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

Abstract

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.

The Multi-field Coupled Vibration Analysis of AT-Cut Quartz Crystal Resonators with Parallelism Error

“…Totally speaking, the vibration response of QCRs is solving the straight-crested waves governed by Eq. ( 5) under consideration of boundary conditions ( 8) and (9). Under the external voltage, the electric field E2 can be calculated via…”

“…However, because of the anisotropy of quartz and structural size, a pure TSh mode is impossible when QCR works, in which other modes like flexural, and face-shear modes are usually coupled together. In order to suppress these unwanted modes, the length/thickness ratio of the quartz plate should be chosen properly [6][7][8][9]. During the structural design process, the frequency spectrum is usually used to analyze the mode coupling of QCRs [6], from which certain length/ thickness ratios that are related to strong mode coupling and must be excluded can be clearly achieved.…”

Surface Roughness Effects on the Vibration Characteristics of AT-Cut Quartz Crystal Plate

“…For instance, some spurious modes with their resonance frequencies extremely close to the resonant frequency are coupled with the TSh mode [4], such as extension (E) mode, face-shear (FS) mode, flexural (F) mode, etc. To avoid these unwanted modes and spurious modes, it is essential to obtain the aspect ratio of the quartz crystal plate [4][5][6][7]. As investigated in ref.…”

“…As investigated in ref. [6], for the straight-crested waves and vibration modes around the operating TSh mode of the quartz crystal plate, it is necessary to exclude certain plate length/thickness ratios. However, most studies on this topic focus on the vibration characteristics of ideal resonators at room temperature [8][9][10][11][12][13][14].…”