Unilateral Contact and Dry Friction in Finite Freedom Dynamics

  • J. J. Moreau
Part of the International Centre for Mechanical Sciences book series (CISM, volume 302)


An approach to the dynamics of mechanical systems with a finite number of degrees of freedom, involving unilateral constraints, is developed. In the n-dimensional linear spaces of forces and velocities, some classical concepts of Convex Analysis are used, but no convexity assumption is made concerning the constraint inequalities. The velocity is not supposed to be a differentiable function of time, but only to have locally bounded variation, so the role of the acceleration is held by a n-dimensional measure on the considered time interval. Dynamics is then governed by measure differential inclusions, which treat possible velocity jumps on the same footing as smooth motions. Possible collisions are described as soft, thus dissipative. Friction is taken into account under a recently proposed expression of Coulomb’s law. These formulations have the advantage of generating numerical algorithms of time-discretization, able to handle, in particular, the nonsmooth effects arising from unilaterality and from dry friction.


Contact Force Differential Inclusion Tangent Cone Covariant Component Polar Cone 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Delassus, E.: Mémoire sur la théorie des liaisons finies unilatérales, Ann. Sci. Ecole Norm. Sup., 34 (1917), 95–179.MATHMathSciNetGoogle Scholar
  2. 2.
    Pérés, J.: Mécanique Générale, Masson, Paris 1953.MATHGoogle Scholar
  3. 3.
    Moreau, J.J.: Les liaisons unilatérales et le principe de Gauss, C.R. Acad. Sci. Paris, 256 (1963), 871–874.MathSciNetGoogle Scholar
  4. 4.
    Moreau, J.J.: Quadratic programming in mechanics: dynamics of one-sided constraints, SIAM J. Control, 4 (1966), 153–158.CrossRefMathSciNetGoogle Scholar
  5. 5.
    Dunford, N. and J.T. Schwartz: Linear Operators, Part 1: General Theory, Interscience Pub. Inc., New York 1957.Google Scholar
  6. 6.
    Dinculeanu, N.: Vector Measures, Pergamon, London, New York 1967.Google Scholar
  7. 7.
    Moreau, J.J.: Bounded variation In time, in: Topics in Nonsmooth Mechanics (Ed. J.J. Moreau, P.D. Panagiotopoulos and G. Strang), Birkhäuser, to appear.Google Scholar
  8. 8.
    Pandit, S.G. and S.G. Deo: Differential Systems Involving Impulses, Lecture Notes in Math., vol. 954, Springer-Verlag, Berlin, Heidelberg, New York 1982.Google Scholar
  9. 9.
    Moreau, J.J.: Une formulation de la dynamique classique, C,R. Acad, Sci. Paris, Sér.l I, 304 (1987), 191-194.Google Scholar
  10. Panagiotopoulos, P.D.: Inequality Problems in Mechanics and Applications, Blrkhäuser, Boston, Basel, Stuttgart 1985.CrossRefGoogle Scholar
  11. 11.
    Monteiro Marques, M.D.P.: Chocs inélastiques standards: un résultat d’existence, Travaux du Séminaire d’Analyse Convexe, Univ. des Sci. et Techniques du Languedoc, vo1.15, exposé n° 4, Montpellier, 1985.Google Scholar
  12. 12.
    Monteiro Marques, M.D.P.: Rafle par un convexe semi-continu inférieurement, d’intérieur non vide, en dimension finie, C.R. Acad. Sci. Paris, Sér. I, 299 (1984), 307–310.Google Scholar
  13. 13.
    Monteiro Marques, M.D.P.: Inclusöes Díferenciaís e Choques Inelâsticos, Doctoral Dissertation, Faculty of Sciences, University of Lisbon, 1988Google Scholar
  14. 14.
    Moreau, J.J.: Liaisons unilatérales sans frottement et chocs inélastiques, C.R. Acad. Sci. Paris, Sér.I1, 296 (1983), 1473–1476.MATHMathSciNetGoogle Scholar
  15. 15.
    Moreau, J.J.: Standard inelastic shocks and the dynamics of unilateral constraints, in: Unilateral Problems in Structural Analysis (Ed. G. Del Piero and F. Maceri), CISM Courses and Lectures No.288, Springer-Verlag, Wien, New York 1985.Google Scholar
  16. 16.
    Schatzman, M.: A class of nonlinear differential equations of second order in time, J. Nonlinear Analysis, Theory, Methods and Appl., 2 (1978), 355–373.MATHMathSciNetGoogle Scholar
  17. 17.
    Buttazzo, G. and D. Percivale: On the approximation of the elastic bounce problem on Riemannian manifolds, J. Diff. Equations, 47 (1983), 227–245.ADSCrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Carriero, M. and E. Pascali: Uniqueness of the one-dimensional bounce problem as a generic property in Li([0,T1;R), Bolletino U.M.I.(6) 1-A (1982), 87-91.Google Scholar
  19. 19.
    Percivale, D.: Uniqueness in the elastic bounce problem, J. Diff, Equations, 56(1985), 206–215Google Scholar
  20. 20.
    Moreau, J.J.: Sur les lois de frottement, de plasticité et de viscosité, C.R. Acad. Sci. Paris, Sér.A, 271 (1970), 608-611.Google Scholar
  21. 21.
    Moreau, J.J.: On unilateral constraints, friction and plasticity, in: New Variational Techniques in Mathematical Physics (Ed. G. Capriz and G. Stampacchia), CIME 2° ciclo 1973, Edizioni Cremonese, Roma, 1974, I, 73–322.Google Scholar
  22. 22.
    Moreau, J.J.: Application of convex analysis to some problems of dry friction, in: Trends in Applications of Pure Mathematics to Mechanics, vol. 2 (Ed, H. ZorskI ), Pitman Pub, Ltd., London 1979, 263–280.Google Scholar
  23. 23.
    Duvaut, G. and J.L. Lions: Les Inéquations en Mécanique et en Physique, Dunod, Paris 1972.MATHGoogle Scholar
  24. 24.
    Moreau, J.J.: Dynamique de systèmes ä liaisons unilatérales avec frottement sec éventuel; essais numériques, Note Technique 85–1, Lab. de Mécanique Générale des Milieux Continus, Univ. des Sci, et Techniques du Languedoc, Montpellier, 1985.Google Scholar
  25. 25.
    Moreau, J.J.: Une formulation du contact à frottement sec; application au calcul numérique, C.R. Acad. Sci, Paris, Sér.11, 302 (1986), 799–801.MATHMathSciNetGoogle Scholar
  26. 26.
    Lecornu, L.: Sur la loi de Coulomb, C.R. Acad. Sci. Paris, 140 (1905), 847-848.Google Scholar
  27. 27.
    Oden, J.T. and J.A.C. Martins: Models and computational methods for dynamic friction phenomena, Computer Methods In Appl. Mech. and Engng., 52 (1985), 527-634.Google Scholar
  28. 28.
    Jean, M. and G. Touzot: Implementation of unilateral contact and dry friction in computer codes dealing with large deformations problems, to appear in: Numerical Methods in Mechanics of Contacts Involving Friction, J. de Mécanique theor. et appl., Special issue, 1988.Google Scholar
  29. 29.
    Abadie, J.: On the Kuhn-Tucker theorem, in: Nonlinear Programming (Ed. J.Abadie), North-Holland Pub, Co., Amsterdam 1967, 19–36.Google Scholar
  30. Jean, M. and J.J. Moreau: Dynamics in the presence of unilateral contacts and dry friction; a numerical approach, in: Unilateral Problems in Structural Analysis 2“ (Ed. G. Del Piero and F. Maceri), CISM Courses and Lectures No 304, Springer-Verlag, Wien 1987, 151–196.Google Scholar
  31. Moreau, J.J.: Evolution problem associated with a moving convex set in a Hilbert space, J. Diff. Equations, 26 (1977), 347–374.ADSCrossRefGoogle Scholar
  32. 32.
    Moreau, J.J.: Sur les mesures différentielles de fonctions vectorielles et certains problèmes d’évolution, C.R. Acad. Sci. Paris, Sér.A, 282 (1976), 837–840.MATHMathSciNetGoogle Scholar
  33. 33.
    Jean, M. and E. Pratt: A system of rigid bodies with dry friction, Int. J. Engng. Sci., 23 (1985), 497–513.CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Rockafellar, R.T.: Convex Analysis, Princeton Univ. Press, Princeton 1970.Google Scholar
  35. 35.
    Moreau, J.J.: Fonctionnelles Convexes, Lecture Notes, Séminaire sur les Equations aux Dérivées Partielles, Collège de France, Paris 1967.Google Scholar
  36. 36.
    Painlevé, P.: Sur les lois du frottement de glissement, C.R. Acad, Sci. Paris, 121 (1895), 1 12-1 15. Same title, ibid. 141 (1905), 401–405 and 141 (1905), 546-552.Google Scholar
  37. 37.
    Delassus, E.: Considérations sur le frottement de glissement, Nouv. Ann. de Math. (4ème série), 20 (1920), 485–496.Google Scholar
  38. 38.
    Delassus, E.: Sur les lois du frottement de glissement, Bull. Soc. Math. France, 51 (1923), 22–33.MATHMathSciNetGoogle Scholar
  39. 39.
    Klein, F.: Zu Painlevés Kritik des Coulombschen Reibungsgesetze, Zeitsch. Math. Phys., 58 (1910), 186–191.Google Scholar
  40. 40.
    Mises, R.v.: Zur Kritik der Reibungsgesetze, ibid.,191–195.Google Scholar
  41. 41.
    Hamel, G.: Bemerkungen zu den vorstehenden Aufsätzen der Herren F. Klein und R. v. Mises, ibid., 195–196.Google Scholar
  42. 42.
    Prandtl, L.: Bemerkungen zu den Aufsätzen der Herren F. Klein, R, v. Mises und G. Hamel, ibid., 196–197.Google Scholar
  43. 43.
    Beghin, H.: Sur certains problèmes de frottement, Nouv. Ann. de Math, 2 (1923–24), 305–312.Google Scholar
  44. 44.
    Beghín, H.: Sur l’indétermination de certains problèmes de frottement, Nouv. Ann. de Math, 3 (1924–25), 343–347.Google Scholar

Copyright information

© Springer-Verlag Wien 1988

Authors and Affiliations

  • J. J. Moreau
    • 1
  1. 1.Université des Sciences et Techniques du LanguedocMontpellierFrance

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