Abstract
The dynamical behavior of a bouncing ball with a sinusoidally vibrating table is revisited in this paper. Based on the equation of motion of the ball, the mapping for period-1 motion is constructured and thereby allowing the stability and bifurcation conditions to be determined. Comparison with Holmes's solution [1] shows that our range of stable motion is wider, and through numerical simulations, our stability result is observed to be more accurate. The Poincaré mapping sections of the unstable period-1 motion indicate the existence of identical Smale horseshoe structures and fractals. For a better understanding of the stable and chaotic motions, plots of the physical motion of the bouncing ball superimposed on the vibration of the table are presented.
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References
Holmes, P. J., ‘The dynamics of repeated impacts with a sinusoidally vibrating table’, Journal of Sound and Vibration 84, 1982, 173–189.
Guckenheimer, J. and Holmes, P. J., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer-Verlag, New York, 1983.
Wood, L. A. and Byrne, K. P., ‘Analysis of a random repeated impact process’, Journal of Sound and Vibration 81, 1981, 329–325.
Everson, R. M., ‘Chaotic dynamics of a bouncing ball’, Physica D 19, 1986, 355–383.
Masri, F. and Caughey, T. K., ‘On the stability of the impact damper’, ASME Journal of Applied Mechanics 33, 1966, 587–592.
Shaw, S. W. and Holmes, P. J., ‘A periodically forced piecewise linear oscillator’, Journal of Sound and Vibration 90, 1983, 123–155.
Reithmeier, E., ‘Periodic solutions on nonlinear dynamical systems with discontinuities’, in Proceedings of the IUTAM Symposium on Nonlinear Dynamics in Engineering Systems, Stuttgart, 1989, pp. 249–256.
Han, R. P. S., Luo, A. C. J., and Deng, W., ‘Chaotic motion of a horizontal impact pair’, Journal of Sound and Vibration 181, 1995, 231–250.
Bapat, C. N. and Bapat, C., ‘Impact-pair under periodic excitation’, Journal of Sound and Vibration 120, 1987, 53–61.
Heiman, M. S., Sherman, P. J., and Bajaj, A. K., ‘On the dynamics and stability of an inclined impact pair’, Journal of Sound and Vibration 114, 1987, 535–547.
Heiman, S., Bajaj, A. K., and Sherman, P. J., ‘Periodic motions and bifurcation in dynamics of an inclined impact pair’, Journal of Sound and Vibration 124, 1988, 55–78.
Shaw, J. and Shaw, S. W., “The onset of chaos in a two degree-of-freedom impact system’, ASME Journal of Applied Mechanics 56, 1989, 168–174.
Bayly, V. and Virgin, L. N., ‘An experimental study of an impacting pendulum’, Journal of Sound and Vibration 164, 1993, 364–374.
Whiston, G. S., ‘Singularities in vibro-impact dynamics’, Journal of Sound and Vibration 152, 1992, 427–460.
Peterka, F. and Vacik, J., ‘Transition to chaotic motion in mechanical systems with impacts’, Journal of Sound and Vibration 154, 1992, 95–115.
Han, R. P. S. and Luo, A. C. J., ‘Period-n motion of a bouncing ball with a harmonically excited table’, in preparation.
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Luo, A.C.J., Han, R.P.S. The dynamics of a bouncing ball with a sinusoidally vibrating table revisited. Nonlinear Dyn 10, 1–18 (1996). https://doi.org/10.1007/BF00114795
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DOI: https://doi.org/10.1007/BF00114795