Resonances of an elastic plate in a compressible confined fluid
AbstractWe 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 to ensure the existence of resonances. Such conditions are validated numerically.