Skip to main content

Qualitative analysis of a forced nonsmooth oscillator with contact and friction


We study the qualitative dynamics of a simple mass-spring system involving non regularized unilateral contact and Coulomb friction and submitted to an oscillating external force. The period-amplitude plane of the excitation appears to be essentially divided into two ranges of sliding solutions. At each point of the lower range there exist infinitely many equilibrium points and all the trajectories go to equilibrium in finite time. In the upper range, there no longer exist equilibria. Different kinds of periodic solutions are shown to exist in different zones and the transitions between these zones are explicitly computed. The upper boundary of this range, where the mass looses contact, is also computed and special attention is paid to the dependence of this upper boundary with respect to the period of the excitation.

This is a preview of subscription content, access via your institution.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18


  1. Let us compare problems (21) and (25). It appears that proving that (21) has a solution everywhere in zone \(\Upomega_5\) amounts to proving that problem (25) has a solution such that u 4 is constant everywhere in a nonzero measure subset of the \(\{T,\varepsilon\}\) plane, which could have been missed by a direct study of problem (25), and which is a result interesting in itself. Moreover distinguishing Zones \(\Upomega_5\) and \(\Upomega_6\) seems easier for an intuitive introduction to the partition of the plane. Nevertheless, the distinction between these two zones would not be necessary if we were dealing only with the transition to the loss of contact.

  2. In fact in this range the complete calculations are obviously also carried out using Maple, but here from explicit formula.

  3. A difficulty has already been encountered when studying the transition from Zone \(\Upomega_5\) to Zone \(\Upomega_6: \) since we don’t know a lot about the qualitative behaviour, another guess could be that the loss of contact arises only through non periodic solutions so that the line \(\rbrack T_{\alpha} + T_{\beta}, + \infty \lbrack \times \{\frac{4\mu \mathcal{A}}{K_t+3\mu W}\}\) would be the boundary for the loss of periodicity, instead of a transition between different types of periodic solutions. The answer is given if problem (27) has a solution.

  4. It is interesting to observe that, while giving the transition between Zone \(\Upomega_1\) and Zone \(\Upomega_2\) only amounts to plotting the graph of an explicit function, calculating all the other curves of Fig. 18 requires an implicit computation that can be very time consuming.


  1. Alart P, Curnier A (1986) Contact discret avec frottement: unicité de la solution, convergence de l’algorithme. Publications du Laboratoire de Mécanique Appliquée. Ecole Polytechnique Fédérale de Lausanne

  2. Ballard P, Léger A, Pratt E (2006) Stability of discrete systems involving shocks and friction. In: Wriggers P, Nackenhorst U (eds) Analysis and simulation of contact problems. Lecture notes in applied and computational mechanics. Springer, Berlin, pp 343–350

    Google Scholar 

  3. Basseville S, Léger A (2006) Stability of equilibrium states in a simple system with unilateral contact and Coulomb friction. Arch Appl Mech 76(7/8):403–428

    MATH  Article  Google Scholar 

  4. Ballard P, Basseville S (2005) Existence and uniqueness for dynamical unilateral contact with Coulomb friction: a model problem. Math Model Num Anal 39(1):57–77

    MathSciNet  Google Scholar 

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

    MATH  Article  Google Scholar 

  6. Brezis H (1973) Operateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert. North Holland, Amsterdam

    Google Scholar 

  7. Geymonat G, Léger A (1993) Role de l’expérimentation numérique dans les problèmes non-linéaires à plusieurs paramètres. Matapli 35:35–51

    Google Scholar 

  8. Jean M (1999) The nonsmooth contact dynamics method. Comput Methods Appl Mech Eng 177:235–257

    MathSciNet  MATH  Article  Google Scholar 

  9. Klarbring A (1990) Examples of non-uniqueness and non-existence of solutions to quasistatic contact problems with friction. Ing Arch 60:529–541

    Google Scholar 

  10. Martins JAC, Monteiro Marques MDP, Gastaldi F (1994) On an example of non-existence of solution to a quasistatic frictional contact problem. Eur J Mech A Solids 13(1):113–133

    MathSciNet  MATH  Google Scholar 

  11. Moreau JJ (1988) Unilateral contact and dry friction in finite freedom dynamics. In: Moreau JJ, Panagiotopoulos PD (eds) Nonsmooth mechanics and applications. CISM courses and lectures 302. Springer, Vienne

    Google Scholar 

  12. Pratt E, Léger A, Jean M (2008) Critical oscillations of mass-spring systems due to nonsmooth friction. Arch Appl Mech 78:89–104

    MATH  Article  Google Scholar 

  13. Pratt E, Léger A, Jean M (2010) About a stability conjecture concerning unilateral contact with friction. Nonlinear Dyn 59:73–94

    MATH  Article  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Elaine Pratt.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Léger, A., Pratt, E. Qualitative analysis of a forced nonsmooth oscillator with contact and friction. Ann. Solid Struct. Mech. 2, 1–17 (2011).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Coulomb friction
  • Mass-spring systems
  • Nonsmooth dynamics
  • Stability
  • Unilateral contact