Nonlinear Dynamics

, Volume 23, Issue 4, pp 335–352 | Cite as

Numerical Study of a Forced Pendulum with Friction

  • C.-H. Lamarque
  • J. Bastien
Article

Abstract

We first describe the model of a forced pendulum with viscousdamping and Coulomb friction. Then we show that a unique local solutionof the mathematically well-posed problem exists. An adapted numericalscheme is built. Attention is devoted to the study of the nonlinearbehaviour of a pendulum via a numerical scheme with small constant timesteps. We describe the global behaviour of the free and forcedoscillations of the pendulum due to friction. We show that chaoticbehaviour occurs when friction is not too large. Lyapunov exponents arecomputed and a Melnikov relation is obtained as a limit of regularisedCoulomb friction. For larger friction, asymptotic behaviour correspondsto equilibrium.

Coulomb friction implicit Euler method numerical scheme maximal monotone operator pendulum chaos vibrations 

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References

  1. 1.
    Ueda, Y., 'Randomly transitional phenomena in systems governed by Duffing's equation', Journal of Statistical Physics 20, 1979, 181–196.Google Scholar
  2. 2.
    Lorenz, E. N., 'Deterministic nonperiodic flow', Journal of the Atmospheric Sciences 20, 1963, 130–141.Google Scholar
  3. 3.
    Sparrow, C., Bifurcations in the Lorenz Equations, Lecture Notes in Applied Mathematics, Vol. 41, Springer-Verlag, New York, 1982.Google Scholar
  4. 4.
    Lamarque, C.-H. and Malasoma, J.-M., 'Computation of basin of attraction for three coexisting attractors', European Journal of Mechanics A/Solids 11, 1992, 781–790.Google Scholar
  5. 5.
    Paoli, L., Schatzman, M., and Panet, M., 'Vibrations with an obstacle and finite number of degrees of freedom', in EUROMECH 280, Proceedings of the International Symposium on Identification of Nonlinear Mechanical Systems from Dynamic Tests, L. Jézéquel and C. H. Lamarque (eds.), Balkema, Rotterdam, 1992, pp. 159–164.Google Scholar
  6. 6.
    Pfeiffer, F. and Prestl, W., 'Hammering in diesel-engine driveline systems', Nonlinear Dynamics 5, 1994, 477–492.Google Scholar
  7. 7.
    Dowell, E.H. and Schwartz, H.B., 'Forced response of a cantilever beam with a dry friction damper attached, Part I: Theory', Journal of Sound and Vibration 91, 1983, 255–267.Google Scholar
  8. 8.
    Dowell, E. H. and Schwartz, H. B., 'Forced response of a cantilever beam with a dry friction damper attached, Part II: Experiment', Journal of Sound and Vibration 91, 1983, 269–291.Google Scholar
  9. 9.
    Whiteman, W. E. and Ferri, A. A., 'Displacement-dependent dry friction damping of a beam-like structure', Journal of Sound and Vibration 198, 1996, 313–329.Google Scholar
  10. 10.
    Li, G. X. and Paidoussis, M. P., 'Impact phenomena of rotor-casing dynamical systems', Nonlinear Dynamics 5, 1994, 53–70.Google Scholar
  11. 11.
    Whiston, G. S., 'Global dynamics of a vibro-impacting linear oscillator', Journal of Sound and Vibration 118, 1987, 395–429.Google Scholar
  12. 12.
    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.Google Scholar
  13. 13.
    Shaw, S. W., 'On the dynamic response of a system with dry friction', Journal of Sound and Vibration 108, 1986, 305–325.Google Scholar
  14. 14.
    Awrejcewicz, J. and Delfs, J., 'Dynamics of a self-excited stick-slip oscillator with two degrees of freedom. Part I. Investigation of equilibria', European Journal of Mechanics, A/Solids 9, 1990, 269–282.Google Scholar
  15. 15.
    Awrejcewicz, J. and Delfs, J., 'Dynamics of a self-excited stick-slip oscillator with two degrees of freedom. Part II. Slip-stick, slip-slip, stick-slip transitions, periodic and chaotic orbits', European Journal of Mechanics, A/Solids 9, 1990, 397–418.Google Scholar
  16. 16.
    Foale, S. and Bishop, S. R., 'Bifurcations in impact oscillations', Nonlinear Dynamics 6, 1994, 285–299.Google Scholar
  17. 17.
    Popp, K., Stelter, P., 'Stick-Slip vibrations ans chaos', Philosophical Transactions of the Royal Society of London, series A 332, 1990, 89–105.Google Scholar
  18. 18.
    Cone, K. M. and Zadoks, R. I., 'A numerical study of an impact oscillator with the addition of dry friction', Journal of Sound and Vibration 188, 1995, 659–683.Google Scholar
  19. 19.
    Deimling, K., Multivalued Differential Equations, de Gruyter, Berlin, 1992.Google Scholar
  20. 20.
    Moreau, J. J., 'Unilateral contact and dry friction in finite freedom dynamics', Non-Smooth Mechanics and Applications, J. J. Moreau and P. Panagiotopoulos (eds.), C.I.S.M. Courses and Lectures, Springer-Verlag, Berlin, 1988, pp. 1–81.Google Scholar
  21. 21.
    Brézis, H., Opérateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert, Mathematics Studies, Vol. 5, North-Holland, Amsterdam, 1973.Google Scholar
  22. 22.
    Schatzman, M., 'A class of nonlinear differential equations of second order in time', Nonlinear Analysis, Theory, Methods & Applications 2, 1978, 355–373.Google Scholar
  23. 23.
    Paoli, L., 'Analyse numérique de vibrations avec contraintes unilatérales', Ph.D. Thesis, University of St-Etienne, France, 1993.Google Scholar
  24. 24.
    Paoli, L. and Schatzman, M., 'A numerical scheme for a dynamical impact problem with loss of energy in finite dimension', Rapport interne 167 de l'Équipe d'Analyse Numérique Lyon Saint-Etienne, 1994.Google Scholar
  25. 25.
    Monteiro Marques, M. D. P., 'An existence, uniqueness and regularity study of the dynamics of systems with one-dimensional friction', European Journal of Mechanics, A/Solids 13, 1994, 277–306.Google Scholar
  26. 26.
    Schatzman, M., Bastien, J., and Lamarque, C.-H., 'An ill-posed mechanical problem with friction', European Journal of Mechanics 18, 1999, 415–420.Google Scholar
  27. 27.
    Chua, L. O., Komuro, M., and Matsumoto, T., 'The double scroll family, Parts I and II', IEEE Transactions on Circuits and Systems CAS-33, 1986, 1072–1118.Google Scholar
  28. 28.
    Doerner, R., Hübinger, B., Heng, H., and Martienssen, W., 'Approaching nonlinear dynamics by studying the motion of a pendulum. II. Analyzing chaotic motion', International Journal of Bifurcation and Chaos 4, 1994, 761–771.Google Scholar
  29. 29.
    Rumpel, R. J., 'On the qualitative behaviour of nonlinear oscillators with dry friction', Zeitschrift für angewandte Mathematik und Mechanik 76, 1996, S1-S2.Google Scholar
  30. 30.
    Müller, P. C., 'Calculation of Lyapunov exponents for dynamic systems with discontinuities', Chaos, Solitons & Fractals 5, 1995, 1671–1681.Google Scholar
  31. 31.
    Nqi, F. and Schatzman, M., 'Computation of Lyapunov exponents for a dynamical system with impact', Publications du Laboratoire d'Analyse Numérique Lyon-Saint Etienne, CNRS-UMR 5585, Université Lyon I, 1997.Google Scholar
  32. 32.
    Awrejcewicz, J. and Holicke, M., 'Detection of stick-slip chaotic oscillations using Melnikov's method', in Proceedings of ICNBC96, Łódz, Dobieszkow, J. Awrejcewicz and C.-H. Lamarque (eds.), ENTPE, 1996, pp. 65–68.Google Scholar
  33. 33.
    Wiggins, S., Global Bifurcation and Chaos, Analytical Methods, AMS, Vol. 73, Springer-Verlag, New York, 1988.Google Scholar
  34. 34.
    Lamarque, C.-H., Malasoma, J.-M., and Ouarzazi, M. N., 'Chaos in the convective flow of a fluid mixture in a porous medium', Nonlinear Dynamics 15, 1998, 83–102.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • C.-H. Lamarque
    • 1
  • J. Bastien
    • 1
  1. 1.Ecole Nationale des Travaux Publics de l'EtatLGM – URA CNRS 1652Vaulx-en-Velin CedexFrance

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