Electrically Forced Vibrations of Partially Electroded Rectangular Quartz Plate Piezoelectric Resonators

We performed a theoretical analysis on electrically forced, coupled thickness-shear, and flexural vibrations of a rectangular AT-cut quartz resonator, which is electroded in its central part only. Mindlin's first-order theory for piezoelectric plates is used and simplified by the flexure-twist approximation due to Mindlin and Spencer. An analytical solution is then obtained. The admittance of the resonator under a time-harmonic driving voltage, which is an important property in resonator design, is calculated and examined. The mode shapes at resonances are also presented and discussed. It is found that for a perfectly symmetric resonator only cylindrical modes depending on one of the two in-plane spatial coordinates can be excited by the applied voltage. This supports the conventional approximate analyses based on the assumption of cylindrical deformations.