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

摘要

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.