Skip to main content
SpringerLink
Log in

The equilibrium stability for a smooth and discontinuous oscillator with dry friction

  • Research paper
  • Published:
Acta Mechanica Sinica Aims and scope Submit manuscript

Abstract

In this paper, we investigate the equilibrium stability of a Filippov-type system having multiple stick regions based upon a smooth and discontinuous (SD) oscillator with dry friction. The sets of equilibrium states of the system are analyzed together with Coulomb friction conditions in both \((f_{\mathrm{n}}, f_{\mathrm{s}})\) and \((x, {\dot{x}})\) planes. In the stability analysis, Lyapunov functions are constructed to derive the instability for the equilibrium set of the hyperbolic type and LaSalle’s invariance principle is employed to obtain the stability of the non-hyperbolic type. Analysis demonstrates the existence of a thick stable manifold and the interior stability of the hyperbolic equilibrium set due to the attractive sliding mode of the Filippov property, and also shows that the unstable manifolds of the hyperbolic-type are that of the endpoints with their saddle property. Numerical calculations show a good agreement with the theoretical analysis and an excellent efficiency of the approach for equilibrium states in this particular Filippov system. Furthermore, the equilibrium bifurcations are presented to demonstrate the transition between the smooth and the discontinuous regimes.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  1. Shaw, S.W.: On the dynamic response of a system with dry friction. J. Sound Vib. 108, 305–325 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  2. Liu, X.J., Wang, D.J., Chen, Y.S.: Self-excited vibration of the shell-liquid coupled system induced by dry friction. Acta Mech. Sin. 11, 373–382 (1995)

    Article  MATH  Google Scholar 

  3. Elmer, F.J.: Nonlinear dynamics of dry friction. J. Phys. A 30, 6057–6063 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  4. Lamarque, C.H., Bastien, J.: Numerical study of a forced pendulum with friction. Nonlinear Dyn. 23, 335–352 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  5. Cheng, G., Zu, J.W.: Dynamics of a dry friction under two frequency excitations. J. Sound Vib. 275, 591–603 (2004)

    Article  Google Scholar 

  6. Wiercigroch, M., Pavlovskaia, E.: Engineering applications of non-smooth dynamics. Nonlinear Dyn. Phenom. Mech. SMIA 181, 211–273 (2012)

    Article  MathSciNet  Google Scholar 

  7. Luo, A.C.J., Gegg, B.C.: Periodic motions in a periodically forced oscillator moving on an oscillating belt with dry friction. ASME J. Commun. Nonlinear Dyn. 1, 212–220 (2006)

    Article  Google Scholar 

  8. Leine, R.I., Van Campen, D.H., De Kraker, A.: Stick-slip vibrations induced by alternate friction models. Nonlinear Dyn. 16, 41–54 (1998)

    Article  MATH  Google Scholar 

  9. Ding, W.J., Fan, S.C., Lu, M.W.: A new criterion for occurrence of stick-slip motion in drive mechanism. Acta Mech. Sin. 16, 273–281 (2000)

  10. van de Wouw, N., van den Heuvel, M.N., Nijmeijer, H.: Performance of an automatic ball balancer with dry friction. Int. J. Bifurc. Chaos 15, 65–82 (2005)

    Article  MATH  Google Scholar 

  11. Leine, R.I., van de Wouw, N.: Stability and Convergence of Mechanical Systems with Unilateral Constraints. Lecture Notes in Applied and Computational Mechanics, vol. 36. Springer, Berlin (2008)

  12. Shevitz, D., Paden, B.: Lyapunov stability theory of nonsmooth systems. IEEE Trans. Autom. Control 39, 1910–1914 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bacciotti, A., Ceragioli, F.: Stability and stabilization of discontinuous systems and nonsmooth lyapupnov functions. ESIAM Control Optim. Calc. Var. 4, 361–376 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Basseville, S., Léger, A., Pratt, E.: Investigation of the equilibrium states and their stability for a simple model with unilateral contact and Coulomb friction. Arch. Appl. Mech. 73, 409–420 (2003)

    Article  MATH  Google Scholar 

  15. Benjamin, Biemond: J.J., van de Wouw, N., Nijmeijer, H.: Bifurcations of equilibrium sets in mechanical systems with dry friction. Physica D 241, 1882–1894 (2012)

    Article  MathSciNet  Google Scholar 

  16. Filippov, A.F.: Differential Equations with Discontinuous Right-hand Sides. Kluwer Acadamic, Dordrecht (1988)

    Book  MATH  Google Scholar 

  17. Yakubovich, V.A., Leonov, G.A., Gelig, AKh: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, Singapore (2004)

    Book  MATH  Google Scholar 

  18. Leine, R.I., van de Wouw, N.: Stability properties of equilibrium sets of nonlinear mechanical systems with dry friction and impact. Nonlinear Dyn. 51, 551–583 (2008)

  19. Cao, Q.J., Wiercigroch, M., Pavlovskaia, E.E., et al.: Archetypal oscillator for smooth and discontinuous dynamics. Phys. Rev. E. 74, 046218 (2006)

  20. Cao, Q.J., Wiercigroch, M., Pavlovskaia, E.E., et al.: The limit case response of the archetypal oscillator for smooth and discontinuous dynamics. Int. J. Nonlinear Mech. 43, 462–473 (2008)

  21. Cao, Q.J., Wercigroch, M., Pavlovskaia, E.E., et al.: Piecewise linear approach to an archetypal oscillator for smooth and discontinuous dynamics. Philos. Trans. R. Soc. A 366, 635–652 (2008)

  22. Tian, R.L., Wu, Q.L., Yang, X.W., et al.: Chaotic threshold for the smooth-and-discontinuous oscillator under constant excitations. Eur. Phys. J. Plus 128, 1–12 (2013)

  23. Léger, A., Pratt, E., Cao, Q.J.: A fully nonlinear oscillator with contact and friction. Nonlinear Dyn. 70, 511–522 (2012)

    Article  MathSciNet  Google Scholar 

  24. Dieci, L., Lopez, L.: Sliding motion in filippov differential systems: theoretical results and a computational approach. SIAM J. Numer. Anal. 47, 2023–2051 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  25. Léger, A., Pratt, E.: On the periodic solutions of a non smooth dynamical system. Rev. de Méca. Appli. et Théor. 2, 501–513 (2011)

    Google Scholar 

  26. Colombo, A., Di Bernardo, M., Hogan, S.J., et al.: Bifurcations of piecewise smooth flows: perspectives, methodologies and open problems. Physica D 241, 1845–1860 (2012)

  27. Di Bernardo, M., Budd, C.J., Champneys, A.R., et al.: Bifurcations in nonsmooth dynamical systems. SIAM Rev. 50, 629–701 (2008)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The project was supported by the National Natural Science Foundation of China (Grant 11372082) and the National Basic Research Program of China (Grant 2015CB057405).

Author information

Authors and Affiliations

  1. School of Astronautics, Centre for Nonlinear Dynamics Research, Harbin Institute of Technology, Harbin, 150001, China

    Zhi-Xin Li & Qing-Jie Cao

  2. Laboratoire de Mécanique et d’ Acoustique, CNRS, 31, Chemin Joseph Aiguier, 13402, Marseille Cedex 20, France

    Alain Léger

Authors
  1. Zhi-Xin Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Qing-Jie Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alain Léger
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Qing-Jie Cao.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, ZX., Cao, QJ. & Léger, A. The equilibrium stability for a smooth and discontinuous oscillator with dry friction. Acta Mech. Sin. 32, 309–319 (2016). https://doi.org/10.1007/s10409-015-0481-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10409-015-0481-y

Keywords

Search

Navigation