Abstract
An analytical investigation is carried out on the free vibration of a two degree of freedom weakly nonlinear oscillator. Namely, the method of multiple time scales is first applied in deriving modulation equations for a van der Pol oscillator coupled with a Duffing oscillator. For the case of non-resonant oscillations, these equations are in standard normal form of a codimension two (Hopf-Hopf) bifurcation, which permits a complete analysis to be performed. Three different types of asymptotic states-corresponding to trivial, periodic and quasiperiodic motions of the original system-are obtained and their stability is analyzed. Transitions between these different solutions are also identified and analyzed in terms of two appropriate parameters. Then, effects of a coupling, a detuning, a nonlinear stiffness and a damping parameter are investigated numerically in a systematic manner. The results are interpreted in terms of classical engineering terminology and are related to some relatively new findings in the area of nonlinear dynamical systems.
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References
Henry, R. F. and Tobias, S. A., ‘Modes at rest and their stability in coupled non-linear systems’, Journal of Mechanical Engineering Science 3, 1961, 163–173.
Gilchrist, A. O., ‘The free oscillations of conservative quasilinear systems with two degrees of freedom’, International Journal of Mechanical Science 3, 1962, 286–311.
Rosenberg, R. M., ‘On non-linear vibrations of systems with many degrees of freedom’, Advances in Applied Mechanics 9, 1966, 155–242.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley, New York, 1979.
Hayashi, C. and Kuramitsu, M., ‘Self-excited oscillations in a system with two degrees of freedom’, Mem. Faculty of Engineering of Kyoto University 36, 1974, 87–104.
Guevara, M. R. and Glass, L., ‘Phase locking, period doubling bifurcations and chaos in a mathematical model of a periodically driven oscillator: A theory of entrainment of biological oscillators and the generation of cardiac dysrhythmias’, Journal of Mathematical Biology 14, 1982, 1–23.
Knobloch, E. and Guckenheimer, J., ‘Convective transitions induced by a varying aspect ratio’, Physical Review A 27, 1983, 408–417.
Wang, X. J. and Nicolis, G., ‘Bifurcation phenomena in coupled chemical oscillators: Normal form analysis and numerical simulations’, Physica D 26, 1987, 140–155.
Rand, R. H. and Holmes, P. J., ‘Bifurcation of periodic motions in two weakly coupled van der Pol oscillators’, International Journal of Non-Linear Mechanics 15, 1980, 387–390.
Storti, D. W. and Rand, R. H., ‘Dynamics of two strongly coupled van der Pol oscillators’, International Journal of Non-Linear Mechanics 17, 1982, 143–152.
Chakraborty, T. and Rand, R. H., ‘The transition from phase locking to drift in a system of two weakly coupled van der Pol oscillators’, International Journal of Non-Linear Mechanics 23, 1988, 369–376.
Bajaj, A. K. and Sethna, P. R., ‘Bifurcations in three-dimensional motions of articulated tubes’, Journal of Applied Mechanics 49, 1982, 606–618.
Shaw, J. and Shaw, S. W., ‘Instabilities and bifurcations in a rotating shaft’, Journal of Sound and Vibration 132, 1989, 227–244.
Aronson, D. G., Doedel, E. J., and Othmer, H. G., ‘An analytical and numerical study of the bifurcations in a system of linearly coupled oscillators’, Physica D 25, 1987, 20–104.
Aronson, D. G., Ermentrout, G. B., and Kopell, N., ‘Amplitude response of coupled oscillators’, Physica D 41, 1990, 403–449.
Kevrekidis, I. G., Schmidt, L. D., and Aris, R., ‘Some common features of periodically forced reacting systems ’, Chemical Engineering Science 41, 1986, 1263–1276.
Taylor, M. A. and Kevrekidis, I. G., ‘Some common dynamic features of coupled reacting systems’, Physica D 51, 1991, 274–292.
Mansour, W. M., ‘Quenching of limit cycles of a van der Pol oscillator’, Journal of Sound and Vibration 35, 1972, 395–405.
Tondl, A., ‘Quenching of self-excited vibrations: Equilibrium aspects’, Journal of Sound and Vibration 43, 1975, 251–260.
Asfar, K. R., ‘Quenching of self-excited vibrations’, ASME Journal of Vibration and Acoustics 111, 1989, 130–133.
Natsiavas, S., ‘Dynamic vibration absorbers for a class of self-excited mechanical systems’, Journal of Applied Mechanics 1992 (accepted for publication).
Nayfeh, A. H. and Balachandran, B., ‘Modal interactions in dynamical and structural systems’, Applied Mechanics Reviews 42, 1989, S175–201.
Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, New York, 1983.
Nayfeh, A. H. and Sanchez, N. E., ‘Bifurcations in a forced softening Duffing oscillator’, International Journal of Non-Linear Mechanics 24, 1989, 483–497.
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Natsiavas, S. Free vibration of two coupled nonlinear oscillators. Nonlinear Dyn 6, 69–86 (1994). https://doi.org/10.1007/BF00045433
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DOI: https://doi.org/10.1007/BF00045433