Lee´s plate equations with new correction factors and analysis of spurious vibrations of rectangular AT-cut quartz plates

Strip AT-cut quartz resonators have been recently developed and widely used in frequency applications because of their small size. Many spurious resonances can be supported in a rectangular AT-cut plate. Therefore, the analysis of the influence of both the X and Z´ dimensions of the plate upon the spurious characteristics is necessary to design the strip resonators. However, it is very difficult to exactly solve three dimensional equations of elasticity. The practical analysis of spurious characteristics has been carried out on the basis of approximate two dimensional equations. In this paper, the applicability of Lee´s two dimensional equations to AT-cut plates is examined. The resonant frequencies are calculated as a function of X-length, and are compared to Koga´s experimental data. The result shows that new correction factors should be introduced in the equations in order to adjust the approximate values of wave numbers for the thickness-flexure waves propagating in X and Z´ directions