Investigation of Coupling Effects in Rectangular Quartz Resonators with Partial Electrodes

Mindlin’s two-dimensional theory has been derived and applied on researches of quartz resonators for many years. However, in most of these works focuses were placed on vibrations varying only in two directions including thickness direction, while the effect of another direction like length or width direction was normally neglected. Besides, researchers often model quartz resonators as fully electroded plates because of the resultant simplicity. Since a real device is finite in all directions and only centrally electroded, results obtained in those papers can’t offer quantitative information of vibrations with enough accuracy. In this paper, a theoretical analysis of a rectangular trapped-energy resonator of AT-cut quartz is studied using 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 energy trapping and coupling phenomenon and are helpful for thorough and accurate understanding of quartz resonators’ vibrations.