A new two-dimensional theory for vibrations of piezoelectric crystal plates with electroded faces

Abstract

A system of two-dimensional (2-D) governing equations for piezoelectric plates with general crystal symmetry and with electroded faces is deduced from the three-dimensional (3-D) equations of linear piezoelectricity by expansion in series of trigonometric functions of thickness coordinate. The essential difference of the present derivation from the earlier studies by trigonometrical series expansion is that the antisymmetric in-plane displacements induced by gradients of the bending deflection (the zero-order component of transverse displacement) are expressed by the linear functions of the thickness coordinate, and the rest of displacements are expanded in cosine series of the 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 for AT-cut quartz plates are calculated from the present 2-D equations as well as from the 3-D equations, and comparison shows that the agreement is very close without introducing any corrections. Predicted frequency spectra by the present equations are shown to agree closely with the experimental data by Koga and Fukuyo [J. Inst. Elec. Comm. Engrs. of Japan 36, 59 (1953)] and those by Nakazawa, Horiuchi, and Ito [Proceedings of 1990 IEEE Ultrasonics Symposium (IEEE, New York, 1990)].