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Purpose <p>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.</p> Methods <p>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.</p> Results <p>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.</p> Conclusion <p>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.</p>

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

  • 期刊论文

  • 发表日期：

    2026年3月18日

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  • Yan Guo,
  • Jack Wu,
  • Zhe Wang,
  • Qiang Zhou,
  • Ji Wang,
  • Heung Soo Kim,
  • Bin Huang

摘要

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.