Abstract
Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1 : 1 resonance in the (1, 2) and (3, 1) plate modes.
Article PDF
Similar content being viewed by others
Explore related subjects
Discover the latest articles, news and stories from top researchers in related subjects.Avoid common mistakes on your manuscript.
References
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Sathyamoorthy, M., ‘Nonlinear vibration of plates — a review’, The Shock and Vibration Digest 15, 1983, 3–16.
Sathyamoorthy, M., ‘Nonlinear vibration analysis of plates: A review and survey of current developments’, Applied Mechanics Review 40, 1987, 1553–1561.
Miles, J. W., ‘Resonant motions of a spherical pendulum’, Physica D 11, 1984, 309–323.
Tousi, S. and Bajaj, A. K., ‘Period-doubling bifurcations and modulated motions in forced mechanical systems’, ASME Journal of Applied Mechanics 107, 1985, 446–452.
Miles, J. W., ‘Resonantly forced motion of two quadratically coupled oscillators’, Physica D 13, 1984, 247–260.
Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, Applied Mechanics Review 42, 1989, S175-S201.
Sridhar, S., Mook, D. T., and Nayfeh, A. H., ‘Nonlinear resonances in the forced responses of plates, Part II: Asymmetric responses of circular plates’, Journal of Sound and Vibration 59, 1978, 159–170.
Maewal, A., ‘Chaos in a harmonically excited elastic beam’, ASME Journal of Applied Mechanics 53, 1986, 625–632.
Yasuda, K. and Torii, T., ‘Multi-mode response of a square membrane’, JSME International Journal 30, 1987, 963–969.
Johnson, J. M. and Bajaj, A. K., ‘Amplitude modulated and chaotic dynamics in resonant motion of strings’, Journal of Sound and Vibration 128, 1989, 87–107.
Yang, X. L. and Sethna, P. R., ‘Nonlinear phenomena in forced vibrations of a nearly square plate-antisymmetric case’, Journal of Sound and Vibration 155, 1992, 413–441.
Yasuda, K. and Asano, T., ‘Nonlinear forced oscillations of a rectangular membrane with degenerate modes’, Bulletin of JSME 29, 1986, 3090–3095.
Hale, J. K., Ordinary Differential Equations, Wiley-Interscience, New York, 1969.
Maewal, A., ‘Miles' evolution equations for axisymmetric shells: Simple strange attractors in structural dynamics’, International Journal of Non-linear Mechanics 21, 1987, 433–438.
Doedel, E., AUTO: Software for continuation and bifurcation problems in ordinary differential equations, Report, Department of Applied Mathematics, California Institute of Technology, 1986.
Pai, P. F. and Nayfeh, A. H., ‘A nonlinear composite plate theory’, Nonlinear Dynamics 2, 1991, 445–477.
Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983.
Bajaj, A. K. and Johnson, J. M., ‘On the amplitude dynamics and ‘crisis’ in resonant motion of stretched strings’, Philosophical Transactions of the Royal Society London A 338, 1992, 1–41.
Grebogi, C., Ott, E., and Yorke, J. A., ‘Crises, sudden changes in chaotic attractors, and transient chaos’, Physica D 7, 1983, 181–200.
Sil'nikov, L. P., ‘A contribution to the problem of the structure of an extended neighborhood of a rough equilibrium state of saddle-focus type’, Mathematics USSR Sbornik 10, 1970, 91–102.
Glendenning, P. and Sparrow, C., ‘Local and global behavior near homoclinic orbits’, Journal of Statistical Physics 35, 1984, 645–696.
Bajaj, A. K. and Johnson, J. M., ‘Asymptotic techniques and complex dynamics in weakly non-linear forced mechanical systems’, International Journal of Non-Linear Mechanics 25, 1990, 211–226.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chang, S.I., Bajaj, A.K. & Krousgrill, C.M. Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance. Nonlinear Dyn 4, 433–460 (1993). https://doi.org/10.1007/BF00053690
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00053690