Abstract
In some parameter ranges, the dynamics of a forced oscillator with Coulomb friction dependent on both displacement and velocity is reducible to the dynamics of a one-dimensional map. In numerical simulations, period-doubling bifurcations are observed for the oscillator. In this bifurcation procedure, the map arising from the Coulomb model may not have ‘standard’ form. The bifurcation sequence of the Coulomb model is compared to that of the standard one-dimensional maps to see if it exhibits ‘universal’ behavior. All observed components of the bifurcation sequence fit the universal sequence, although some universal events are not witnessed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Grabec, I., ‘Chaos generated by the cutting process’, Physics Letters A 117 (8), 1986, 384–386.
Popp, K. and Stelter, P., ‘Nonlinear oscillations of structures induced by dry friction’, in W. Schiehlen (ed.), Nonlinear Dynamics in Engineering Systems, Berlin-Heidelberg: Springer-Verlag, 1990.
Carlson, J. M. and Langer, J. S., ‘Mechanical model of an carthquake fault’, Physical Review A 40 (11), 1989, 6470–6484.
Den Hartog, J. P., ‘Forced vibrations with combined Coulomb and viscous friction’, Transactions of the ASME 83, 1931, 107–115.
Shaw, S. W., ‘On the dynamic response of a system with dry friction’, Journal of Sound and Vibration 108(2), 1986, 305–325.
Anderson, J. R. and Ferri, A. A., ‘Behavior of a single-degree-of-freedom system with a generalized friction law’, Journal of Sound and Vibration 140(2), 1990, 287–304.
Feeny, B., ‘Chaos and Friction’, PhD Thesis, Theoretical and Applied Mechanics, Cornell University, 1990.
Feeny, B. and Moon, F. C., ‘Chaos in a dry-friction oscillator: experiment and numerical modeling’, submitted to the Journal of Sound and Vibration.
Feeny, B., ‘A nonsmooth Coulomb friction oscillator’, to appear in Physica D.
Feeny, B. F. and Moon, F. C., ‘Autocorrelation on symbol dynamics for a chaotic dry-friction oscillator’, Physics Letters A 141(8, 9), 1989, 397–400.
Devaney, R. L., An Introduction to Chaotic Dynamical Systems, Addison-Wesley, Redwood City, California, 1987.
Feigenbaum, M. J., ‘Quantitative universality for a class of nonlinear transformations’, Journal of Statistical Physics 19(1), 1978, 25–52.
Metropolis, N., Stein, M. L., and Stein, P. R., ‘On finite limit sets for transformations on the unit interval’, Journal of Combinatorial Theory 15(1), 1973, 25–44.
Schuster, H. G., Deterministic Chaos: an Introduction, VCH Publishers, Weinheim, Federal Republic of Germany, 1988.
Szczygielski, W. and Schweitzer, G., ‘Dynamics of a high-speed rotor touching a boundary’, in G. Bianchi and W. Schichlen (eds.), Dynamics of Multibody Systems, IUTAM/IFToMM Symposium Udine 1985, Springer, Berlin Heidelberg 1986, 287–298.
Oka, H. and Kokubu, H., ‘An approach to constrained equations and strange attractors’, Patterns and Waves-Qualitative Analysis of Nonlinear Differential Equations, 1986, 607–630.
Zak, Michail, ‘Terminal attractors in neural networks’, Neural Networks 2, 1989, 259–274.
Coffman, K., McCormick, W. D., and Swinney, L., ‘Multiplicity in a chemical reaction with one-dimensional dynamics’, Physical Review Letters 56(10), 1986, 999–1002.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Feeny, B.F., Moon, F.C. Bifurcation sequences of a Coulomb friction oscillator. Nonlinear Dyn 4, 25–37 (1993). https://doi.org/10.1007/BF00047119
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF00047119