Three dimensional finite element modeling of quartz crystal strip resonators

A general and powerful finite element code has been developed for the analysis of quartz crystal strip resonators utilizing a three dimensional hexahedral element with cubic interpolations of the geometry and field quantities. This higher order interpolation has been found to be better for computing high frequency vibration modes than the twenty node quadratic elements normally used in three dimensional analysis. The difficulty in designing strip resonators comes mainly from the existence of coupling of unwanted modes such as flexure, face shear, and extension, with the thickness shear mode, and moreover, this coupling is very strongly influenced by crystal blank dimensions. Therefore, in order to properly design these devices, an analysis tool is necessary which can predict these modes with adequate precision. The general development of the theory and finite element equations are presented along with results. An approximate method is employed to remove the electric variables from the formulation, greatly reducing the storage and CPU requirements of the program. A convergence analysis is also presented which quantifies the precision differences between the quadratic and cubic elements