Finite element analysis of the high frequency vibrations of contoured crystal plates with higher-order plate theory

The advantages of contouring in crystal resonator, such as energy trapping and reducing the displacements in the edges, have been well observed and utilized. Analytical efforts for thorough understanding and precise prediction of these effects have been made through the simplified equations with few strongly coupled vibration modes and prescribed thickness variations by solving the differential equations for solutions in infinite series. These solutions have been useful in revealing and explaining some well-known phenomena such as the weakening of the couplings, but many real devices have contours which cannot be effectively expressed in simple functions and this has made it is impossible to solve the equations. In this paper, we started with the derivation of power series based Mindlin plate theory specifically for plates with variable thickness, finding that the effect of thickness variation is only limited to the face-traction terms of the two-dimensional equations of motion. The equations are further implemented in the finite element analysis by taking into consideration of the variation of the thickness through the integration of each element over the plate. Consequently, the finite element analysis is formulated in a manner similar to uniform plates with the exception that the thickness is no longer a constant. The numerical results from several thickness variation cases are presented and analyzed to show the effects of the contours.