Parametric instabilities of in-plane bending vibrations of a rotating ring coupled to multiple, discrete, rotating, time-varying stiffness spring-sets of general geometric description are investigated in this work. Instability boundaries are identified analytically using perturbation analysis and given as closed-form expressions in the system parameters. Ring rotation and time-varying stiffness significantly affect instability regions. Different configurations with a rotating and nonrotating ring, and rotating spring-sets are examined. Simple relations governing the occurrence and suppression of instabilities are discussed for special cases with symmetric circumferential spacing of spring-sets. These results are applied to identify possible conditions of ring gear instability in example planetary gears.
Issue Section:
Technical Papers
1.
Hoppe
, R.
, 1871, “The Bending Vibration of a Circular Ring
,” Crelles J. Math.
, 73
, pp. 158
–170
.2.
Love
, A. E. H.
, 1927, 1944, A Treatise of the Mathematical Theory of Elasticity
, Dover
, New York
.3.
Seidel
, B. S.
, and Erdelyi
, E. A.
, 1964, “On the Vibration of a Thick Ring in Its Own Plane
,” ASME J. Eng. Ind.
0022-0817, 86
, pp. 240
–244
.4.
Evensen
, D. A.
, 1966, “Nonlinear Flexural Vibrations of Thin Circular Rings
,” ASME J. Appl. Mech.
0021-8936, 33
, pp. 553
–560
.5.
Rao
, S. S.
, and Sundararajan
, V.
, 1969, “In-Plane Flexural Vibrations of Circular Rings
,” ASME J. Appl. Mech.
0021-8936, 91
(3
), pp. 620
–625
.6.
Rao
, S. S.
, 1971, “Three-Dimensional Vibrations of a Ring on an Elastic Foundation
,” Aeronaut. J.
0001-9240, 75
, pp. 417
–419
.7.
Kirkhope
, J.
, 1976, “Simple Frequency Expression for the In-Plane Vibration of Thick Circular Rings
,” J. Acoust. Soc. Am.
0001-4966, 59
(1
), pp. 86
–89
.8.
Kirkhope
, J.
, 1977, “In-Plane Vibration of a Thick Circular Ring
,” J. Sound Vib.
0022-460X, 50
(2
), pp. 219
–227
.9.
Kuhl
, W.
, 1942, “Measurements to the Theories of Resonant Vibrations of Circular Rings of Arbitrary Wall Thickness
,” Akust. Z.
0365-382X, 7
, pp. 125
–152
.10.
Lincoln
, J. W.
, and Volterra
, E.
, 1967, “Experimental and Theoretical Determination of Frequencies of Elastic Toroids
,” Exp. Mech.
0014-4851, 24
, pp. 211
–217
.11.
Hawkings
, D. L.
, 1977, “A Generalized Analysis of the Vibration of Circular Rings
,” J. Sound Vib.
0022-460X, 54
(1
), pp. 67
–74
.12.
Wu
, X.
, and Parker
, R. G.
, “Vibration of Rings on a General Elastic Foundation
,” Journal of Sound and Vibration (in press).13.
Carrier
, G. F.
, 1945, “On the Vibrations of the Rotating Ring
,” Q. Appl. Math.
0033-569X, 3
(3
), pp. 235
–245
.14.
Johnson
, D. C.
, 1952, “Free Vibrations of a Rotating Elastic Body
,” Aircr. Eng.
0002-2667, 24
, pp. 234
–236
.15.
Bert
, C. W.
, and Chen
, T. L. C.
, 1978, “On Vibration of a Thick Flexible Ring Rotating at High Speed
,” J. Sound Vib.
0022-460X, 61
(4
), pp. 517
–570
.16.
Endo
, M.
, Hatamura
, K.
, Sakata
, M.
, and Taniguchi
, O.
, 1984, “Flexural Vibration of a Thin Rotating Ring
,” J. Sound Vib.
0022-460X, 92
(2
), pp. 261
–272
.17.
Bickford
, W. B.
, and Reddy
, E. S.
, 1985, “On the In-Plane Vibrations of Rotating Rings
,” J. Sound Vib.
0022-460X, 101
(1
), pp. 13
–22
.18.
Kung
, L. E.
, and Soedel
, W.
, 1986, “Free Vibration of a Pneumatic Tire-Wheel Unit Using a Rings on an Elastic Foundation and a Finite Element Model
,” J. Sound Vib.
0022-460X, 107
(2
), pp. 181
–194
.19.
Huang
, S. C.
, and Hsu
, B. S.
, 1992, “An Approach to the Dynamic Analysis of Rotating Tire-Wheel-Suspension Units
,” J. Sound Vib.
0022-460X, 156
(3
), pp. 505
–519
.20.
Dohrmann
, C. R.
, 1998, “Dynamics of a Tire-Wheel-Suspension Assembly
,” J. Sound Vib.
0022-460X, 210
(5
), pp. 627
–642
.21.
Morgan
, C. D.
, 2000, “Comments on Dynamics of a Tire-Wheel-Suspension Assembly
,” J. Sound Vib.
0022-460X, 232
(2
), pp. 473
–476
.22.
Kim
, W.
, and Chung
, J.
, 2002, “Free Non-Linear Vibration of a Rotating Thin Ring With the In-Plane and Out-of-Plane Motions
,” J. Sound Vib.
0022-460X, 258
(1
), pp. 167
–178
.23.
Huang
, S. C.
, and Soedel
, W.
, 1987, “Effect of Coriolis Acceleration on the Free and Forced In-Plane Vibrations of Rotating Rings on Elastic Foundation
,” J. Sound Vib.
0022-460X, 115
(2
), pp. 253
–274
.24.
Huang
, S. C.
, and Soedel
, W.
, 1987, “Response of Rotating Rings to Harmonic and Periodic Loading and Comparison With the Inverted Problem
,” J. Sound Vib.
0022-460X, 118
(2
), pp. 253
–270
.25.
Metrikin
, A. V.
, and Tochilin
, M. V.
, 2000, “Steady-State Vibrations of an Elastic Ring Under Moving Load
,” J. Sound Vib.
0022-460X, 232
(3
), pp. 511
–524
.26.
Mote
, C. D.
, Jr., 1970, “Stability of Circular Plates Subjected to Moving Loads
,” J. Franklin Inst.
0016-0032, 290
(4
), pp. 329
–344
.27.
Iwan
, W. D.
, and Stahl
, K. J.
, 1973, “The Response of an Elastic Disk With a Moving Mass System
,” ASME J. Appl. Mech.
0021-8936, 40
(2
), pp. 445
–451
.28.
Iwan
, W. D.
, and Moeller
, T. L.
, 1976, “The Stability of a Spinning Elastic Disk With a Transverse Load System
,” ASME J. Appl. Mech.
0021-8936, 43
(3
), pp. 485
–490
.29.
Shen
, I. Y.
, and Mote
, C. D.
, Jr., 1991, “On the Mechanisms of Instability of a Circular Plate Under a Rotating Spring-Mass-Dashpot System
,” J. Sound Vib.
0022-460X, 148
(2
), pp. 307
–318
.30.
Shen
, I. Y.
, and Mote
, C. D.
, Jr., 1991, “Parametric Resonances of a Circular Plate With Inclusion Subjected to a Rotating Spring
,” J. Sound Vib.
0022-460X, 149
(1
), pp. 164
–169
.31.
Shen
, I. Y.
, and Mote
, C. D.
, Jr., 1992, “Parametric Excitation Under Multiple Excitation Parameters: Asymmetric Plates Under a Rotating Spring
,” Int. J. Solids Struct.
0020-7683, 29
(8
), pp. 1019
–1032
.32.
Vangipuram Canchi
, S.
, and Parker
, R. G.
, “Parametric Instability of a Circular Ring Subjected to Moving Springs
,” Journal of Sound and Vibration
(in press).33.
Vangipuram Canchi
, S.
, 2004, “Parametric Instability of a Circular Ring Subjected to Moving Springs
,” Masters thesis, Ohio State University.34.
Vangipuram Canchi
, S.
, and Parker
, R. G.
, “Effect of Ring-Planet Mesh Phasing and Contact Ratio on Parametric Instabilities of a Planetary Gear Ring
,” (in preparation).35.
Bickford
, W. B.
, and Maganty
, S. P.
, 1986, “On the Out-of-Plane Vibrations of Thick Rotating Rings
,” J. Sound Vib.
0022-460X, 110
(1
), pp. 121
–127
.36.
Parker
, R. G.
, and Lin
, Y.
, 2001, “Parametric Instability of Axially Moving Media Subjected to Multifrequency Tension and Speed Fluctuation
,” ASME J. Appl. Mech.
0021-8936, 68
(1
), pp. 49
–57
.37.
Ozguven
, N. H.
, and Houser
, D. R.
, 1988, “Mathematical Models Used in Gear Dynamics
,” J. Sound Vib.
0022-460X, 121
(3
), pp. 383
–411
.38.
Parker
, R. G.
, Vijayakar
, S. M.
, and Imajo
, T.
, 2000, “Non-Linear Dynamic Response of a Spur Gear Pair: Modeling and Experimental Comparisons
,” J. Sound Vib.
0022-460X, 237
(3
), pp. 435
–455
.39.
Blankenship
, W. G.
, and Kahraman
, A.
, 1996, “Gear Dynamics Experiments, Part I: Characterization of Forced Response
,” Proc. ASME Power Transmission and Gearing Conference, San Diego.40.
Parker
, R. G.
, and Lin
, J.
, 2004, “Mesh Phasing Relationships in Planetary and Epicyclic Gears
,” J. Mech. Des.
1050-0472, 126
, pp. 365
–370
.Copyright © 2006
by American Society of Mechanical Engineers
You do not currently have access to this content.
