Use of 1-D Finite Elements for the Temperature Behavior of a Contoured and Partially Plated SC-Cut Resonator

摘要

A two-dimensional, Langragean variational equation of motion for the incremental displacements superposed on homogenous thermal strains in an electrode plated crystal resonator was derived. A one-dimensional form of the equation was considered and solved using 1-D, cubic finite elements. The program was applied to the study of the f-T behavior of a contoured and partially plated SC-Cut quartz strip. The variable thickness in a contoured strip was interpolated using the same cubic finite element shape functions. Numerical convergence of the fundamental thickness shear frequency and its f-T curve was examined in meshes of increasing fineness. A preliminary study was performed for the thickness shear mode and its f-T curve with respect to the electrode thickness and contour, including both the plano -convex and plano-gaussian contoured surfaces. The plano-gaussian strip has more parameters than the plano-convex strip in the control of f-T behavior and energy trapping. Keywords: Quartz crystal resonators, Frequency temperature behavior, Finite element analysis, Reprints. (jhd)