Analysis of Doubly Rotated Quartz Resonators Utilizing Essentially Thickness Modes with Transverse Variation

摘要

Closed-form asymptotic expressions for the frequency-wavenumber dispersion relations in doubly rotated quartz plates vibrating in the vicinity of the odd pure thickness frequencies are derived from the equations of linear piezoelectricity and the associated boundary conditions on the major surfaces. The usual assumptions of small piezoelectric coupling and small wavenumbers along the plates are made and it is supposed that the pure thickness frequencies are sufficiently different that one pure thickness wave is dominant at a time. In the treatment the mechanical displacement is decomposed along the eigenvector triad of the pure thickness solution to facilitate the asymptotic analysis. The fact that the wavenumbers along the plate are restricted to be small significantly reduces the complexity of the equations without neglecting any transformed elastic constants. The resulting asymptotic dispersion equation enables the construction of a scalar differential equation describing the transverse behavior of essentially thickness modes of vibration in doubly rotated quartz plates.