Resonances of an elastic plate coupled with a compressible confined flow

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 κ solve the equations λ(κ) = κ², where λ is the eigenvalue of a self-adjoint operator of the form A + κB. 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.