A new 2-D theory for vibrations of piezoelectric crystal plates with electroded faces

A system of two-dimensional governing equations for piezoelectric plates with general crystal symmetry and with electroded faces are deduced from the three-dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of thickness coordinate. In the cosine-series expansion for the mechanical displacements, the antisymmetric in-plane displacements induced by the gradients of deflection of plate is separated from the rest terms and is expressed by a linear function of thickness coordinate. For the electric potential, a sine-series expansion is used for it is well suited for satisfying the electrical conditions at the faces covered with conductive electrodes. A system of approximate first-order equations is extracted from the infinite system of 2-D equations. Dispersion curves for thickness-shear, flexure, and face-shear modes varying along x₁ and those for thickness-twist and face-shear varying along x₃ are calculated for AT-cut quartz plates and they are compared very closely with the corresponding ones computed from the 3-D equations, without introducing any corrections. Predicted frequency spectra by the present equations are shown to agree closely with the experimental data by Koga and Fukuyo and that by Nakazawa, Koriuchi, and Ito