Mechanical effects of electrodes on the vibrations of quartz crystal plates

A system of approximate first-order equations is extracted from an infinite system of 2-D equations for piezoelectric crystal plates with thickness-graded material properties, which is deduced from the 3-D equations of linear piezoelectricity. These equations are used to study mechanical effects on the thickness-shear (TS), flexural (F), and face-shear (FS) vibrations of an AT-cut quartz 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 gold and aluminum electrodes with R, the ratio of the mass of the electrodes to that of the plate, equal to 0.05. without introducing any correction factors. In order to assess electrode effects, spectra of omega vs. a/bq (length-to-thickness ratio of the quartz) are computed for plates with gold and aluminum electrodes and different R ratios. And the spectrum of omega vs. R is computed for plates with aluminum electrodes and a given a/bq ratio. For a plate with gold electrodes, the frequencies of predominant TS, F, and FS modes are decreasing as R increases, but the amount of frequency changes for the TS mode is much greater than those for the other two modes. However, for a plate with aluminum electrodes, the frequencies of the TS and FS modes are decreasing, but those of the F modes are increasing as R increases.