Force Frequency Effects in Third Overtone Thickness Shear Quartz Resonators

The behavior of vibrating crystal plates under the action of external forces was studied. The plates were respectively subjected to diametrical compression force and bending force in different configurations: (a) a clamped cantilever, (b) a cantilever with displaced knife edges, and (c) a dual support mounting with displaced knife edge applied midway between. Finite element models were developed using the theory of small deformations superposed on finite initial deformations in Lagrangian formulation. The model results compared well with the experimental results by Mingins, Ballato, et. al., [3, 4]. For the AT-cut plate, the azimuth similar to % angles of frequency insensitivity due to compressional force were found at psi similar to 65 degrees and 115 degrees. For the plate bending in a clamped cantilever configuration a W-shaped plot of negative frequency deviations was observed when similar to% was varied from 0 degrees to 180, while the maximum positive frequency deviations occurred at psi similar to 270 degrees. For the plate bending of cantilever with displaced knife edges, the results were similar to the plate bending in a clamped cantilever configuration. For the plate bending of dual support mounting with displaced knife edge applied in midway, the results for psi = 0 degrees to 180 degrees were very similar to psi = 180 degrees to 360 degrees. By searching out the effects of frequency deviations, optimal angles can be obtained to minimize or eliminate them.