Three-dimensional perturbation solution of the natural vibrations of piezoelectric rectangular plates

Abstract The paper discusses a perturbation solution of the natural frequencies and mode shapes of a piezoelectric rectangular plate modelled as a three-dimensional body. The coupled theory of piezoelectricity is used, with the governing equations consisting of one electrostatic and three mechanical equations coupled through the piezoelectric effect. Analytical perturbation formulas up to the first-order terms have been derived and used. An important difference of the present analysis as compared to the classical perturbation method consists in that the small parameter enters not only the governing equations but the boundary conditions as well. To address this complication an efficient new approach that makes use of generalized functions has been proposed. Results of the natural frequencies and mode shapes obtained by the perturbation method are discussed for a thin piezoelectric rectangular plate, a thick plate and a piezoelectric parallelepiped. All the results obtained using the perturbation method have been compared with the exact solutions of the coupled electromechanical problem. The proposed perturbation approach furnishes an efficient approximate method of studying the coupled piezoelectric vibration problem. The main advantage of the method derives from the fact that only the elastic solution is required, the effect of piezoelectric coupling being accounted for at a post-processing stage.