Firstly, a finite element formulation of the coupled electromechanical problem of an elastic structure equipped with piezoelectric patches is introduced. Its originality lies in the fact that the electrical state of the system is fully described by very few discrete unknowns: only a couple of variables per piezoelectric patch, namely (i) the electric charge contained in the electrodes and (ii) the voltage between the electrodes. A first advantage of this formulation is that, since the electrical state is fully discretized at the weak formulation step, any standard elastic finite element formulation can be easily modified to include the piezoelectric patches. A second advantage is that, since global electrical variables are used, realistic electrical boundary conditions, such as equipotentiality on the electrodes and prescribed global charges, naturally appear. Moreover, global charge-voltage variables are intrinsically adapted to include any external electrical circuit into the electromechanical problem and to simulate shunted piezoelectric patches.

The second part of the work is devoted to the introduction of a reduced order model by expanding the solution onto the normal modes of the electro-mechanical problem with all patches short circuited. It is shown that the classical effective electromechanical coupling factors, which directly influence the efficiency of vibration damping, naturally appear as the main coupling parameters in this reduced order model.

The proposed approach is finally applied to vibration reduction by means of a resonant shunt system to (i) a cantilever beam and (ii) a rotor blade structure. For the beam structure, numerical and experimental results are compared showing an excellent agreement and validating the formulation. In the second case, which corresponds to an application to a complex three-dimensional structure, an optimization of the piezoelectric patches dimensions as well as their position on the structure is proposed in order to maximize the damping effect.

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