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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on October 4, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(4):397-421; doi:10.1093/qjmam/hbm015
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Q. Jl Mech. Appl. Math, Vol. 60. No. 4 © The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

Resonances of an elastic plate in a compressible confined fluid

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

( Laboratoire POEMS, UMR 2706 CNRS/ENSTA/INRIA, ENSTA, 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 24 May 2006. Revise 15 May 2007.
  Abstract

We present a theoretical study of the resonances of a fluid–structure problem, an elastic plate placed in a duct in the presence of a compressible fluid. The case of a rigid plate has been largely studied. Acoustic resonances are then associated to resonant modes trapped by the plate. Due to the elasticity of the plate, we need to solve a quadratic eigenvalue problem in which the resonance frequencies k solve the equations {gamma} (k) = k2, where {gamma} are the eigenvalues of a self-adjoint operator of the form A + kB. First, we show how to study the eigenvalues located below the essential spectrum by using the min–max principle. Then, we study the fixed-point equations. We establish sufficient conditions on the characteristics of the plate and of the fluid to ensure the existence of resonances. Such conditions are validated numerically.


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