Three dimensional finite element calculations of an experimental quartz rotation sensor

A 3-D, finite-element (FE) study of an AT-cut quartz stripline resonator for use as a rotation sensor is presented. Modeling of all three dimensions is required to understand the coupling between the thickness-shear motion and thickness-length-type flexure and to consider geometric variations in the length direction. Experimental and predicted vibrational frequencies agree with less than 1% error. The relative strengths of electromechanical coupling also show good agreement with experiment. Thickness-shear motions in both transverse directions are well modeled. Because of the thin geometry, many high-order flexure modes must also be considered. the resulting 3-D spectrum is considerably richer in modes than the corresponding 2-D results. The model mesh is quite fine, requiring more than 65000 degrees of freedom. The high-order modes (the shear modes of interest are greater than the 100th mode in the spectrum) are carefully classified using a straightforward procedure that gives the effective piezoelectric coupling to specific electrode geometries. The FE method allows quick analysis of the influence of variations in geometric parameters