Multimodal Coupled Vibration Model for Force-Frequency Effect Analysis of Rectangular Quartz Plates

초록

Purpose In view of the application advantages of rectangular quartz plates and their application potential in developing force sensors, this paper proposes a numerical method based on the multimodal coupled vibration model to investigate the force-frequency effect of rectangular quartz plates. It aims to improve numerical calculation methods and provide a theoretical reference for the accurate calculation of force-frequency effect in quartz plates. Methods A theoretical framework of three-dimensional incremental theory considering piezoelectric effects was established. The three-dimensional model was simplified using Mindlin's two-dimensional plate theory to derive the equations of motion, incorporating geometric nonlinearity and higher-order material coefficients. Through series truncation of displacements, the zero-order and first-order five-mode coupled equations were obtained. Considering the influence of initial stress, steady-state analysis was first performed to solve the initial stress distributions of the rectangular plate under axial load. By defining a reasonable stress extraction area, accurate mean values of stress or displacement gradient were acquired as initial inputs for the governing equations. Results With the aid of a partial differential equation module, the eigenvalue problem of five-mode coupled vibrations was solved, yielding the frequency variations under different loads. The influence of factors such as different plate thicknesses and aspect ratios on the force-frequency effect was investigated. The results reveal that the force-frequency coefficient gradually converges as the aspect ratio increases, and that thinner plates exhibit larger force-frequency coefficients. The numerical results were verified experimentally. Conclusion The findings clarify the linear force-frequency response mechanism of quartz plates and the regulatory role of geometric parameters, and provide important guidance for the design and optimization of resonant sensors based on thickness-shear mode.

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