In this paper, we study bifurcations in systems with impact and friction, modeled with a rigid multibody approach. Knowledge from the field of nonlinear dynamics is therefore combined with theory from the field of non-smooth mechanics. We study the nonlinear dynamics of three commercial wooden toys. The toys show complex dynamical behavior but can be studied with one-dimensional maps, which allows for a thorough analysis of the bifurcations.
Begley, C. J. and Virgin L. N., 1997, “A detailed study of the low-frequency periodic behavior of a dry friction oscillator”, Journal of Dynamic Systems, Measurement, and Control119, 491-497.
2.
Beitelschmidt, M., 1999, Reibstöβe in Mehrkörpersystemen, Fortschr.-Ber. VDI Reihe 11, Nr. 275, VDI Verlag, Düsseldorf.
3.
Blazejczyk-Okolewska, B. and Kapitaniak, T., 1996, “Dynamics of Impact Oscillator with Dry Friction”, Chaos, Solitons and Fractals7(9), 1455-1459.
Canudas de Wit, C., Olsson, H., Åström, K. J., and Lipschinsky, P., 1995, “A new model for control of systems with friction”, IEEE Transactions on Automatic Control40(3), 419-425.
6.
Cottle, R. W. and Dantzig, G. B., 1968, “Complementary pivot theory of mathematical programming”, Linear Algebra and its Applications1, 103-125.
7.
Dankowicz, H. and Nordmark, A. B., 2000, “On the origin and bifurcations of stick-slip oscillations”, Physica D136(3-4), 280-302.
8.
di Bernardo, M., Feigin, M. I., Hogan, S. J., and Homer, M. E., 1999, “Local analysis of C-bifurcations in n-dimensional piecewise-smooth dynamical systems”, Chaos, Solitons and Fractals10(11), 1881-1908.
9.
Foale, S. and Bishop, R., 1994, “Bifurcations in impact oscillations”, Nonlinear Dynamics6, 285-299.
10.
Galvanetto, U. and Knudsen, C., 1997, “Event maps in a stick-slip system”, Nonlinear Dynamics13(2), 99-115.
11.
Glocker, Ch., 1995, Dynamik von Starrkörpersystemen mit Reibung und Stöβen, Fortschr.-Ber. VDI Reihe 18, Nr. 182, VDI Verlag, Düsseldorf.
12.
Glocker, Ch., 2000, “Scalar force potentials in rigid multibody systems” in Multibody Dynamics with Unilateral Contacts, eds. F. Pfeiffer, Ch. Glocker, CISM Courses and Lectures Vol. 421, pp. 69-146, Springer, Wien.
13.
Glocker, Ch., 2001, Set-Valued Force Laws, Dynamics of Non-Smooth Systems, Springer-Verlag, Berlin.
14.
Guckenheimer, J., and Holmes, P., 1983, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Vol. 42, Springer-Verlag, New York.
15.
Ivanov, A. P., 1996, “Bifurcations in Impact Systems”, Chaos, Solitons and Fractals7(10), 1615-1634.
16.
Leine, R. I. and Van Campen, D. H., 1999, “Fold bifurcations in discontinuous systems”, in Proceedings of DETC 99 ASME Design Engineering Technical Conferences, September 12-15, Las Vegas, CD-ROM, DETC99/VIB-8034.
17.
Leine, R. I. and Van Campen, D. H., 2000, “Discontinuous bifurcations of periodic solutions”, accepted for publication in Mathematical Modelling of Nonlinear Systems.
18.
Leine, R. I., Van Campen, D. H., and Van de Vrande, B. L., 2000, “Bifurcations in nonlinear discontinuous systems”, Nonlinear Dynamics23(2), 105-164.
19.
Leine, R. I., 2000, Bifurcations in Discontinuous Mechanical Systems of Filippov-Type. Ph.D. thesis, Eindhoven University of Technology, The Netherlands.
20.
Meijaard, J. P., 1996, “A mechanism for the onset of chaos in mechanical systems with motion-limiting stops”, Chaos, Solitons and Fractals7(10), 1649-1658.
21.
Moreau, J. J., 1988, “Unilateral contact and dry friction in finite freedom dynamics” in Non-Smooth Mechanics and Applications, J. J. Moreau, and P. D. Panagiotopoulos, eds. CISM Courses and Lectures Vol. 302, pp. 1-82, Springer, Wien.
22.
Natsiavas, S. and Gonzalez, H., 1992, “Vibration of harmonically excited oscillators with asymmetric constraints”, ASME Journal of Applied Mechanics59, 284-290.
23.
Nordmark, A. B., 1997, “Universal limit mapping in grazing bifurcations”, Physical Review E55(1), 266-270.
24.
Peterka, F., 1996, “Bifurcations and Transition Phenomena in an Impact Oscillator”, Chaos, Solitons and Fractals7(10), 1635-1647.
25.
Pfeiffer, F., 1984, “Mechanische Systeme mit unstetigen Übergängen”, Ingenieur-Archiv54, 232-240.
26.
Pfeiffer, F., 1991, “Dynamical systems with time-varying or unsteady structure”, Zeitschrift für Angewandte Mathematik und Mechanik71(4), T6-T22.
27.
Pfeiffer, F. and Glocker, Ch., 1996, Multibody Dynamics with Unilateral Contacts, Wiley, New York.
28.
Popp, K., Hinrichs, N., and Oestreich, M., 1995, “Dynamical behaviour of a friction oscillator with simultaneous self and external excitation”, in Sādhāna: Academy Proceedings in Engineering Sciences, Indian Academy of Sciences, Banglore, India, Part 2-4, Vol. 20, pp. 627-654.
29.
Roßmann, T., 1998, Eine Laufmachine für Rohre, Fortschr.-Ber. VDI Reihe 8, Nr. 732, VDI Verlag, Düsseldorf.
30.
Van de Vrande, B. L., Van Campen, D. H., and De Kraker, A., 1999, “An approximate analysis of dry-friction-induced stick-slip vibrations by a smoothing procedure”, Nonlinear Dynamics19(2), 157-169.
31.
Wiercigroch, M., 1996, “On modelling discontinuities in dynamic systems”, Machine Vibration5, 112-119.
32.
Yoshitake, Y. and Sueoka, A., 2000, “Forced self-exited vibration accompanied by dry friction”, in Applied Nonlinear Dynamics and Chaos of Mechanical Systems with Discontinuities, M. Wiercigroch and A. de Kraker, eds. World Scientific, Singapore.
