Characteristics of a Lagrangian, high-frequency plate element for the static temperature behavior of low-frequency quartz resonators

Finite-element matrix equations based on the Lagrangian, first-order, incremental plate equations of motion superposed on homogeneous thermal strains were formulated using virtual work principles. A program for an isoparametric, four-node quadrilateral element was written and applied to the study of the frequency-temperature (FT) behavior of flexure-mode quartz resonators. The lumped-mass and consistent-mass matrices were found to yield practically the same FT curves. For simple prismatic resonators, two schemes, reduced/selective integration and incompatible modes, produced relatively similar FT curves. The incompatible modes scheme yielded better results for resonators of more complex shapes, such as the tuning fork. It is concluded that the six-degree-of-freedom per node element is needed for the analysis of the FT behavior of a fully anisotropic flexure-mode resonator.<>