The transverse vibrations of a circular disk of uniform thickness rotating about its axis with constant angular velocity are analyzed when the disk is subject to a space-fixed linear spring. Combination resonances are shown to occur at rotational speeds different from the classical critical speeds at which, in the linear analysis, the spinning disk is unable to support arbitrary spatially fixed transverse loads. Using the method of multiple scales, the authors determine a set of eight nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of four interacting modes. The symmetry of the system and the loading conditions are reflected in the symmetry of the modulation equations. Using convenient variable transformations, the authors obtain an equivalent set of equations, which they use to prove that no single- mode solution is possible for this case. The stability characteristics of the trivial solution for all of the possible interactions among four of the lowest modes are studied.
Benson, R.C., 1983, "Observations on the steady-state solution of an extremely flexible spinning disk with a transverse load," Journal of Applied Mechanics50, 525-530.
2.
Chen, J.S., 1994, "Stability analysis of a spinning elastic disk under a stationary concentrated edge load," Journal of Applied Mechanics61, 788-792.
3.
Chen, J.S., 1997, "Parametric resonance of a spinning disk under space-fixed pulsating edge loads," Journal of Applied Mechanics64, 139-143.
4.
Chen, J.S. and Bogy, D.B., 1992, "Effects of load parameters on the natural frequencies and stability of a flexible spinning disk with a stationary load system," Journal of Applied Mechanics59, S230-S235.
5.
Cole, K.A. and Benson, R.C., 1988, "A fast eigenfunction approach for computing spinning disk deflections," Journal of Applied Mechanics55, 453-457.
6.
Eftstathiades, G.J., 1971, "A new approach to the large-deflection vibrations of imperfect circular disks using Galerkin's procedure," Journal of Sound and Vibration16, 231-253.
7.
Hadian, J. and Nayfeh, A.H., 1990, "Modal interaction in circular plates," Journal of Sound and Vibration142, 279-292.
8.
Huang, S.C. and Chiou W.J., 1997, "Modeling and vibration analysis of spinning-disk and moving-head assembly in computer storage systems," Journal of Vibration and Acoustics119, 185-191.
9.
Iwan, W.D. and Moeller, T.L., 1976, "The stability of a spinning elastic disk with a transverse load system," Journal of Applied Mechanics43, 485-490.
10.
Iwan, W.D. and Stahl K.J., 1973, "The response of an elastic disk with a moving mass system ," Journal of Applied Mechanics40, 445-450.
11.
Leissa, A.W., 1969, Vibrations of Plates, NASA SP-160, Washington, DC.
12.
Mignolet, M.P., Eick, C.D., and Harish, M.V., 1996, "Free vibrations of flexible rotating disk," Journal of Sound and Vibration196, 537-577.
13.
Nayfeh, A.H., 1973, Perturbation Methods, Wiley , New York.
14.
Nayfeh, A.H., 1981, Introduction to Perturbation Techniques, Wiley, New York.
15.
Nayfeh, A.H., 2000, Nonlinear Interactions, Wiley , New York.
16.
Nayfeh, A.H. and Balachandran, B. , 1995, Applied Nonlinear Dynamics , Wiley, New York.
17.
Nayfeh, A.H. and Chin, C.-M., 1999a, Perturbation Methods With Maple, http://www./esm.vt.edu/∼anayfeh/.
18.
Nayfeh, A.H. and Chin, C.-M., 1999b, Perturbation Methods With Mathematica, http://www./esm.vt.edu/∼anayfeh/.
19.
Nayfeh, A.H., Jilani, A., and Manzione P., 1999, "Transverse vibrations of a centrally clamped rotating circular disk," in ASME 17th Biennial Conference on Mechanical Vibration and Noise, Symposium on Systems With Nonsmooth Dynamics, DETC99/VIB-8177, Las Vegas, NV, September 12-15.
20.
Nayfeh, A.H. and Mook, D.T., 1979, Nonlinear Oscillations, Wiley , New York.
21.
Nayfeh, T.A. and Vakakis, A.F., 1992, "Subharmonic travelling waves in a geometrically non-linear circular plates," International Journal ofNon-Linear Mechanics29, 233-245.
22.
Nowinski, J.L., 1964, "Nonlinear transverse vibrations of a spinning disk," Journal of Applied Mechanics31, 72-78.
23.
Raman, A. and Mote Jr.,C. D. , 1999, "Non-linear oscillations of circular plates near a critical speed resonance," International Journal of Non-Linear Mechanics34, 139-157.
24.
Renshaw, A.A. and Mote Jr, C.D., 1995, "A perturbation solution for the flexible rotating disk: Nonlinear equilibrium and stability under transverse loading," Journal of Sound and Vibration183, 309-326.
25.
Shen, I.Y. and Mote Jr., C.D., 1991, "On the mechanism of instability of a circular plate under a rotating spring-mass-dashpot system," Journal of Sound and Vibration178, 307-318.
26.
Shen, I.Y. and Song, Y., 1996, "Stability and vibration of a rotating circular plate subjected to stationary in plane edge loads," Journal of Applied Mechanics63, 121-127.
27.
Sridhar, S., Mook, D.T., and Nayfeh, A.H., 1975, "Nonlinear resonances in the forced responses of plates, Part 1: Symmetric responses of circular plates," Journal of Sound and Vibration41, 359-373.
28.
Sridhar, S., Mook, D.T., and Nayfeh, A.H., 1978, "Non-linear resonances in the forced responses of plates, Part II: Asymmetric responses of circular plates," Journal of Sound and Vibration59, 159-170.
29.
Yang, L. and Hutton, S.G., 1998, "Nonlinear vibrations of elastically-constrained rotating disks," Journal of Vibration and Acoustics120, 475-483.
30.
Yasuda, K., Torii, T., and Shimizu, T., 1992, "Self-excited oscillations of a circular disk rotating in air," JSME International Journal35, 347-352.
31.
Yasuda, K. and Uno, H., 1983, "Multimode response of a circular membrane," Bulletin of the JSME26, 1050-1058.
