The Two-dimensional Analysis of Thickness-shear Vibrations of a Convex-convex Quartz Plates
Low frequency piezoelectric quartz crystals working on the first overtone of thickness-shear mode are popular sources of a precious and highly stable frequencies. The analysis of such crystals is more complicated compared to flat crystals. Usually the energy trapping method is used instead of full three-dimensional analysis. In this paper the new theory of shells  is employed. The crystal plate is assumed to be a symmetrical shell. Two-dimensional anisotropic material tensors are obtained. Mode shape of the fundamental harmonic and anharmonic overtones are numerically calculated.