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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on March 19, 2009
The Quarterly Journal of Mechanics and Applied Mathematics 2009 62(2):105-129; doi:10.1093/qjmam/hbp004
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© The author 2009. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Resonances of an elastic plate coupled with a compressible confined flow

Anne-Sophie Bonnet-Ben Dhia{dagger} and Jean-François Mercier{ddagger}

( Laboratoire Propagations d'Ondes : Étude Mathématique et Simulation (POEMS), UMR 7231 CNRS/ENSTA/INRIA, École Nationale Supérieure de Techniques Avancées, 32, Boulevard Victor, 75739 Paris cedex 15, France )

{dagger} <Anne-Sophie.Bonnet-Bendhia{at}ensta.fr>

{ddagger} Corresponding author. <Jean-Francois.Mercier{at}ensta.fr>

Received 18 January 2008. Revise 4 February 2009.
  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 {lambda}(k) = k2, where {lambda} 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.


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