# Resonances of an elastic plate coupled with a compressible confined flow

1 POEMS - Propagation des Ondes : Étude Mathématique et Simulation
Inria Saclay - Ile de France, UMA - Unité de Mathématiques Appliquées, CNRS - Centre National de la Recherche Scientifique : UMR7231
Abstract : A theoretical study of the resonances of an elastic plate in a compressible flow in a two-dimensional duct is presented. Due to the fluid-structure coupling, a quadratic eigenvalue problem is involved, in which the resonance frequencies k solve the equations λ(k) = k2, where λ is the eigenvalue of a self-adjoint operator of the form A + kB. In a previous paper, we have proved that a linear eigenvalue problem is recovered if the plate is rigid or the fluid at rest. We focus here on the general problem for which elasticity and flow are jointly present and derive a lower bound for the number of resonances. The expression of this bound, based on the solution of two linear eigenvalue problems, points out that the coupling between elasticity and flow generally reduces the number of resonances. This estimate is validated numerically. © The author 2009. Published by Oxford University Press; all rights reserved.
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Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2009, 62 (2), pp.105-129. 〈10.1093/qjmam/hbp004〉
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Soumis le : jeudi 17 octobre 2013 - 10:10:32
Dernière modification le : jeudi 7 février 2019 - 17:49:34

### Citation

Anne-Sophie Bonnet-Ben Dhia, Jean-François Mercier. Resonances of an elastic plate coupled with a compressible confined flow. Quarterly Journal of Mechanics and Applied Mathematics, Oxford University Press (OUP), 2009, 62 (2), pp.105-129. 〈10.1093/qjmam/hbp004〉. 〈hal-00873072〉

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