Resonances of an elastic plate in a compressible confined fluid

Abstract

SummaryWe 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 γ(k) = k 2 where γ 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 that ensure the existence of resonances. Such conditions are validated numerically.

A Sufficient Condition for the Absence of Two-Dimensional Instabilities of an Elastic Plate in a Duct with Compressible Flow

“…In particular we proved the existence of eigenvalues λ i (ω), i = 1, This study of real resonances has revealed an interesting behavior: when the fluidstructure problem is decoupled (rigid plate or fluid at rest), the resonance frequencies are necessarily real [1] (see also lemma 7). But when both effects are present, fluid in flow and elastic plate, the resonance frequencies can be complex [2].…”

“…from X = −∞ on the plate and if we look for the scattered field, the problem is found well-posed except for a sequence of so-called resonance frequencies [1,2]. These resonances correspond to the existence of eigenmodes (called also trapped modes): solutions in L 2 (Ω) of the problem without source.…”

“…The case of real resonance frequencies ( m(ω) = 0) has already been studied [1,2]. These frequencies are obtained as eigenvalues of a self-adjoint operator.…”

Dynamic Localization Phenomena in Elasticity, Acoustics and Electromagnetism

“…Trapped modes were introduced more than fifty years ago (see for instance Jones (1953)) and since then have induced an important amount of works (Callan et al, 1991;Evans et al, 1994;Kaplunov and Sorokin, 1995;Granot, 2002;Bonnet-BenDhia and Mercier, 2007). They provide a very powerful tool to understand the response of wave systems when excited by a source because they represent an intrinsic basis corresponding to various kind of resonances.…”

Numerical Continuation of Resonances and Bound States in Coupled Channel Schrödinger Equations

“…Such systems allow the interior field to couple to the external domain which leaves a characteristic fingerprint on the far-field pattern of the scattered wave. Many examples appear naturally in acoustic scattering [2] and in fluid-mechanical structure interaction [3]. In all cases the appearance of resonant states has a profound and important influence on the system's dynamics.…”