In the context of vibration reduction using piezoelectric semi-passive techniques, this paper presents an analytical solution for the dynamic response of a thin plate with various boundary conditions fitted with a shunted piezoelectric patch. The proposed formulation is based on the classical plate theory and takes into account the effect of the mass and the stiffness of the piezoelectric patch. The Rayleigh-Ritz modal projection method is applied to convert the partial differential equations into ordinary differential equations. An example of a vibration attenuation application using an RL shunt circuit is presented.