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

doi:10.1007/s10483-025-3255-6

Abstract

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.

Key words: square quartz, quartz crystal resonator (QCR), force-frequency effect, eccentrically concentrated load, distributed load, force sensor

2010 MSC Number:

O343

Lixia MA, Qiang ZHOU, Lijun YI, Ji WANG. A semi-analytical model and mechanism analysis for force-frequency effect and coefficient of square quartz. Applied Mathematics and Mechanics (English Edition), 2025, 46(6): 1089-1106.

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Figures/Tables 18

Fig. 1

Fig. 2

Fig. 3

Fig. 4

Table 1

Values of R1 and R3 for square quartz under eccentrically concentrated loads based on Mathieu's exact solution"

$M$	Parallel to the $x 1$ -axis		Perpendicular to the $x 1$ -axis
$M$	$R 1$	$R 3$	$R 1$	$R 3$
0	-1.922	0.627	0.627	-1.922
1	-1.640	0.451	0.451	-1.640
2	-1.018	0.051	0.051	-1.018
3	-0.401	-0.406	-0.406	-0.401
4	0.084	-0.863	-0.863	0.084

Table 1

Table 2

Values of R′1 and R′3 for square quartz under eccentrically concentrated loads based on Mathieu's exact solution and central stress results from anisotropic FEM simulations"

$M$	Parallel to $x 1$ -axis		Perpendicular to $x 1$ -axis
$M$	$R ′ 1$	$R ′ 3$	$R ′ 1$	$R ′ 3$
0	-1.527	0.787	0.719	-1.636
1	-1.436	0.606	0.542	-1.525
2	-1.129	0.113	0.079	-1.147
3	-0.612	-0.547	-0.493	-0.516
4	-0.041	-1.249	-1.074	0.208

Table 2

Table 3

Values of R″1 and R″3 for square quartz under eccentrically concentrated loads based on Mathieu's exact solution and central stress results from anisotropic FEM simulations"

$M$	$R ″ 1$	$R ″ 3$
0	-1.582	0.753
1	-1.481	0.574
2	-1.138	0.096
3	-0.564	-0.520
4	0.084	-1.162

Table 3

Table 4

Comparison among FFC results from our semi-analytical model (see Eq. (8) and Table 1 for concentrated loads), our modified anisotropic semi-analytical model (see Eq. (8) and Table 2 for concentrated loads, and Eq. (11) for distributed load), our nonlinear FEM solution, and published experimental results"

Load type	$t$ /mm	$f 0$ /MHz	$a$ /mm	FFC/(Hz·N^-1)		Semi-analytical model/(Hz·N^-1)	Modified anisotropic semi-analytical model/(Hz·N^-1)	Nonlinear FEM simulation/(Hz·N^-1)
Perpendicular, eccentrically concentrated load	0.46	3.58	8	$M = 0$	-11.11^[23]	$− 15.32$	$− 17.91$	$− 16.30$
				$M = 1$	-8.89^[23]	$− 10.81$	$− 13.31$	$− 10.76$
				$M = 2$	1.11^[23]	$− 0.72$	$− 1.35$	1.07
				$M = 3$	11.11^[23]	10.98	13.32	12.59
Perpendicular, symmetrically concentrated load	0.43	3.58	8	-12^[31]		$− 15.29$	$− 17.91$	$− 14.78$
Parallel, distributed load	0.020	75	1	101 010^{* [19]}		92 542.50	92 539.12	101 729.03
Parallel, distributed load	0.041 7	37.88	2	7 952.1^{* [18]}		11 804.06	11 803.01	10 674.91
$ψ = 34.8 °$ , distributed load	0.1	16.11	3	1 819.6^{* [24]}		970.93	970.96	1 110.62
	0.1	16.11	3	804^{* [38]}		970.93	970.96	1110.62
	0.1	16.11	3	962.57^{* [39]}		970.93	970.96	1110.62
	0.1	16.11	3	595.96^{* [40]}		970.93	970.96	1110.62
* denotes the results obtained through load transfer efficiency.

Table 4

Fig. 5

Fig. 6

Fig. 7

Fig. 8

Fig. 9

Fig. 10

Fig. 11

Fig. 12

Fig. 13

Fig. 14

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