Two-dimensional equations of motion of successively higher-order approximations for piezoelectric crystal plates with triclinic symmetry are deduced from the three-dimensional equations of linear piezoelectricity by expansion in series of trigonometric functions of the thickness coordinate of the plate. These equations, complemented by two additional relations: one, the usual relation of face tractions to the mass of electrodes, and the other relating face charges to face potentials and face displacements, can accommodate either the traction and charge boundary conditions at the faces of the plate without electrodes or the traction and potential boundary conditions at the faces of the plate with electrodes. Dispersion curves are obtained from the first- to fourth-order approximate plate equations for a rotated 45° Y-cut lithium tantalate plate without electrodes, and these curves are compared with those from the frequency equation of the three-dimensional equations with close agreement. Solutions of forced vibrations of an AT-cut quartz plate with electrodes are obtained from the first-order plate equations complemented by the two additional relations. It is shown that cutoff frequencies of the fundamental thickness-shear vibrations from the approximate equations are identical to the ‘‘exact’’ ones obtained from the three-dimensional equations.

A system of two-dimensional first-order equations for piezoelectric crystal plates with general symmetry and with electroded faces was recently deduced from the three-dimensional equations of linear piezoelectricity. Solutions of these equations for AT-cut plates of quartz were shown to give accurate dispersion curves without corrections, and the resonances predicted agree closely with the experimental data of Koga and Fukuyo [I. Koga and H. Fukuyo, J. Inst. Electr. Commun. Eng. Jpn. 36, 59 (1953)] and that of Nakazawa, Horiuchi, and Ito (M. Nakazawa, K. Horiuchi, and H. Ito, Proceedings of the 1990 IEEE Ultrasonics Symposium, pp. 547–555). In this article, these equations are employed to study the free as well as the forced vibrations of doubly rotated quartz plates. Solutions of straight-crested vibrational modes varying in the x1 and x3 directions of SC-cut quartz plates of infinite extent are obtained and from which dispersion curves are computed. Comparison of those dispersion curves with those from the three-dimensional equations shows that the agreement is very close without any corrections. Resonance frequencies for free vibrations and capacitance ratios for piezoelectrically forced vibrations are computed and examined for various length-to-thickness or width-to-thickness ratios of rectangular SC-cut quartz plates. The capacitance ratio as a function of forcing frequency is computed for a rectangular AT-cut quartz and compared with the experimental data of Seikimoto, Watanabe, and Nakazawa (H. Sekimoto, Y. Watanabe, and M. Nakazawa, Proceedings of the 1992 IEEE Frequency Control Symposium, pp. 532–536) and is in close agreement.