Frequency-temperature relations of thickness-shear and flexural vibrations of contoured quartz resonators

The two-dimensional first-order equations of incremental vibrations superposed on the steady and homogeneous thermal deformations of crystal plates by Lee and Yong are reduced to a set of four coupled equations for frequencies up to and including those of the fundamental x ₁x₂ (or slow) thickness-shear mode. These equations are employed to study the frequency-temperature relations of the thickness-shear and flexural vibrations of beveled plates of AT-cut of quartz. Closed form solutions for the uniform portion and analytical solutions in terms of infinite power series for the contoured portions are obtained. By requiring these solutions to satisfy the continuity conditions and traction-fret edge conditions, the frequency equation is obtained. Effects of the contouring and the orientation angle of the plate are examined systematically by computing the frequency changes as a function of temperature increment for various values of geometric parameters and the orientation angle of the plate. Similar results for the uniform and finite plates are also computed and presented for comparison