A semi-analytical model and mechanism analysis for force-frequency effect and coefficient of square quartz.

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Bibliographic Details
Title:	A semi-analytical model and mechanism analysis for force-frequency effect and coefficient of square quartz.
Authors:	Ma, Lixia^1,2,3 (AUTHOR), Zhou, Qiang⁴ (AUTHOR), Yi, Lijun^1,2 (AUTHOR) yilijun@nbu.edu.cn, Wang, Ji¹ (AUTHOR)
Source:	Applied Mathematics & Mechanics. Jun2025, Vol. 46 Issue 6, p1089-1106. 18p.
Subjects:	Crystal oscillators, Crystal resonators, Quartz crystals, Finite element method, Azimuth, Quartz
Abstract:	This study presents a closed-form solution for central stress, a semi-analytical model, and a modified anisotropic semi-analytical model to efficiently calculate the force-frequency coefficients (FFCs) of square quartz crystal resonators (QCRs) with different side lengths and azimuth angles under eccentrically concentrated and distributed loads. The semi-analytical model is validated by comparisons between the experimental results and the nonlinear finite element method (FEM) simulation results. Based on the semi-analytical model for the FFC and nonlinear FEM simulations, the FFC variations of square QCRs under external loads and the related mechanisms are investigated. Among the initial stresses caused by external loads, the central stress parallel to the x-crystallographic axis is the primary factor influencing the FFC of quartz. Our findings can provide practical tools for calculating the FFC, and help the design and development of square quartz force sensors. [ABSTRACT FROM AUTHOR]
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Database:	Engineering Source

Description
Abstract:	This study presents a closed-form solution for central stress, a semi-analytical model, and a modified anisotropic semi-analytical model to efficiently calculate the force-frequency coefficients (FFCs) of square quartz crystal resonators (QCRs) with different side lengths and azimuth angles under eccentrically concentrated and distributed loads. The semi-analytical model is validated by comparisons between the experimental results and the nonlinear finite element method (FEM) simulation results. Based on the semi-analytical model for the FFC and nonlinear FEM simulations, the FFC variations of square QCRs under external loads and the related mechanisms are investigated. Among the initial stresses caused by external loads, the central stress parallel to the x-crystallographic axis is the primary factor influencing the FFC of quartz. Our findings can provide practical tools for calculating the FFC, and help the design and development of square quartz force sensors. [ABSTRACT FROM AUTHOR]
ISSN:	02534827
DOI:	10.1007/s10483-025-3255-6