Vibrations of a beam between obstacles. Convergence of a fully discretized approximation
Dumont, Yves ; Paoli, Laetitia
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006), p. 705-734 / Harvested from Numdam
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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.

Publié le : 2006-01-01
DOI : https://doi.org/10.1051/m2an:2006031
Classification:  35L85,  65M12,  74H45 
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     author = {Dumont, Yves and Paoli, Laetitia},
     title = {Vibrations of a beam between obstacles. Convergence of a fully discretized approximation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {40},
     year = {2006},
     pages = {705-734},
     doi = {10.1051/m2an:2006031},
     mrnumber = {2274775},
     zbl = {1106.74057},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2006__40_4_705_0}
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Dumont, Yves; Paoli, Laetitia. Vibrations of a beam between obstacles. Convergence of a fully discretized approximation. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 705-734. doi : 10.1051/m2an:2006031. http://gdmltest.u-ga.fr/item/M2AN_2006__40_4_705_0/

[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 1011.74031

[3] Y. Dumont, Some remarks on a vibro-impact scheme. Numer. Algorithms 33 (2003) 227-240. | Zbl 1134.74392

[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 102178 | Zbl 0077.11202

[6] J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems and bifurcations of vector fields. Springer-Verlag, New York, Berlin, Heidelberg (1983). | MR 709768 | Zbl 0515.34001

[7] T. Hughes, The finite element method. Linear static and dynamic finite element analysis. Prentice-Hall International, Englewood Cliffs (1987). | MR 1008473 | Zbl 0634.73056

[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 1013.74033

[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 1056.74059

[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 1067.70012

[13] L. Paoli, An existence result for non-smooth vibro-impact problems. Math. Mod. Meth. Appl. S. (M3AS) 15 (2005) 53-93. | Zbl 1079.34006

[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 0792.34012

[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 1021.65065

[17] P. Ravn, A continuous analysis method for planar multibody systems with joint clearance. Multibody Syst. Dynam. 2 (1998) 1-24. | Zbl 0953.70517

[18] R.T. Rockafellar, Convex analysis. Princeton University Press, Princeton (1970). | MR 274683 | Zbl 0193.18401

[19] M. Schatzman and M. Bercovier, Numerical approximation of a wave equation with unilateral constraints. Math. Comp. 53 (1989) 55-79. | Zbl 0683.65088

[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 0666.70030

[21] J. Simon, Compact sets in the space L p (0,T;B). Ann. Mat. Pur. Appl. 146 (1987) 65-96. | Zbl 0629.46031

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