Skip to main content
SpringerLink
Log in

Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact

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

Abstract

In this paper, we will give conditions under which the equilibrium set of multi-degree-of-freedom non-linear mechanical systems with an arbitrary number of frictional unilateral constraints is attractive. The theorems for attractivity are proved by using the framework of measure differential inclusions together with a Lyapunov-type stability analysis and a generalisation of LaSalle’s invariance principle for non-smooth systems. The special structure of mechanical multi-body systems allows for a natural Lyapunov function and an elegant derivation of the proof. Moreover, an instability theorem for assessing the instability of equilibrium sets of non-linear mechanical systems with frictional bilateral constraints is formulated. These results are illustrated by means of examples with both unilateral and bilateral frictional constraints.

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. Adly, S., Goeleven, D.: A stability theory for second-order nonsmooth dynamical systems with application to friction problems. J. Mathematiques Pures et Appliquées 83, 17–51 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  2. Alvarez, J., Orlov, I., Acho, L.: An invariance principle for discontinuous dynamic systems with application to a Coulomb friction oscillator. J. Dyn. Syst. Meas. Control 122, 687–690 (2000)

    Article  Google Scholar 

  3. Anosov, D.V.: Stability of the equilibrium positions in relay systems. Autom. Remote Control 20, 130–143 (1959)

    MATH  MathSciNet  Google Scholar 

  4. Aubin, J.-P., Ekeland, I.: Applied Nonlinear Analysis. Pure and Applied Mathematics. Wiley, Berlin (1984)

    Google Scholar 

  5. Aubin, J.-P., Frankowska, H.: Set-Valued Analysis, vol. 2, Systems and Control: Foundations and Applications. Birkhäauser, Boston (1990)

    Google Scholar 

  6. Bacciotti, A., Ceragioli, F.: Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. ESIAM: Control Optim. Calculus Variations 4, 361–376 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Bainov, D.D., Simeonov, P.S.: Systems with Impulse Effect: Stability, Theory and Applications. Wiley, New York (1989)

    MATH  Google Scholar 

  8. Brogliato, B.: Nonsmooth Mechanics, 2nd edn. Springer, London (1999)

    MATH  Google Scholar 

  9. Brogliato, B.: Absolute stability and the Lagrange—Dirichlet theorem with monotone multivalued mappings. Syst. Control Lett. 51, 343–353 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  10. Brogliato, B., Goeleven, D.: The Krakovskii—LaSalle invariance principle for a class of unilateral dynamical systems. Math. Control Signals Syst. 17(1), 57–76 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  11. Brogliato, B., Niculescu, S.-I., Orhant, P.: On the control of finite-dimensional mechanical systems with unilateral constraints. IEEE Trans. Autom. Control 42(2), 200–215 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Camlibel, M., Heemels, W., van der Schaft, A., Schumacher, J.: Switched networks and complementarity. IEEE Trans. Circuits Syst. 50(8), 1036–1046 (2003)

    Article  Google Scholar 

  13. Chareyron, S., Wieber, P.B.: A LaSalle's invariance theorem for nonsmooth Lagrangian dynamical systems. In: Proceedings of the ENOC-2005 Conference, 7–12 August, Eindhoven, The Netherlands (2005)

  14. Chareyron, S., Wieber, P.B.: Stability and regulation of nonsmooth dynamical systems. IEEE Trans. Autom. Control (2007) submitted

  15. Davrazos, G.N., Koussoulas, N.T.: A review of stability results for switched and hybrid systems. In: 9th Mediterranean Conference on Control and Automation, 27–29 June, Dubrovnik, Croatia (2001)

    Google Scholar 

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

    MATH  Google Scholar 

  17. Glocker, Christoph.: On frictionless impact models in rigid-body systems. Phil. Trans. R. Soc. Lond. A 359, 2385–2404 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  18. Glocker, Christoph.: Set-Valued Force Laws, Dynamics of Non-Smooth Systems, vol. 1, Lecture Notes in Applied Mechanics. Springer-Verlag, Berlin (2001)

    Google Scholar 

  19. Glocker, Christoph.: An introduction to impacts. In: Haslinger, J., Stavroulakis, G.E. (eds.), Nonsmooth Mechanics of Solids, vol. 485, CISM Courses and Lectures. Springer, Wien, New York, pp. 45–101 (2006)

  20. Goeleven, D., Brogliato, B.: Stability and instability matrices for linear evolution variational inequalities. IEEE Trans. Autom. Control 49, 521–534 (2004)

    Article  MathSciNet  Google Scholar 

  21. Goeleven, D., Brogliato, B.: Necessary conditions of asymptotic stability for unilateral dynamical systems. Nonlinear Anal. 61(6), 961–1004 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Haslinger, J., Stavroulakis, G.E. (eds.): Nonsmooth Mechanics of Solids, vol. 485, CISM Courses and Lectures. Springer, Wien, New York (2006)

    Google Scholar 

  23. Johansson, M., Rantzer, A.: Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans. Autom. Control 43(4), 555–559 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Kane, T.R., Levinson, D.A.: Dynamics, Theory and Applications. McGraw-Hill, New York (1985)

    Google Scholar 

  25. Khalil, H.K.: Nonlinear Systems. Prentice-Hall, Englewood Cliffs, NJ (1996)

    Google Scholar 

  26. Kunze, M.: Non-Smooth Dynamical Systems. Springer, Berlin (2000)

    MATH  Google Scholar 

  27. Leine, R.I.: Bifurcations in discontinuous mechanical systems of Filippov-Type. Ph.D. Thesis, Eindhoven University of Technology, The Netherlands (2000)

  28. Leine, R.I.: On the stability of motion in non-smooth mechanical systems. Habilitation Thesis, ETH Zäurich, Switzerland (2007)

  29. Leine, R.I., Brogliato, B., Nijmeijer, H.: Periodic motion and bifurcations induced by the Painleváe paradox. Eur. J. Mech. A Solids 21(5), 869–896 (2002)

    MATH  MathSciNet  Google Scholar 

  30. Leine, R.I., Glocker, Christoph.: A set-valued force law for spatial Coulomb—Contensou friction. Eur. J. Mech. A Solids 22, 193–216 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  31. Leine, R.I., Nijmeijer, H.: Dynamics and Bifurcations of Non-Smooth Mechanical Systems, vol. 18, Lecture Notes in Applied and Computational Mechanics. Springer-Verlag, Berlin (2004)

    Google Scholar 

  32. Moreau, J.J.: Liaisons unilatáerales sans frottement et chocs inelastiques. Comptes Rendu des Sáeances de l'Academie des Sciences 296, 1473–1476 (1983)

    MATH  MathSciNet  Google Scholar 

  33. Moreau, J.J.: Bounded variation in time. In: Moreau, J.J., Panagiotopoulos, P.D., Strang, G. (eds.), Topics in Nonsmooth Mechanics, pp. 1–74. Birkhäuser Verlag, Basel, Boston, Berlin (1988)

    Google Scholar 

  34. Moreau, J.J.: Unilateral contact and dry friction in finite freedom dynamics. In: Moreau, J.J., Panagiotopoulos, P.D. (eds.) Non-Smooth Mechanics and Applications, vol. 302, CISM Courses and Lectures, pp. 1–82. Springer, Wien, New York (1988)

    Google Scholar 

  35. Moreau, J.J.: Some numerical methods in multibody dynamics: Application to granular materials. Eur. J. Mech. A Solids 13(4, suppl.), 93–114 (1994)

    MATH  MathSciNet  Google Scholar 

  36. Papastavridis, J. G.: Analytical Mechanics: A Comprehensive Treatise on the Dynamics of Constrained Systems: For Engineers, Physicists and Mathematicians. Oxford University Press, New York (2002)

    MATH  Google Scholar 

  37. Payr, M., Glocker, C.: Oblique frictional impact of a bar: Analysis and comparison of different impact laws. Nonlinear Dyn. 41, 361–383 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  38. Pettersson, S., Lennartson, B.: Hybrid system stability and robustness verification using linear matrix inequalities. Int. J. Control 75(16–17), 1335–1355 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  39. Pfeiffer, F., Glocker, Christoph.: Multibody Dynamics with Unilateral Contacts. Wiley, New York (1996)

    MATH  Google Scholar 

  40. Quittner, P.: An instability criterion for variational inequalities. Nonlinear Anal. 15(12), 1167–1180 (1990)

    Google Scholar 

  41. Rockafellar, R.T.: The Theory of Subgradients and its Applications to Problems of Optimization. Convex and Nonconvex Functions, vol. 1, Research and Education in Mathematics. Heldermann Verlag, Berlin (1970)

    Google Scholar 

  42. Rockafellar, R.T., Wets, R.-B.: Variational Analysis. Springer, Berlin (1998)

    MATH  Google Scholar 

  43. Shevitz, D., Paden, B.: Lyapunov stability of nonsmooth systems. IEEE Trans. Autom. Control 39(9), 1910–1914 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  44. Tornambè, A.: Modeling and control of impact in mechanical systems: theory and experimental results. IEEE Trans. Autom. Control 44(2), 294–309 (1999)

    Article  MATH  Google Scholar 

  45. Van de Wouw, N., Leine, R.I.: Attractivity of equilibrium sets of systems with dry friction. Int. J. Nonlinear Dyn. Chaos Eng. Syst. 35(1), 19–39 (2004)

    MATH  Google Scholar 

  46. van de Wouw, N., van den Heuvel, M., van Rooij, J., Nijmeijer, H.: Performance of an automatic ball balancer with dry friction. Int. J. Bif. Chaos 15(1), 65–82 (2005)

    Article  MATH  Google Scholar 

  47. Yakubovich, V.A., Leonov, G.A., Gelig, A.K.: Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities, vol. 14, Series on Stability, Vibration and Control of Systems, Series A. World Scientific, Singapore (2004)

    Google Scholar 

  48. Ye, H., Michel, A., Hou, L.: Stability theory for hybrid dynamical systems. IEEE Trans. Autom. Control 43(4), 461–474 (1998)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMES – Center of Mechanics, ETH Zürich, CH-8092, Zürich, Switzerland

    R. I. Leine

  2. Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands

    N. van de Wouw

Authors
  1. R. I. Leine
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. N. van de Wouw
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. I. Leine.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Leine, R.I., van de Wouw, N. Stability properties of equilibrium sets of non-linear mechanical systems with dry friction and impact. Nonlinear Dyn 51, 551–583 (2008). https://doi.org/10.1007/s11071-007-9244-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-007-9244-z

Keywords

Search

Navigation