A Homotopy Shape Solution for Thickness-Vibration of Centrally Partially Electroded Regular Polygonal At-Cut Quartz Resonators

In this paper, the homotopy polygonal electrode is applied in the quartz crystal resonator, whose shapes are determined by homotopy transform. For theoretical analysis, the ST two-dimensional scalar differential equation is used and the displacement is assumed as summation of trigonometric series. Two-dimensional Gaussian quadrature is used to solve the complex integral part of the equation. Tikhonov regularization is used to approximate the result when the ill-conditioned matrices that are difficult to solve emerge. The frequencies and the mode shapes of thickness-shear vibration obtained from theoretical calculation are compared with the simulation results of Comsol to validate the correctness of both methods.