Vibrations of a beam between obstacles. Convergence of a fully discretized approximation
ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 705-734.

We consider mathematical models describing dynamics of an elastic beam which is clamped at its left end to a vibrating support and which can move freely at its right end between two rigid obstacles. We model the contact with Signorini's complementary conditions between the displacement and the shear stress. For this infinite dimensional contact problem, we propose a family of fully discretized approximations and their convergence is proved. Moreover some examples of implementation are presented. The results obtained here are also valid in the case of a beam oscillating between two longitudinal rigid obstacles.

DOI : 10.1051/m2an:2006031
Classification : 35L85, 65M12, 74H45
Mots-clés : dynamics with impact, Signorini's conditions, space and time discretization, convergence 
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     title = {Vibrations of a beam between obstacles. {Convergence} of a fully discretized approximation},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
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Dumont, Yves; Paoli, Laetitia. Vibrations of a beam between obstacles. Convergence of a fully discretized approximation. ESAIM: Modélisation mathématique et analyse numérique, Tome 40 (2006) no. 4, pp. 705-734. doi : 10.1051/m2an:2006031. http://archive.numdam.org/articles/10.1051/m2an:2006031/

[1] B. Brogliato, A.A. ten Dam, L. Paoli, F. Genot and M. Abadie, Numerical simulation of finite dimensional multibody nonsmooth mechanical systems. ASME Appl. Mechanics Rev. 55 (2002) 107-149.

[2] Y. Dumont, Vibrations of a beam between stops: Numerical simulations and comparison of several numerical schemes. Math. Comput. Simul. 60 (2002) 45-83. | Zbl

[3] Y. Dumont, Some remarks on a vibro-impact scheme. Numer. Algorithms 33 (2003) 227-240. | Zbl

[4] Y. Dumont and L. Paoli, Simulations of beam vibrations between stops: comparison of several numerical approaches, in Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference (ENOC-2005), CD Rom (2005).

[5] L. Fox, The numerical solution of two-point boudary values problems in ordinary differential equations, Oxford University Press, New York (1957). | MR | Zbl

[6] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag, New York, Berlin, Heidelberg (1983). | MR | Zbl

[7] T. Hughes, The finite element method. Linear static and dynamic finite element analysis. Prentice-Hall International, Englewood Cliffs (1987). | MR | Zbl

[8] K. Kuttler and M. Shillor, Vibrations of a beam between two stops. Dynamics of continuous, discrete and impulsive systems, Series B, Applications and Algorithms 8 (2001) 93-110. | Zbl

[9] C.H. Lamarque and O. Janin, Comparison of several numerical methods for mechanical systems with impacts. Int. J. Num. Meth. Eng. 51 (2001) 1101-1132. | Zbl

[10] F.C. Moon and S.W. Shaw, Chaotic vibration of a beam with nonlinear boundary conditions. Int. J. Nonlinear Mech. 18 (1983) 465-477.

[11] L. Paoli, Analyse numérique de vibrations avec contraintes unilatérales. Ph.D. thesis, University of Lyon 1, France (1993).

[12] L. Paoli, Time-discretization of vibro-impact. Phil. Trans. Royal Soc. London A. 359 (2001) 2405-2428. | Zbl

[13] L. Paoli, An existence result for non-smooth vibro-impact problems. Math. Mod. Meth. Appl. S. (M3AS) 15 (2005) 53-93. | Zbl

[14] L. Paoli and M. Schatzman, Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales : cas avec perte d'énergie. RAIRO Modél. Math. Anal. Numér. 27 (1993) 673-717. | Numdam | Zbl

[15] L. Paoli and M. Schatzman, Ill-posedness in vibro-impact and its numerical consequences, in Proceedings of European Congress on COmputational Methods in Applied Sciences and engineering (ECCOMAS), CD Rom (2000).

[16] L. Paoli and M. Schatzman, A numerical scheme for impact problems, I and II. SIAM Numer. Anal. 40 (2002) 702-733; 734-768. | Zbl

[17] P. Ravn, A continuous analysis method for planar multibody systems with joint clearance. Multibody Syst. Dynam. 2 (1998) 1-24. | Zbl

[18] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton (1970). | MR | Zbl

[19] M. Schatzman and M. Bercovier, Numerical approximation of a wave equation with unilateral constraints. Math. Comp. 53 (1989) 55-79. | Zbl

[20] S.W. Shaw and R.H. Rand, The transition to chaos in a simple mechanical system. Int. J. Nonlinear Mech. 24 (1989) 41-56. | Zbl

[21] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pur. Appl. 146 (1987) 65-96. | Zbl

[22] D. Stoianovici and Y. Hurmuzlu, A critical study of applicability of rigid body collision theory. ASME J. Appl. Mech. 63 (1996) 307-316.

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