Stochastic variational inequalities for elasto-plastic oscillators

Alain Bensoussan; Janos Turi

doi:10.1016/j.crma.2006.08.008

Partial Differential Equations/Probability Theory

Stochastic variational inequalities for elasto-plastic oscillators

Presented by: Philippe G. Ciarlet

Alain Bensoussan ¹; Janos Turi ²

¹ International Center for Decision and Risk Analysis, School of Management, P.O.Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688, USA
² Programs in Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083-0688, USA

Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 399-406.

Abstract
Résumé

The purpose of this Note is to show that models used in the literature for the hysteresis effect of non-linear elasto-plastic oscillators submitted to random vibrations are equivalent to stochastic variational inequalities. This powerful tool allows to study the ergodic properties of the Markov process related to the displacement. We characterize completely the invariant measure by a partially degenerate elliptic partial differential equation, with new Dirichlet coupling conditions.

On montre que les modèles représentant l'effet d'hystérésis pour les oscillateurs non-linéaires elasto-plastiques sont équivalents à une inéquation variationnelle stochastique. Cette technique puissante permet d'étudier complétement les propriétés ergodiques du processus de Markov relatif au déplacement. On caractérise complétement la mesure invariante par une équation aux dérivées partielles elliptique partiellement dégénerée, avec des conditions de Dirichlet nouvelles.

Received: 2006-08-15
Accepted: 2006-08-17
Published online: 2006-09-15

DOI: 10.1016/j.crma.2006.08.008

Author's affiliations:

Alain Bensoussan ¹; Janos Turi ²

¹ International Center for Decision and Risk Analysis, School of Management, P.O.Box 830688, SM 30, University of Texas at Dallas, Richardson, TX 75083-0688, USA
² Programs in Mathematical Sciences, University of Texas at Dallas, Richardson, TX 75083-0688, USA

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     author = {Alain Bensoussan and Janos Turi},
     title = {Stochastic variational inequalities for elasto-plastic oscillators},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {399--406},
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     doi = {10.1016/j.crma.2006.08.008},
     language = {en},
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Alain Bensoussan; Janos Turi. Stochastic variational inequalities for elasto-plastic oscillators. Comptes Rendus. Mathématique, Volume 343 (2006) no. 6, pp. 399-406. doi : 10.1016/j.crma.2006.08.008. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2006.08.008/

References
Cited by

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[4] D. Karnopp; T.D. Scharton Plastic deformation in random vibration, J. Acoust. Soc. Amer., Volume 39 (1966), pp. 1154-1161

[5] R.Z. Khasminskii Stochastic Stability of Differential Equations, Sijthoff and Noordhoff, The Netherlands, 1980

[6] A. Preumont Random Vibration and Spectral Analysis, Kluwer, Boston, 1994

[7] J.B. Roberts; P.D. Spanos Random Vibration and Statistical Linearization, Wiley and Sons, Chichester, UK, 1990

[8] E.H. Vanmarcke Random Fields: Analysis and Synthesis, The MIT Press, 1984

Cited by Sources:

^⁎ This research was partially supported by a grant from CEA, Commissariat à l'énergie atomique.

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