Thickness-shear and flexural vibrations of contoured crystal strip resonators

Abstract

A system of two-dimensional equations of motion of successively higher-order approximations for contoured crystal plates are deduced from three-dimensional equations of elasticity by expansion in a series of trigonometrical functions. By removing the first-order thickness-stretch and x2x3 thickness-shear modes and all the higher modes, a set of first-order equations of motion is obtained for contoured crystal plates and for frequencies up to and including those of the fundamental thickness-shear modes. Exact solutions in terms of infinite power series are obtained for the coupled thickness-shear and flexural vibrations of quartz strip resonators with linearly varying thickness in the x1 direction. Frequency spectra and mode shapes of strip resonators are computed for various values of bevel. The effect of the bevel on the frequencies and modes is examined