Skip to main content
SpringerLink
Log in

Existence and stability of limit cycles in a delayed dry-friction oscillator

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

Periodic, chaotic, chattering, and bifurcation behavior are fundamental consequences of the nonsmooth nature of systems with dry friction. This work is concerned with the analysis of a single degree of freedom system which is additionally damped by a delayed dry friction device. We get a complete set of closed-form expressions to describe the dynamics of the delay-induced phenomena exhibited by the system. The conditions to determine the existence and stability of limit cycles are clearly defined. This analysis is addressed in the context of both classic stability theory for nonlinear systems and the qualitative theory of Piecewise Smooth Dynamical Systems. Through exhaustive numerical simulations the effectiveness of the set of closed-form expressions is confirmed. Excellent agreement was found between the numerical and analytical results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chatterjee, S.: Time-delayed feedback control of friction-induced instability. Int. J. Non-Linear Mech. 42(9), 1127–1143 (2007). doi:10.1016/j.ijnonlinmec.2007.08.002

    MATH  Google Scholar 

  2. Cheng, G., Zu, J.W.: A numerical study of a dry friction oscillator with parametric and external excitations. J. Sound Vib. 287(1–2), 329–342 (2005). doi:10.1016/j.jsv.2004.11.003

    Article  Google Scholar 

  3. Colombo, A., di Bernardo, M., Hogan, S.J., Kowalczyk, P.: Complex dynamics in a hysteretic relay feedback system with delay. J. Nonlinear Sci. 17(2), 85–108 (2007). doi:10.1007/s00332-005-0745-y

    Article  MathSciNet  MATH  Google Scholar 

  4. Dankowicz, H., Nordmark, A.B.: On the origin and bifurcations of stick-slip oscillations. Physica D 136(3–4), 280–302 (2000). doi:10.1016/S0167-2789(99)00161-X

    Article  MathSciNet  MATH  Google Scholar 

  5. Das, J., Mallik, A.: Control of friction driven oscillation by time-delayed state feedback. J. Sound Vib. 297(3–5), 578–594 (2006). doi:10.1016/j.jsv.2006.04.013

    Article  Google Scholar 

  6. di Bernardo, M., Budd, C.J., Champneys, A.R.: Normal form maps for grazing bifurcations in n-dimensional piecewise-smooth dynamical systems. Physica D 160(3–4), 222–254 (2001). doi:10.1016/S0167-2789(01)00349-9

    Article  MathSciNet  MATH  Google Scholar 

  7. di Bernardo, M., Kowalczyk, P., Nordmark, A.: Bifurcations of dynamical systems with sliding: derivation of normal-form mappings. Physica D 170(3–4), 175–205 (2002). doi:10.1016/S0167-2789(02)00547-X

    Article  MathSciNet  MATH  Google Scholar 

  8. di Bernardo, M., Kowalczyk, P., Nordmark, A.: Sliding bifurcations: a novel mechanism for the sudden onset of chaos in dry friction oscillators. Int. J. Bifurc. Chaos 13(10), 2935–2948 (2003). doi:10.1142/S021812740300834X

    Article  MATH  Google Scholar 

  9. di Bernardo, M., Budd, C., Champneys, A., Kowalczyk, P.: Piecewise-smooth Dynamical Systems: Theory and Applications. Applied Mathematical Science, vol. 163. Springer, Berlin (2008)

    MATH  Google Scholar 

  10. Feeny, B., Guran, A., Hinrichs, N., Popp, K.: A historical review on dry friction and stick-slip phenomena. Appl. Mech. Rev. 51(5), 321–341 (1998). doi:10.1115/1.3099008

    Article  Google Scholar 

  11. Feigin, M.: Forced Oscillations in Systems with Discontinuous Nonlinearities. Nauka, Moscow (1994) (in Russian)

    Google Scholar 

  12. Hartog, J.D.: Forced vibrations with combined coulomb and viscous friction. ASME Appl. Mech. 53(9), 107–115 (1931)

    Google Scholar 

  13. Heckl, M.A., Abrahams, I.D.: Active control of friction driven oscillations. J. Sound Vib. 193(1), 417–426 (1996). doi:10.1006/jsvi.1996.0285

    Article  Google Scholar 

  14. Hundal, M.: Response of a base excited system with coulomb and viscous friction. J. Sound Vib. 64(3), 371–378 (1979). doi:10.1016/0022-460X(79)90583-2

    Article  MATH  Google Scholar 

  15. Ibrahim, R.A.: Friction-induced vibration, chatter, squeal, and chaos. Part I: mechanics of contact and friction. Appl. Mech. Rev. 47(7), 209–226 (1994). doi:10.1115/1.3111079

    Article  Google Scholar 

  16. Ibrahim, R.A.: Friction-induced vibration, chatter, squeal, and chaos. Part II: dynamics and modeling. Appl. Mech. Rev. 47(7), 227–253 (1994). doi:10.1115/1.3111080

    Article  Google Scholar 

  17. Ji, J.C., Leung, A.Y.T.: Resonances of a non-linear SDOF system with two time-delay in linear feedback control. J. Sound Vib. 253(5), 985–1000 (2002). doi:10.1006/jsvi.2001.3974

    Article  MathSciNet  Google Scholar 

  18. Khalil, H.: Nonlinear Systems, 3rd edn. Prentice-Hall, Upper Saddle River (2000)

    Google Scholar 

  19. Levitan, E.S.: Forced oscillation of a spring-mass system having combined coulomb and viscous damping. J. Acoust. Soc. Am. 32(10), 1265–1269 (1960). doi:10.1121/1.1907893

    Article  MathSciNet  Google Scholar 

  20. Li, Y., Feng, Z.C.: Bifurcation and chaos in friction-induced vibration. Commun. Nonlinear Sci. Numer. Simul. 9(6), 633–647 (2004). doi:10.1016/S1007-5704(03)00058-3

    Article  MathSciNet  MATH  Google Scholar 

  21. Lopez, I., Busturia, J.M., Nijmeijer, H.: Energy dissipation of a friction damper. J. Sound Vib. 278(3), 539–561 (2004). doi:10.1016/j.jsv.2003.10.051

    Article  Google Scholar 

  22. Maccari, A.: Vibration control for the primary resonance of a cantilever beam by a time delay state feedback. J. Sound Vib. 259(2), 241–251 (2003). doi:10.1006/jsvi.2002.5144

    Article  MathSciNet  Google Scholar 

  23. Oestreich, M., Hinrichs, N., Popp, K.: Bifurcation and stability analysis for a non-smooth friction oscillator. Arch. Appl. Mech. 66(5), 301–314 (1996). doi:10.1007/BF00795247

    Article  MATH  Google Scholar 

  24. Osorio, G.: Complex behavior in impacting systems. Ph.D. thesis, Department of Computer and Systems Engineering. University of Naples Federico II, Naples, Italy (2007)

  25. Plaut, R.H., Hsieh, J.C.: Non-linear structural vibrations involving a time delay in damping. J. Sound Vib. 117(3), 497–510 (1987). doi:10.1016/S0022-460X(87)80068-8

    Article  MathSciNet  Google Scholar 

  26. Press, W.H., Teukolsky, S.A., Vettering, W.T., Flannery, B.R.: Numerical Recipes in C: The Art of Scientific Computing, 2nd edn. Cambridge University Press, Cambridge (1992)

    MATH  Google Scholar 

  27. Shampine, L.F., Thompson, S.: Solving ddes in matlab. Appl. Numer. Math. 37(3), 441–458 (2001). doi:10.1016/S0168-9274(00)00055-6

    Article  MathSciNet  MATH  Google Scholar 

  28. Sipahi, R., Olgac, N.: Active vibration suppression with time delayed feedback. J. Vib. Acoust. 125(3), 384–388 (2003). doi:10.1115/1.1569942

    Article  Google Scholar 

  29. Slotine, J.J.E., Li, W.: Applied Nonlinear Control. Englewood Cliffs, Prentice-Hall (1991)

    MATH  Google Scholar 

  30. Vidyasagar, M.: Nonlinear Systems Analysis, 2nd edn. Classics in Applied Mathematics, vol. 42. Prentice-Hall, Upper Saddle River (1992)

    MATH  Google Scholar 

  31. Wang, Y., Wang, D., Chai, T.: Modeling and control compensation of nonlinear friction using adaptive fuzzy systems. Mech. Syst. Signal Process. 23(8), 2445–2457 (2009). doi:10.1016/j.ymssp.2009.05.006

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dep. of Structural Engineering, University of Naples “Federico II”, Via Claudio 21, 80125, Naples, Italy

    Julián M. Londoño & Giorgio Serino

  2. Dep. of Systems and Computer Engineering, University of Naples “Federico II”, Via Claudio 21, 80125, Naples, Italy

    Mario di Bernardo

  3. Department of Engineering Mathematics, University of Bristol, Queen’s Building, University Walk, BS8 1TR, Bristol, UK

    Mario di Bernardo

Authors
  1. Julián M. Londoño
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giorgio Serino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Mario di Bernardo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Julián M. Londoño.

Additional information

The first author thanks the Programme Alβan, the European Union Programme of High Level scholarships for Latin America, for the financial support under the Scholarship E06D101622CO.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Londoño, J.M., Serino, G. & di Bernardo, M. Existence and stability of limit cycles in a delayed dry-friction oscillator. Nonlinear Dyn 67, 483–496 (2012). https://doi.org/10.1007/s11071-011-9997-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-011-9997-2

Keywords

Search

Navigation