Thickness-shear and flexural vibrations of contoured crystal strip resonators

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