Archive of Applied Mechanics

, Volume 79, Issue 6–7, pp 667–677 | Cite as

Estimation of Lyapunov exponents for a system with sensitive friction model

  • Jerzy Wojewoda
  • Andrzej Stefański
  • Marian Wiercigroch
  • Tomasz Kapitaniak
Special Issue


Mathematical modelling and numerical analysis of a vibrating system with dry friction is presented. Three qualitatively different friction characteristics are considered. One of them is an example of so-called sensitive friction characteristic. Their influence on the dynamics on the attractor of the friction oscillator is investigated through bifurcational analysis. This analysis is supported by Lyapunov exponents estimated using approach for the systems with discontinuities. Theoretical background for such a synchronisation-based method of determining the largest Lyapunov exponent is explained. The results obtained through the proposed approach approximate the LLE with a good precision.


Friction Stick-slip Lyapunov exponents Synchronisation 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al-Bender F., Lampaert V., Swevers J.: A novel generic model at asperity level for dry friction force dynamics. Tribol. Lett. 16, 81–93 (2004)CrossRefGoogle Scholar
  2. 2.
    Armstrong-Hélouvry B.: Control of Machines with Friction. Kluwer Academic Press, Massachussets (1991)MATHGoogle Scholar
  3. 3.
    Armstrong-Hélouvry B., Dupont P., Canudas de Wit C.: A survey of models, analysis tools and compensation methods for the control of machines with friction. Automatica 30(7), 1083–1138 (1994)MATHCrossRefGoogle Scholar
  4. 4.
    Babakov, I.M.: Theory of vibrations. Gos. Izd. Tech.-Teoret. Lit. Moscow (in Russian) (1968)Google Scholar
  5. 5.
    Benettin G., Galgani L., Giorgilli A., Strelcyn J.M.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems: a method for computing all of them. Part I. Theory. Meccanica 15, 9–20 (1980)MATHCrossRefGoogle Scholar
  6. 6.
    Benettin G., Galgani L., Giorgilli A., Strelcyn J.M.: Lyapunov exponents for smooth dynamical systems and Hamiltonian systems: a method for computing all of them. Part II. Numerical application. Meccanica 15, 21–30 (1980)CrossRefGoogle Scholar
  7. 7.
    Bliman, P.-A., Sorine, M.: Easy-to-use realistic dry friction models for automatic control. In: Proceedings of 3rd European Control Conference, Rome, Italy, pp. 3788–3794 (1995)Google Scholar
  8. 8.
    Bo L.C., Pavelescu D.: The friction–speed relation and its influence on the critical velocity of stick–slip motion. Wear 82, 277–289 (1982)CrossRefGoogle Scholar
  9. 9.
    Canudas de Wit C., Olsson H., Åström K.J., Lischinsky P.: A new model for control of systems with friction. IEEE Trans. Autom. Control 40(3), 419–425 (1995)MATHCrossRefGoogle Scholar
  10. 10.
    Coulomb, C.A.: Théorie des machines simples. Mémoires de Mathématique et de Physique de l’Academie des Sciences, pp. 161–331 (1785)Google Scholar
  11. 11.
    Courtney-Pratt J.S., Eisner E.: The effect of a tangential force on the contact of metallic bodies. Proc. R. Soc. A238, 529–550 (1956)Google Scholar
  12. 12.
    Dahl, P.: A solid friction model. Technical Report TOR-0158(3107–18)-1, The Aerospace Corporation, El Segundo, CA (1968)Google Scholar
  13. 13.
    Den Hartog J.P.: Forced vibrations with combined Coulomb and viscous friction. Trans. ASME 53, 107–115 (1931)Google Scholar
  14. 14.
    De Souza S.L.T., Caldas I.L.: Calculation of Lyapunov exponents in systems with impacts. Chaos, Solitons Fractals 19(3), 569–579 (2004)MATHCrossRefGoogle Scholar
  15. 15.
    Haessig D.A., Friedland B.: On the modelling and simulation of friction. J. Dyn. Syst. Meas. Control Trans. ASME 113(3), 354–362 (1991)CrossRefGoogle Scholar
  16. 16.
    Hess D.P., Soom A.: Friction at a lubricated line contact operating at oscillating sliding velocities. J. Tribol. 112, 147–152 (1990)CrossRefGoogle Scholar
  17. 17.
    Hinrichs N., Oestreich M., Popp K.: Dynamics of oscillators with impact and friction. Chaos Solitons Fractals 4(8), 535–558 (1997)CrossRefGoogle Scholar
  18. 18.
    Hunt J.B., Torbe I., Spencer G.C.: The phase-plane analysis of sliding motion. Wear 8, 455–465 (1965)CrossRefGoogle Scholar
  19. 19.
    Jin L., Lu Q.-S., Twizell E.H.: A method for calculating the spectrum of Lyapunov exponents by local maps in non-smooth impact-vibrating systems. J. Sound Vib. 298(4–5), 1019–1033 (2006)CrossRefMathSciNetGoogle Scholar
  20. 20.
    Johannes V.I., Green M.A., Brockley C.A.: The role of the rate of application of the tangential force in determining the static friction coefficient. Wear 24, 381–385 (1973)CrossRefGoogle Scholar
  21. 21.
    Liang J.-W., Feeny B.F.: Dynamical friction behavior in a forced oscillator with a compliant contact. J. Appl. Mech. 65, 250–257 (1998)CrossRefGoogle Scholar
  22. 22.
    McMillan A.J.: A non-linear friction model for self-excited vibrations. J. Sound Vib. 205(3), 323–335 (1997)CrossRefMathSciNetGoogle Scholar
  23. 23.
    Müller P.: Calculation of Lyapunov exponents for dynamical systems with discontinuities. Chaos Solitons Fractals 5(9), 1671–1681 (1995)MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Oestreich, M.: Untersuchung von Schwingern mit nichtglatten Kennlinien. Fortsrchritt-Berichte VDI, Reiche 11: Schwingungstechnik, Nr. 258 (in German) (1998)Google Scholar
  25. 25.
    Olsson H., Åström K.J., Canudas de Wit C., Gäfvert M., Lischinsky P.: Friction models and friction compensation. Eur. J. Control 4(3), 176–195 (1998)MATHGoogle Scholar
  26. 26.
    Oseledec V.I.: A multiplicative ergodic theorem: Lyapunov characteristic numbers for dynamical systems. Trans. Moscow Math. Soc. 19, 197–231 (1968)MathSciNetGoogle Scholar
  27. 27.
    Popp K., Stelter P.: Non-linear oscillations of structures induced by dry friction. In: Schiehlen, W. (eds) Non-linear Dynamics in Engineering Systems, Springer, New York (1990)Google Scholar
  28. 28.
    Powell J., Wiercigroch M.: Influence of non-reversible Coulomb characteristics on the response of a harmonically excited linear oscillator. Mach. Vib. 1(2), 94–104 (1992)Google Scholar
  29. 29.
    Rabinowicz E.: The nature of the static and kinetic coefficients of friction. J. Appl. Phys. 22(12), 1373–1379 (1951)CrossRefGoogle Scholar
  30. 30.
    Shimada I., Nagashima T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys. 61(6), 1605–1616 (1979)MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Stefański A., Kapitaniak T.: Using chaos synchronization to estimate the largest Lyapunov exponent of non-smooth systems. Discrete Dyn. Nat. Soc. 4, 207–215 (2000)MATHCrossRefGoogle Scholar
  32. 32.
    Stefański A.: Estimation of the largest Lyapunov exponent in systems with impacts. Chaos Solitons Fractals 11(15), 2443–2451 (2000)MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    Stefański A., Wojewoda J., Wiercigroch M., Kapitaniak T.: Chaos caused by non-reversible dry friction. Chaos Solitons Fractals 16, 661–664 (2003a)MATHCrossRefGoogle Scholar
  34. 34.
    Stefański A., Kapitaniak T.: Estimation of the dominant Lyapunov exponent of non-smooth systems on the basis of maps synchronization. Chaos Solitons Fractals 15, 233–244 (2003b)MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Stefański A., Da̧browski A., Kapitaniak T.: Evaluation of the largest Lyapunov exponent in dynamical systems with time delay. Chaos Solitons Fractals 23, 1651–1659 (2005)MATHMathSciNetGoogle Scholar
  36. 36.
    Stefański, A., Wojewoda, J., Wiercigroch, M., Kapitaniak, T.: Regular and chaotic oscillations of friction force. Proc. Instn Mech. Engrs 220(C) 09305, 1–12 (2006)Google Scholar
  37. 37.
    Stribeck, R.: Die wesentlichen Eigenschaften der Gleit- und Rollenlager—The key qualities of sliding and roller bearings. Zeitschrift des Vereines Deutscher Ingenieure 46(38,39), 1342–1348, 1432–1437 (1902)Google Scholar
  38. 38.
    Wiercigroch M.: Comments on the study of a harmonically excited linear oscillator with a Coulomb damper. J. Sound Vib. 167(3), 560–563 (1993)CrossRefGoogle Scholar
  39. 39.
    Wiercigroch, M., Sin, V.W.T., Liew, Z.F.K.: Non-reversible dry friction oscillator: design and measurements. Proc. Instn Mech. Engrs 213(C), 527–534 (1999)Google Scholar
  40. 40.
    Wojewoda, J., Stefański, A., Wiercigroch, M., Kapitaniak, T.: Hysteretic effects of dry friction: modelling and experimental studies. Phil. Trans. of the R. Soc. A. (2007) doi:10.1098/rsta.2007.2125.
  41. 41.
    Wolf A.: Quantifying chaos with Lyapunov exponents. In: Holden, V. (eds) Chaos, pp. 273–290. Manchester University Press, Manchester (1986)Google Scholar

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Jerzy Wojewoda
    • 1
  • Andrzej Stefański
    • 1
  • Marian Wiercigroch
    • 2
  • Tomasz Kapitaniak
    • 1
  1. 1.Division of DynamicsTechnical University of ŁódźŁódźPoland
  2. 2.Centre for Applied Dynamics Research, School of Engineering, King’s CollegeUniversity of AberdeenAberdeenScotland, UK

Personalised recommendations