A set of hierarchical finite elements for quartz plate resonators

Formulation of a finite strip element for high frequency quartz resonator plate vibrations is presented. The finite strip formulation is equivalent to the nth order Mindlin plate theory depending on the order of the strip in the thickness direction. Consequently the finite strip element could be used to model any high frequency resonator plate vibrations by choosing the appropriate nth order plate theory. This choice could be incorporated into the finite strip program so that the user may model higher overtones of thickness shear vibrations without regeneration of finite element meshes. This forms the basis of the set of hierarchical finite elements for high frequency quartz plate vibrations. The mass effect of electrodes is included in the formulation. Numerical results for the first and third order Mindlin plate theory are obtained and compared with Koga´s experimental results. Good comparisons are obtained