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)].

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.

An infinite system of two-dimensional (2-D) equations for piezoelectric plates with general symmetry and faces in contact with vacuum is derived from the 3-D equations of linear piezoelectricity in a manner similar to that of previous work, in which an infinite system of 2-D equations for plates with electroded faces was derived.By using a new truncation procedure, second-order equations for piezoelectric plates with faces in contact with either vacuums or electrodes are extracted from the aforementioned infinite systems of equations, respectively. The second-order equations for plates with or without electrodes are shown to predict accurate dispersion curves by comparing to the corresponding curves from the 3-D equations in a range up to the cut-off frequencies of the first symmetric thickness-stretch and the second symmetric thicknessshear modes without introducing any correction factors. Furthermore, a system of 1-D second-order equations for strips with rectangular cross section is deduced from the 2-D second-order equations by averaging variables across the narrow width of the plate. The present 1-D equations are used to study the extensional vibrations of barium titanate strips of finite length and narrow rectangular cross section. Predicted frequency spectra are compared with previously calculated results and experimental data.

A set of two-dimensional (2-D), second-order approximate equations for extensional, thickness-stretch and symmetric thickness-shear vibrations of piezoelectric ceramic plates with electroded faces is extracted from the infinite system of 2-D equations deduced previously. The new truncation procedure developed recently is used for it improves the accuracy of calculated dispersion curves. Closed-form solutions are obtained for free vibrations of circular disks of barium titanate. Dispersion curves calculated from the present approximate 2-D equations are compared with those obtained from the 3-D equations, and the predicted resonance frequencies are compared with experimental data. Both comparisons show good agreement without any corrections. The frequencies of the edge modes calculated from the present 2-D equations are very close to the experimental data. Furthermore, mode shapes at various frequencies are calculated in order to identify the frequency segments of the spectrum at which one of the coupled modes-i.e., the radial extension (R), edge mode (Eg), thickness-stretch (TSt), and symmetric thicknessshear (s.TSh)-is predominant.

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