Keywords: structural-acoustic, piezoelectric patches, variational formulation, finite element, noise and vibration attenuation, shunt damping, reduced order model.

For noise and vibration reduction, various approaches can be employed depending on the frequency range to attenuate. Generally, active or passive piezoelectric techniques are effective in the low frequency range, while dissipative materials (such as viscoelastic treatments or porous insulation) are efficient for a higher frequency domain. In this work, we propose a reduced order model to describe a fully coupled electro-mechanical-acoustic problem in the low frequency range. The system consists of an elastic structure with surface-mounted piezoelectric patches coupled with an inviscid, compressible and barotropic fluid, gravity effects being neglected. The piezoelectric elements, connected with a resonant shunt circuit (eventually with switch), are used for the vibration damping of the coupled system.

First, a non-symmetric finite element formulation of the coupled system is derived from a variational principle involving structural displacement, electrical voltage of the piezoelectric elements, and acoustic pressure inside the fluid cavity. This formulation, with only one couple of electric variables per patch, is well adapted to practical applications since realistic electrical boundary conditions, such that equipotentiality on the electrodes and prescribed global electric charges, naturally appear. The global charge/voltage variables are intrinsically adapted to include any external electrical circuit into the electromechanical problem and to simulate the effect of resistive or resonant shunt damping techniques.

The second part of this work is devoted to the introduction of a reduced-order model of the coupled problem. The proposed methodology, based on a normal mode expansion, requires the computation of the eigenmodes of (i) the structure with short-circuited piezoelectric patches, and (ii) the rigid acoustic cavity. It is shown that the projection of the full-order coupled finite element model on the uncoupled bases, leads to a reduced order model in which the main parameters are the classical fluid-structure and eletromechanical modal coupling factors. Despite its reduced size, this model is proved to be very efficient for simulations of steady-state and transient analyses of the coupled structural-acoustic system with shunt damping.

In the last part of the paper, a three-dimensional numerical example is investigated. The problem consists of an elastic plate equipped with piezoelectric elements and coupled with an acoustic cavity. This example is first analyzed in order to show that the reduced order model is capable of capturing the main characteristics of the system dynamic behavior, notably in terms of attenuation. Then, sensitivity analyses concerning the piezoelectric patch (size and position) and the electric circuit (resistance and inductance) are performed in order to highlight the role of each parameter on the performance of the shunt damping techniques.

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