A system of approximate first-order governing equations is extracted from an infinite system of two-dimensional equations for piezoelectric crystal plates with thickness-graded material properties, which is deduced from the three-dimensional equations of linear piezoelectricity using a similar approach as that in Lee and Yu [IEEE Trans. UFFC, 45 (1), 1998] with the new series expansion of Lee, Yu, and Lin [J. Appl. Phys., 83 (3), 1998]. These equations are employed to study mechanical effects on the thickness-shear (TS), flexural (F), and face-shear (FS) vibrations of an AT-cut quartz plate plated with two identical electrodes. Dispersion curves are calculated from the present 2-D equations as well as the 3-D equations. The comparison of these curves shows that the agreement is very close for all three frequency branches of TS, F, and FS modes in a range up to the 1.5 times the fundamental TS frequency and for various values of mass ratio of electrodes to the plate, R, without introducing any correction factors. Mass ratio, thickness ratio, density ratio, and stiffness ratio of electrodes to the crystal plate are the parameters affecting the resonance frequencies Ω of the composite plate. In order to assess their effects, Ω vs a/b_(q) (length-to-thickness ratio of the quartz) are computed for various R of gold electrodes, and Ω vs R for a given a/b_(q) and electrodes of gold and aluminum. It is found that for plate with gold electrodes, the frequencies of predominant TS, F, and FS modes are decreasing as R increasing, but the amount of frequency changes for the TS mode is much greater than those for the other two modes. However, for the plate with aluminum electrodes the frequency of the TS mode is decreasing, but those of the F and FS modes are increasing as R increasing, reflecting the relative effects among the mass ratio, thickness ration, as well as the stiffness ratio for different electrode materials.
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