Force-frequency coefficient of symmetrical incomplete circular quartz crystal resonator

The changes in the resonance frequencies of the thickness-shear vibration of symmetrical incomplete circular AT-cut quartz crystal resonators (QXRs), which are used as sensing elements in digital force sensors or pressure transducers, by the application of diametrical forces are discussed by considering the piezoelectric effect and the anistropic characteristics of crystal plates. Two-dimensional motion equations for predicting the frequency changes are derived from three-dimensional piezoelectricity equations, and equations deduced by Janiaud for solving the stress distribution in crystal plates are used to calculate the force-frequency coefficients of incomplete circular resonators. The results show that the piezoelectric effect of crystal plates decreases the force-frequency coefficient of the azimuth angle 0 degrees by 7% as compared with no piezoelectricity assumption. The incomplete circular shape can be applied to a larger load than the circular disc because the flat regions of the incomplete shape provide a distributed load application without the problems of stress concentration of a point force inherent to a circular disc. The incomplete shape gives an increased force sensitivity when compared with the complete disc near azimuth angle 0 degrees. The theoretical curves agree well with Ratajski's experimental results.