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Mindlin's two-dimensional theory has been derived and applied to research on quartz resonators for a long time. However, most works have focused on vibrations varying only in two directions, including thickness direction, while the effect of other directions like the length or width direction is normally neglected. Besides, researchers often model quartz resonators as fully electroded plates because of the resulting simplicity. Since a real device is finite in all directions and is only centrally electroded, results obtained in such works cannot offer quantitative information on vibrations with enough accuracy. In this paper, a theoretical analysis of a rectangular trapped-energy resonator of AT-cut quartz is studied using the Ritz method, associated with the variational formulation of Mindlin's first-order equations. Frequency spectra and mode shapes of a real-scaled trapped-energy resonator, which is finite in all directions, are obtained with the consideration of mode couplings among thickness-shear mode, thickness-twist mode, and flexural mode. Results show the existence of an energy-trapping and coupling phenomenon and are helpful for thorough and accurate understanding of quartz resonator vibrations. Detailed discussions on the effects of structural parameters on mode couplings and energy trapping are provided, and the results can helpfully guide the selection of aspect ratio, length/thickness ratio, and electrode inertia in device design.

Coupling Vibration Analysis of Trapped-Energy Rectangular Quartz Resonators by Variational Formulation of Mindlin’s Theory