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.

References

References
1.
Kahraman
,
A.
,
Kharazi
,
A. A.
, and
Umrani
,
M.
, 2003, “
A Deformable Body Dynamic Analysis of Planetary Gears With Thin Rims
,”
J. Sound Vib.
,
262
, pp.
752
768
.
2.
Hidaka
,
T.
,
Terauchi
,
Y.
, and
Nagamura
,
K.
, 1979, “
Dynamic Behavior of Planetary Gear (7th Report, Influence of the Thickness of the Ring Gear)
,”
Bull. JSME
,
22
, pp.
1142
1149
.
3.
Wu
,
X.
, and
Parker
,
R. G.
, 2008, “
Modal Properties of Planetary Gears With an Elastic Continuum Ring Gear
,”
ASME J. Appl. Mech.
,
75
, p.
031014
.
4.
Parker
,
R. G.
, and
Wu
,
X.
, 2010, “
Natural Modes of Planetary Gears With Unequally Spaced Planets and an Elastic Ring Gear
,”
J. Sound Vib.
,
329
, pp.
2265
2275
.
5.
Cunliffe
,
F.
,
Smith
,
J. D.
, and
Welbourn
,
D. B.
, 1974, “
Dynamic Tooth Loads in Epicyclic Gears
,”
J. Eng. Ind.
,
95
, pp.
578
584
.
6.
Botman
,
M.
, 1976, “
Epicyclic Gear Vibrations
,”
J. Eng. Ind.
,
96
, pp.
811
815
.
7.
August
,
R.
, and
Kasuba
,
R.
, 1986, “
Torsional Vibrations and Dynamic Loads in a Basic Planetary Gear System
,”
J. Vib., Acoust., Stress, Reliab. Des.
,
108
, pp.
348
353
.
8.
Kahraman
,
A.
, 1994, “
Natural Modes of Planetary Gear Trains
,”
J. Sound Vib.
,
173
, pp.
125
130
.
9.
Lin
,
J.
, and
Parker
,
R. G.
, 1999, “
Analytical Characterization of the Unique Properties of Planetary Gear Free Vibration
,”
ASME J. Vibr. Acoust.
,
121
, pp.
316
321
.
10.
Lin
,
J.
, and
Parker
,
R. G.
, 1999, “
Sensitivity of Planetary Gear Natural Frequencies and Vibration Modes to Model Parameters
,”
J. Sound Vib.
,
228
, pp.
109
128
.
11.
Lin
,
J.
, and
Parker
,
R. G.
, 2000, “
Structured Vibration Characteristics of Planetary Gears With Unequally Spaced Planets
,”
J. Sound Vib.
,
233
, pp.
921
928
.
12.
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
, pp.
305
311
.
13.
Lin
,
J.
, and
Parker
,
R. G.
, 2001, “
Natural Frequency Veering in Planetary Gears Under Design Parameter Variations
,”
Mech. Struct. Mach.
,
29
, pp.
411
429
.
14.
Ambarisha
,
V. K.
, and
Parker
,
R. G.
, 2006, “
Suppression of Planet Mode Response in Planetary Gear Dynamics Through Mesh Phasing
,”
ASME J. Vibr. Acoust.
,
128
, pp.
133
142
.
15.
Kiracofe
,
D.
, and
Parker
,
R. G.
, 2007, “
Structured Vibration Modes of General Compound Planetary Gear Systems
,”
ASME J. Vibr. Acoust.
,
129
, pp.
1
16
.
16.
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
.
17.
Kahraman
,
A.
, and
Blankenship
,
G. W.
, 1996, “
Interactions Between Commensurate Parametric and Forcing Excitations in a System With Clearance
,”
J. Sound Vib.
,
194
, pp.
317
336
.
18.
Parker
,
R. G.
,
Vijayakar
,
S. M.
, and
Imajo
,
T.
, 2000, “
Nonlinear Dynamic Response of a Spur Gear Pair: Modeling and Experimental Comparisons
,”
J. Sound Vib.
,
237
, pp.
435
455
.
19.
Bollinger
,
J. G.
, and
Harker
,
R. J.
, 1967, “
Instability Potential of High Speed Gearing
,”
J Ind. Math. Soc
,
17
(2)
, pp.
39
55
.
20.
Amabili
,
M.
, and
Rivola
,
A.
, 1997, “
Dynamic Analysis of Spur Gear Pairs: Steady-State Response and Stability of the SDOF Model With Time-Varying Meshing Damping
,”
Mech. Syst. Signal Process.
,
11
, pp.
375
390
.
21.
Benton
,
M.
, and
Seireg
,
A.
, 1981, “
Factors Influencing Instability and Resonances in Geared Systems
,”
J. Mech. Des.
,
103
, pp.
372
378
.
22.
Nataraj
,
C.
, and
Whitman
,
A. M.
, 1997, “
Parameter excitation effects in gear dynamics
,” American Society for Mechanical Engineers Design Engineering Technical Conferences, Sacramento, CA.
23.
Tordion
,
G. V.
, and
Gauvin
,
R.
, 1977, “
Dynamic Stability of a Two-Stage Gear Train Under the Influence of Variable Meshing Stiffnesses
,”
J. Eng. Ind.
,
99
, pp.
785
791
.
24.
Benton
,
M.
, and
Seireg
,
A.
, 1980, “
Normal Mode Uncoupling of Systems With Time Varying Stiffness
,”
J. Mech. Des.
,
102
, pp.
379
383
.
25.
Lin
,
J.
, and
Parker
,
R. G.
, 2002, “
Mesh Stiffness Variation Instabilities in Two-Stage Gear Systems
,”
ASME J. Vibr. Acoust.
,
124
, pp.
68
78
.
26.
Liu
,
G.
, and
Parker
,
R. G.
, 2008, “
Nonlinear Dynamics of Idler Gearsets
,”
Nonlinear Dyn.
,
53
, pp.
345
367
.
27.
Velex
,
P.
, and
Flamand
,
L.
, 1996, “
Dynamic Response of Planetary Trains to Mesh Parametric Excitations
,”
J. Mech. Des.
,
118
, pp.
7
14
.
28.
Lin
,
J.
, and
Parker
,
R. G.
, 2002, “
Planetary Gear Parametric Instability Caused by Mesh Stiffness Variation
,”
J. Sound Vib.
,
249
, pp.
129
145
.
29.
Bahk
,
C. J.
, and
Parker
,
R. G.
, 2011, “
Analytical Solution for the Nonlinear Dynamics of Planetary Gears
,”
ASME J. Comput. Nonlinear Dyn.
,
6
, p.
021007
.
30.
Wu
,
X.
, and
Parker
,
R. G.
, 2006, “
Vibration of Rings on a General Elastic Foundation
,”
J. Sound Vib.
,
295
, pp.
194
213
.
31.
Parker
,
R. G.
, and
Lin
,
J.
, 2004, “
Mesh Phasing Relationships in Planetary and Epicyclic Gears
,”
J. Mech. Des.
,
126
, pp.
365
370
.
32.
Parker
,
R. G.
, 2000, “
A Physical Explanation for the Effectiveness of Planet Phasing to Suppress Planetary Gear Vibration
,”
J. Sound Vib.
,
236
, pp.
561
573
.
33.
Yichao
Guo
and
Parker
,
R. G.
, 2011, “
Analytical Determination of Mesh Phase Relations in General Compound Planetary Gears
,”
Mechanism and Machine Theory
,
46
, pp.
1869
1887
.
34.
Nayfeh
,
A. H.
, and
Mook
,
D. T.
, 1979,
Nonlinear Oscillations
,
John Wiley
,
New York
.
You do not currently have access to this content.