexibility on the Quasi-Static Behavior of a Planetary Gear Set
,”
J. Mech. Des.
,
123
(
3
), pp.
408
415
.
41.
Thomsen
,
J.
, 2003,
Vibration and Stability: Advanced Theory, Analysis, and Tools
,
2nd ed.
,
Springer
,
New York
.
42.
Grippo
,
L.
,
Lampariello
,
F.
, and
Lucidi
,
S.
, 1986, “
A Nonmonotone Line Search Technique for Newton’s Method
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
,
23
(
4
), pp.
707
716
.
43.
Seydel
,
R.
, 1994,
Practical Bifurcation and Stability Analysis
,
Springer
,
New York
.
44.
Friedmann
,
P. P.
, 1990, “
Numerical Methods for the Treatment of Periodic Systems With Applications to Structural Dynamics and Helicopter Rotor Dynamics
,”
Comput. Struct.
,
35
(
4
), pp.
329
347
.
45.
Lin
,
J.
, and
Parker
,
R. G.
, 2000, “
Structured Vibration Characteristics of Planetary Gears With Unequally Spaced Planets
,”
J. Sound Vib.
,
233
(
5
), pp.
921
928
.
46.
Meirovitch
,
L.
, 1997,
Principles and Techniques of Vibrations
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
47.
Blankenship
,
G. W.
, and
Kahraman
,
A.
, 1996, “
Gear Dynamics Experiments, Part I: Characterization of Forced Response
,”
ASME Power Transmission and Gearing Conference
, San Diego.
48.
Blankenship
,
G. W.
, and
Kahraman
,
A.
, 1995, “
Steady State Force Response of a Mechanical Oscillator With Combined Parametric Excitation and Clearance Type Non-Linearity
,”
J. Sound Vib.
,
185
(
5
), pp.
743
765
.
49.
Swift
,
J. W.
, and
Wiesenfeld
,
K.
, 1984, “
Suppress of Period Doubling in Symmetric Systems
,”
Phys. Rev. Lett.
,
52
(
9
), pp.
705
708
.
You do not currently have access to this content.