Abstract
The behaviour of a multi-degree-of-freedom vibro-impact system is studied using a 2 degree-of-freedom impact oscillator as a motivating example. A multi-modal model is used to simulate the behaviour of the system, and examine the complex dynamics which occurs when both degrees of freedom are subjected to a motion limiting constraint. In particular, the chattering and sticking behaviour which occurs for low forcing frequencies is discussed. In this region, a variety of non-smooth events can occur, including newly studied phenomena such as sliding bifurcations. In this paper, the multiple non-smooth events which can occur in the 2 degree-of-freedom system are categorised, and demonstrated using numerical simulations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Wagg, D. J. and Bishop, S. R., ‘Dynamics of a two degree of freedom vibro-impact system with multiple motion limiting constraints’, International Journal of Bifurcation and Chaos 14(1), 2004, 119–140.
Shaw, J. and Shaw, S. W., ‘The onset of chaos in a two-degree of freedom impacting system’, Journal of Applied Mechanics 56, 1989, 168–174.
Hogan, S. J. and Homer, M. E., ‘Graph theory and piecewise smooth dynamical systems of arbitrary dimension’, Chaos, Solitons and Fractals 10(11), 1999, 1869–1880.
Lenci, S. and Rega, G., ‘Regular nonlinear dynamics and bifurcations of an impacting system under general periodic excitation’, Nonlinear Dynamics 34, 2004, 249–268.
Masri, S. F., ‘Theory of the dynamic vibration neutraliser with motion-limiting stops’, ASME, Journal of Applied Mechanics 39, 1972, 563–568.
Chatterjee, S., Mallik, A. K., and Ghosh, A., ‘On impact dampers for non-linear vibrating systems’, Journal of Sound and Vibration, 187(3), 1995, 403–420.
Neilson, R. D. and Gonsalves, D. H., ‘Chaotic motion of a rotor system with a bearing clearance’, in Applications of Fractals and Chaos, A. J. Crilly, R. A. Earnshaw, and H. Jones (eds.), Springer-Verlag, Berlin, 1993, pp. 285–303.
Theodossiades, S. and Natsiavas, S., ‘Periodic and chaotic dynamics of motor-driven gear-pair system with backlash’, Chaos, Solitons and Fractals 12, 2001, 2427–2440.
Cusumano, J. P. and Bai, B.-Y., ‘Period-infinity periodic motions, chaos and spatial coherence in a 10 degree of freedom impact oscillator’, Chaos, Solitons and Fractals 3, 1993, 515–536.
Nigm, M. M. and Shabana, A. A., ‘Effect of an impact damper on a multi-degree of freedom system’, Journal of Sound and Vibration 89(4), 1983, 541–557.
Pfeiffer, F. and Glocker, C., Multibody Dynamics with Unilateral Contacts, Wiley, New York, 1996.
Lamarque, C. H. and Janin, O., ‘Modal analysis of mechanical systems with impact non-linearities: Limitations to a modal superposition’, Journal of Sound and Vibration 235(4), 2000, 567–609.
Natsiavas, S., ‘Dynamics of multiple-degree-of-freedom oscillators with colliding components’, Journal of Sound and Vibration 165(3), 1993, 439–453.
Pun, D., Lua, S. L., Law, S. S., and Cao, D. Q., ‘Forced vibration of a multidegree impact oscillator’, Journal of Sound and Vibration 213(3), 1998, 447–466.
Shaw, S. W. and Holmes, P. J., ‘A periodically forced piecewise linear oscillator’, Journal of Sound and Vibration 90(1), 1995, 129–155.
Luo, G. W. and Xie, J. H., ‘Hopf bifurcations and chaos of a two-degree-of-freedom vibro-impact system in two strong resonance cases’, Non-linear Mechanics 37, 2002, 19–34.
Budd, C. J. and Dux, F., ‘Chattering and related behaviour in impact oscillators’, Philosophical Transactions of the Royal Society of London, Series A 347, 1994, 365–389.
Nordmark, A. B. and Kisitu, R. E. M., ‘On Chattering Bifurcations in 1DOF Impact Oscillator Models’, Preprint Royal Institute of Technology, Sweden, 2003.
Wagg, D. J. and Bishop, S. R., ‘Chatter, sticking and chaotic impacting motion in a two-degree of freedom impact oscillator’, International Journal of Bifurcation and Chaos 11(1), 2001, 57–71.
Toulemonde, C. and Gontier, C., ‘Sticking motions of impact oscillators’, European Journal of Mechanics A: Solids 17(2), 1998, 339–366.
Di Bernardo, M., Johansson, K. H., and Vasca, F., ‘Self-oscillations and sliding in relay feedback systems: Symmetry and bifurcations’, International Journal of Bifurcation and Chaos 4(11), 2001, 1121–1140.
Wagg, D. J., ‘Rising phenomena and the multi-sliding bifurcation in a two-degree of freedom impact oscillator’, Chaos, Solitions and Fractals 22(3), 2003, 541–548.
Meirovitch, L., Analytical Methods in Vibration, McGraw-Hill, New York, 1967.
Thompson, J. M. T. and Stewart, H. B., Nonlinear Dynamics and Chaos, Wiley, Chichester, 2002.
Janin, O. and Lamarque, C. H., ‘Comparison of several numerical methods of mechanical systems with impacts’, International Journal for Numerical Methods in Engineering 51, 2001, 1101–1132.
Wagg, D. J., ‘Periodic sticking motion in a two-degree of freedom impact oscillator’, International Journal of Non-Linear Mechanics 40(8), 2005, 1076–1087.
Di Bernardo, M., Kowalczyk, P., and Nordmark, A., ‘Sliding bifurcations: A novel mechanism for the onset of chaos in dry friction oscillators’, International Journal of Bifurcation and Chaos 13(10), 2003, 2935–2948.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Wagg, D.J. Multiple Non-Smooth Events in Multi-Degree-of-Freedom Vibro-Impact Systems. Nonlinear Dyn 43, 137–148 (2006). https://doi.org/10.1007/s11071-006-0757-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11071-006-0757-7