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A convergence result for a vibro-impact problem with a general inertia operator

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Abstract

We consider a mechanical system with a finite number of degrees of freedom and non-trivial inertia matrix, submitted to a single perfect unilateral constraint. We assume that the local impact law consists in the transmission of the tangential component of the velocity and the reflexion of the normal component which is multiplied by the restitution coefficient e∈[0,1]. Then, starting from the measure-differential formulation of the problem given by J.J. Moreau, we propose a velocity-based time-stepping method, reminiscent of the catching-up algorithm for sweeping processes and we prove that the numerical solutions converge to a solution of the problem.

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References

  1. Ballard, P.: The dynamics of discrete mechanical systems with perfect unilateral constraints. Arch. Ration. Mech. Anal. 154, 199–274 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  2. Dzonou, R., Monteiro Marques, M.D.P.: Sweeping process for inelastic impact problem with a general inertia operator. Eur. J. Mech. A, Solids 26(3), 474–490 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Dzonou, R., Monteiro Marques, M.D.P., Paoli, L.: Algorithme de type ‘sweeping process’ pour un problème de vibro-impact avec un opérateur d’inertie non trivial. C.R. Acad. Sci. Paris Sér. 2B 335, 56–60 (2007)

    MATH  Google Scholar 

  4. Jeffery, R.L.: Non-absolutely convergent integrals with respect to functions of bounded variations. Trans. Am. Math. Soc. 34, 645–675 (1932)

    Article  MathSciNet  Google Scholar 

  5. Mabrouk, M.: A unified variational model for the dynamics of perfect unilateral constraints. Eur. J. Mech. A, Solids 17, 819–842 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. Monteiro Marques, M.D.P.: Chocs inélastiques standards: un résultat d’existence. Séminaire d’Analyse Convexe, USTL, Montpellier, 15, exposé n. 4 (1983)

  7. Monteiro Marques, M.D.P.: Differential Inclusions in Non-Smooth Mechanical Problems: Shocks and Dry Friction. Birkhauser, Boston (1993)

    Google Scholar 

  8. Moreau, J.J.: Un cas de convergence des itérées d’une contraction d’un espace hilbertien. C.R. Acad. Sci. Paris, Sér. A 286, 143–144 (1978)

    MATH  MathSciNet  Google Scholar 

  9. Moreau, J.J.: Liaisons unilatérales sans frottement et chocs inélastiques. C.R. Acad. Sci. Paris, Sér. II 296, 1473–1476 (1983)

    MATH  MathSciNet  Google Scholar 

  10. Paoli, L.: Analyse numérique de vibrations avec contraintes unilatérales. PhD Thesis, Université Claude Bernard-Lyon I, France (1993)

  11. Paoli, L.: Continuous dependence on data for vibro-impact problems. Math. Models Methods Appl. Sci. 15, 53–93 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  12. Paoli, L., Schatzman, M.: Mouvement à nombre fini de dégrés de liberté avec contraintes unilatérales: cas avec perte d’énergie. Modél. Math. Anal. Numér 27(6), 673–717 (1993)

    MATH  MathSciNet  Google Scholar 

  13. Paoli, L., Schatzman, M.: A numerical scheme for impact problems I and II. SIAM, J. Numer. Anal. 40(2), 702–733, 734-768 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Schatzman, M.: A class of nonlinear differential equations of second order in time. Non-linear Anal. Theory Methods Appl. 2, 355–373 (1978)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. LaMUSE (Laboratoire de Mathématiques de l’Université de Saint-Etienne), Université Jean Monnet, 23 Rue du Docteur Paul Michelon, 42023, St-Etienne Cedex 2, France

    Raoul Dzonou & Laetitia Paoli

  2. CMAF and Faculdade de Ciencias, Universidade de Lisboa, Av Prof. Gama Pinto 2, 1649-003, Lisboa, Portugal

    Manuel D. P. Monteiro Marques

Authors
  1. Raoul Dzonou
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  2. Manuel D. P. Monteiro Marques
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  3. Laetitia Paoli
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Correspondence to Manuel D. P. Monteiro Marques.

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Dzonou, R., Monteiro Marques, M.D.P. & Paoli, L. A convergence result for a vibro-impact problem with a general inertia operator. Nonlinear Dyn 58, 361–384 (2009). https://doi.org/10.1007/s11071-009-9484-1

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  • DOI: https://doi.org/10.1007/s11071-009-9484-1

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