Abstract
Dynamics of a class of strongly nonlinear single degree of freedom oscillators is investigated. Their common characteristic is that they possess piecewise linear damping properties, which can be expressed in a general asymmetric form. More specifically, the damping coefficient and a constant parameter appearing in the equation of motion are functions of the velocity direction. This class of oscillators is quite general and includes other important categories of mechanical systems as special cases, like systems with Coulomb friction. First, an analysis is presented for locating directly exact periodic responses of these oscillators to harmonic excitation. Due to the presence of dry friction, these responses may involve intervals where the oscillator is stuck temporarily. Then, an appropriate stability analysis is also presented together with some quite general bifurcation results. In the second part of the work, this analysis is applied to several example systems with piecewise linear damping, in order to reveal the most important aspects of their dynamics. Initially, systems with symmetric characteristics are examined, for which the periodic response is found to be symmetric or asymmetric. Then, dynamical systems with asymmetric damping characteristics are also examined. In all cases, emphasis is placed on investigating the low forcing frequency ranges, where interesting dynamics is noticed. The analytical predictions are complemented with results obtained by proper integration of the equation of motion, which among other responses reveal the existence of quasiperiodic, chaotic and unbounded motions.
Similar content being viewed by others
References
Den Hartog, J. P., 'Forced vibration with combined Coulomb and viscous damping', Transactions of the American Society of Mechanical Engineers 53, 1930, 107–115.
Pratt, T. K. and Williams, R., 'Non-linear analysis for stick/slip motion', Journal of Sound and Vibration 74, 1981, 531–542.
Pierre, C., Ferri, A. A., and Dowell, E. H., 'Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method', ASME Journal of Applied Mechanics 52, 1985, 958–964.
Shaw, S. W., 'On the dynamic response of a system with dry friction', Journal of Sound and Vibration 108, 1986, 305–325.
Popp, K. and Stelter, P., 'Stick-slip vibrations and chaos', Philosophical Transactions of the Royal Society of London A332, 1990, 89–105.
Ibrahim, R. A., 'Friction-induced vibration, chatter, squeal and chaos', Applied Mechanics Reviews 47, 1994, 209–253.
Feeny, B. and Moon, F. C., 'Chaos in a forced dry friction oscillator: experiments and numerical modelling', Journal of Sound and Vibration 170, 1994, 303–323.
Leamy, M. J. and Perkins, N. C., 'Nonlinear periodic response of engine accessory drives with dry friction tensioners', ASME Journal of Vibration and Acoustics 120, 1998, 909–916.
Leine, R. I., Van Campen, D. H., De Kraker, A., and Van Den Steen, L., 'Stick-slip vibrations induced by alternate friction models', Nonlinear Dynamics 16, 1998, 41–54.
Natsiavas, S., 'Stability of piecewise linear oscillators with viscous and dry friction damping', Journal of Sound and Vibration 217, 1998, 507–522.
Gillespie, T. D., Fundamentals of Vehicle Dynamics, Society of Automotive Engineers, Warrendale, PA, 1992.
Dixon, J. C., Tires, Suspension and Handling, Society of Automotive Engineers, Warrendale, PA, 1996.
Nayfeh, A. H. and Mook, D. T., Nonlinear Oscillations, Wiley-Interscience, New York, 1979.
Nayfeh, A. H. and Balachandran, B., Applied Nonlinear Dynamics, Wiley-Interscience, New York, 1995.
Aronson, D. G., Chory, M. A., Hall, G. R., and McGehee, R. P., 'Bifurcations form an invariant circle for two-parameter families of maps of the plane: A computer assisted study', Communications in Mathematical Physics 83, 1982, 303–354.
Chang, S. I., Bajaj, A. K., and Krousgrill, C. M., 'Nonlinear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance', Nonlinear Dynamics 4, 1993, 433–460.
Mitsi, S., Natsiavas, S., and Tsiafis, I., 'Dynamics of nonlinear oscillators under simultaneous internal and external resonances', Nonlinear Dynamics 16, 1998, 23–39.
Natsiavas, S., 'Dynamics of multiple degree of freedom oscillators with colliding components', Journal of Sound and Vibration 165, 1993, 439–453.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Natsiavas, S., Verros, G. Dynamics of Oscillators with Strongly Nonlinear Asymmetric Damping. Nonlinear Dynamics 20, 221–246 (1999). https://doi.org/10.1023/A:1008398813070
Issue Date:
DOI: https://doi.org/10.1023/A:1008398813070