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Some energy conservative schemes for vibro-impacts of a beam on rigid obstacles*

Published online by Cambridge University Press:  04 July 2011

C. Pozzolini  and
M. Salaun
C. Pozzolini
Affiliation:
Pôle de Mathématiques, INSA de Lyon, 20 rue Albert Einstein, 69621 Villeurbanne Cedex, France. cedric.pozzolini@insa-lyon.fr Centre National d'Études Spatiales, 18 avenue Édouard Belin, 31401 Toulouse, France.
M. Salaun
Affiliation:
Université de Toulouse; INSA, UPS, EMAC, ISAE; ICA (Institut Clément Ader); 10 avenue Édouard Belin, 31055 Toulouse, France. michel.salaun@isae.fr
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Abstract

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

Keywords

Variational inequalitiesfinite element methodelastic beamdynamicsunilateral constraintsrestitution coefficient
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 6 , November 2011 , pp. 1163 - 1192
Copyright
© EDP Sciences, SMAI, 2011

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References

Ahn, J. and Stewart, D.E., Euler-Bernoulli, An beam with dynamic contact: discretization, convergence and numerical results. SIAM J. Numer. Anal. 43 (2005) 1455–1480. CrossRef
Carpenter, N.J., Lagrange constraints for transient finite element surface contact. Internat. J. Numer. Methods Engrg. 32 (1991) 103–128. CrossRef
Deuflhard, P., Krause, R. and Ertel, S., A contact-stabilized Newmark method for dynamical contact problems. Internat. J. Numer. Methods Engrg. 73 (2007) 1274–1290. CrossRef
Dumont, Y. and Paoli, L., Vibrations of a beam between obstacles: convergence of a fully discretized approximation. ESAIM: M2AN 40 (2006) 705–734. CrossRef
Dumont, Y. and Paoli, L., Numerical simulation of a model of vibrations with joint clearance. Int. J. Comput. Appl. Technol. 33 (2008) 41–53. CrossRef
Hauret, P. and Le Tallec, P., Energy controlling time integration methods for nonlinear elastodynamics and low-velocity impact. Comput. Methods Appl. Mech. Eng. 195 (2006) 4890–4916. CrossRef
Khenous, H.B., Laborde, P. and Renard, Y., Mass redistribution method for finite element contact problems in elastodynamics. Eur. J. Mech. A. Solids 27 (2008) 918–932. CrossRef
Kuttler, K. and Shillor, M., Vibrations of a beam between two stops, Dynamics of Continuous, Discrete and Impulsive Systems, Series B. Applications and Algorithms 8 (2001) 93–110.
Laursen, T.A. and Chawla, V., Design of energy conserving algorithms for frictionless dynamic contact problems. Internat. J. Numer. Methods Engrg. 40 (1997) 863–886. 3.0.CO;2-V>CrossRef
Laursen, T.A. and Love, G.R., Improved implicit integrators for transient impact problems-geometric admissibility within the conserving framework. Internat. J. Numer. Methods Engrg. 53 (2002) 245–274. CrossRef
Paoli, L., Time discretization of vibro-impact. Philos. Trans. Roy. Soc. London A 359 (2001) 2405–2428. CrossRef
Paoli, L. and Schatzman, M., A numerical scheme for impact problems. I. The one-dimensional case. SIAM J. Numer. Anal. 40 (2002) 702–733. CrossRef
Paoli, L. and Schatzman, M., Numerical simulation of the dynamics of an impacting bar. Comput. Methods Appl. Mech. Eng. 196 (2007) 2839–2851. CrossRef
A. Petrov and M. Schatzman, Viscolastodynamique monodimensionnelle avec conditions de Signorini. C. R. Acad. Sci. Paris, I 334 (2002) 983–988.
A. Petrov and M. Schatzman, A pseudodifferential linear complementarity problem related to a one dimensional viscoelastic model with Signorini condition. Arch. Rational Mech. Anal., to appear.
Renard, Y., The singular dynamic method for constrained second order hyperbolic equations. Application to dynamic contact problems. J. Comput. Appl. Math. 234 (2010) 906–923. CrossRef
Taylor, R.L. and Papadopoulos, P., On a finite element method for dynamic contact-impact problems. Internat. J. Numer. Methods Engrg. 36 (1993) 2123–2140. CrossRef