Frequency-temperature behavior of thickness vibrations of doubly rotated quartz plates affected by plate dimensions and orientations

Abstract

Three-dimensional linear equations of motion for small vibrations superposed on thermal deformations induced by steady, uniform temperature change in quartz are obtained. The material properties of quartz, such as the elastic stiffnesses and thermal expansion coefficients, are assumed temperature dependent and expressible by third-degree polynomials in temperature change. From the solutions of third-order perturbations of these equations for the thickness resonances of infinite quartz plates, six values of the effective third temperature derivatives of elastic stiffnesses C̃(3)pq are calculated by the use of the measured temperature coefficients of frequency by Bechmann, Ballato, and Lukaszek [Proc. IRE 50, 1812 (1962)] for various doubly rotated cuts and the values of the first temperature derivatives C(1)pq and the effective second temperature derivatives C̃(2)pq obtained in a previous study. An infinite system of two-dimensional equations of motion is derived by Mindlin’s method of power-series expansion for crystal plates subject to a steady, uniform temperature change. Four equations, governing the coupled thickness-shear, thickness-twist, thickness-stretch, and flexural vibrations, are extracted from the infinite set and employed to study the frequency-temperature behavior of thickness vibrations of finite SC-cut quartz plates with a pair of free edges. Changes in the thickness-shear resonance frequencies as a function of temperature are predicted and plotted for various values of orientation angles θ and φ, and length-to-thickness ratio a/b. Effects on the frequency-temperature behavior of the plates due to changes in the values of θ, φ, and a/b are observed and discussed.