VIBRATIONS OF A BEAM BETWEEN OBSTACLES. CONVERGENCE OF A FULLY DISCRETIZED APPROXIMATION

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.

[1]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[2]  Laetitia Paoli,et al.  Time discretization of vibro‐impact , 2001, Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[3]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[4]  O. Janin,et al.  Comparison of several numerical methods for mechanical systems with impacts , 2001 .

[5]  D. Stoianovici,et al.  A Critical Study of the Applicability of Rigid-Body Collision Theory , 1996 .

[6]  Steven W. Shaw,et al.  Chaotic vibrations of a beam with non-linear boundary conditions , 1983 .

[7]  Laetitia Paoli,et al.  A numerical scheme for impact problems , 1999 .

[8]  Laetitia Paoli,et al.  An existence result for non-smooth vibro-impact problems , 2005 .

[9]  L. Fox The Numerical Solution of Two-Point Boundary Problems in Ordinary Differential Equations , 1957 .

[10]  Steven W. Shaw,et al.  The transition to chaos in a simple mechanical system , 1989 .

[11]  M. Schatzman,et al.  Numerical approximation of a wave equation with unilateral constraints , 1989 .

[12]  P. Holmes,et al.  Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields , 1983, Applied Mathematical Sciences.

[13]  Laetitia Paoli Analyse numérique de vibrations avec contraintes unilatérales , 1993 .

[14]  Laetitia Paoli CONTINUOUS DEPENDENCE ON DATA FOR VIBRO-IMPACT PROBLEMS , 2005 .

[15]  L. Paoli,et al.  ILL-POSEDNESS IN VIBRO-IMPACT AND ITS NUMERICAL CONSEQUENCES , 2000 .

[16]  Peter Ravn,et al.  A Continuous Analysis Method for Planar Multibody Systems with Joint Clearance , 1998 .

[17]  Yves Dumont Vibrations of a beam between stops: numerical simulations and comparison of several numerical schemes , 2002, Math. Comput. Simul..

[18]  Yves Dumont,et al.  Some Remarks on a Vibro-Impact Scheme , 2003, Numerical Algorithms.

[19]  B. Brogliato,et al.  Numerical simulation of finite dimensional multibody nonsmooth mechanical systems , 2001 .

[20]  Laetitia Paoli,et al.  A Numerical Scheme for Impact Problems I: The One-Dimensional Case , 2002, SIAM J. Numer. Anal..

[21]  Laetitia Paoli,et al.  Mouvement à un nombre fini de degrés de liberté avec contraintes unilatérales: cas avec perte d'énergie , 1993 .

[22]  J. Simon Compact sets in the spaceLp(O,T; B) , 1986 .