Piezoelectrically forced thickness-shear and flexural vibrations of contoured quartz resonators

In a previous paper, a set of first-order equations of motion was obtained for contoured crystal plates and for frequencies up to and including those of the fundamental thickness-shear modes. In the present paper, the governing equations of contoured plates are extended to include the electric potential which is coupled to the mechanical fields by the piezoelectric effect. These equations are, then, employed for the study of piezoelectrically forced thickness-shear and flexural vibrations of beveled AT-cut quartz plate, i.e., the plate with a portion of uniform thickness between the two wings of the double wedge. Analytical solutions are obtained by the Frobenius method. Displacements, stresses, and electric potential are derivable from six independent functions which are in the form of an infinite power series. In addition to the calculations of resonance frequencies and mode shapes, the effect of the contouring on the forced mechanical displacements, electric potential, surface charge, and capacitance ratio are examined.