Stability analysis of a 2-D acoustic/structure model

Abstract

Reliably modeling noise attenuation through interaction with vibrating boundary structures is fundamental to the formulation of effective active noise control systems. In this paper we investigate, through numerical approximation, uniform exponential stability of two systems which model the acoustic/structure interaction of an air-filled, rectangular cavity. The first model assumes dissipative boundary conditions along one side of the boundary, while the second assumes dissipative boundary conditions along all four sides of the cavity. We obtain weak variational formulations for these models, express each as finite dimensional systems, and use the Galerkin technique to transform the distributed parameter systems into systems of ordinary differential equations. We analyze the stability of the finite dimensional systems in order to gain insight into the stability of the original infinite dimensional systems. Essentially, our analysis consists of solving a generalized eigenvalue problem and observing where the eigenvalues lie within the complex plane. This stability analysis leads us to conclude that one model is better suited for use 5n the formulation of the noise control problem.