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Research on Virtual Calibration Method for Strain Gauge Balance Based on Non-Linear FEA Cover

Research on Virtual Calibration Method for Strain Gauge Balance Based on Non-Linear FEA

Measurement Science Review
Volume 26 (2026): Issue 5 (October 2026)
By: ,  ,  ,  ,   and    
Open Access
|Oct 2026
Figures & tables

Figures & Tables

Fig. 1.

The Wheatstone bridge.

Fig. 2.

Photograph of the investigated balance.

Table 1.

Design loads for the investigated balance.

ComponentFy [N]Mz [N·m]Fx [N]Mx [N·m]Fz [N]My [N·m]
Design load1800018001500120040001200
Fig. 3.

FEA mesh of the investigated balance.

Table 2.

The VC matrix of the investigated balance.

ComponentFyMzFxMxFzMy
Csen12236.951379.643506.152638.975350.37603.80
Fy00.00042750.0222331−0.0001753−0.0000060.0000006
Mz0.01309280−0.0229588−0.0008566−0.00005070.0000066
Fx−0.00385660.000000200.0000015−0.00000230.0000003
Mx0.00082020.00006220.0015130−0.00020150.0043991
Fz0.00009040.000006−0.000049−0.000750200.0000102
My−0.000352−0.0000829−0.0012826−0.1637126−0.00088860
Fy*Fy0.00E+007.94E–109.00E–08−3.50E–111.46E–101.46E–12
Mz*Mz1.21E–070.00E+00−6.20E–06−3.57E–09−1.14E–08−1.77E–09
Fx*Fx−5.31E–09−8.31E–100.00E+00−1.66E–11−3.10E–10−3.50E–11
Mx*Mx4.91E–06−4.58E–07−2.80E–050.00E+004.36E–093.17E–10
Fz*Fz−8.43E–09−1.53E–101.56E–071.75E–100.00E+001.18E–10
My*My1.71E–06−2.99E–073.35E–061.08E–098.23E–080.00E+00
Fy*Mz6.96E–096.36E–091.27E–06−6.90E–10−1.27E–091.64E–10
Fy*Fx1.64E–08−7.12E–098.65E–071.45E–09−3.23E–09−1.29E–11
Fy*Mx3.81E–09−6.22E–102.41E–08−7.98E–081.96E–061.64E–07
Fy*Fz6.26E–10−1.45E–10−8.48E–11−1.09E–08−3.21E–09−5.60E–10
Fy*My1.06E–08−2.21E–09−3.97E–092.80E–07−1.97E–07−3.23E–08
Mz*Fx−1.27E–061.94E–07−1.82E–071.45E–081.40E–082.52E–09
Mz*Mx−7.10E–089.59E–094.58E–08−4.76E–07−1.33E–05−1.98E–06
Mz*Fz6.26E–09−4.00E–103.85E–092.18E–07−1.43E–07−8.15E–09
Mz*My−6.54E–084.22E–09−1.98E–084.63E–066.34E–07−7.66E–09
Fx*Mx8.29E–09−6.48E–10−3.25E–08−1.58E–08−1.89E–07−2.18E–08
Fx*Fz1.51E–095.69E–11−4.87E–092.52E–09−7.28E–081.72E–08
Fx*My1.10E–086.32E–118.57E–094.94E–092.06E–062.84E–07
Mx*Fz−1.98E–06−6.51E–082.35E–07−1.64E–098.28E–099.96E–10
Mx*My−4.26E–056.91E–063.20E–051.50E–062.04E–085.51E–09
Fz*My3.13E–07−1.33E–08−2.05E–061.24E–064.60E–092.94E–09
Table 3.

The SC matrix of the investigated balance.

ComponentFyMzFxMxFzMy
Csen11922.801347.853332.672568.625228.89589.48
Fy00.0000390.023591−0.0004420.00341−0.00033
Mz0.0121080−0.023699−0.0014810.031882−0.00421
Fx−0.004616−0.0000540−0.0001420.00025−0.000064
Mx0.0136680.0026060.0030220−0.007704−0.003265
Fz0.007823−0.001112−0.0014610.00056100.000723
My0.046364−0.010228−0.003349−0.1606410.0227330
Fy*Fy0.00E+001.48E–091.16E–070.00E+008.63E–090.00E+00
Mz*Mz9.22E–070.00E+00−5.63E–06−9.33E–081.14E–073.07E–08
Fx*Fx−1.54E–078.53E–090.00E+000.00E+00−9.21E–086.00E–09
Mx*Mx−2.52E–069.03E–08−2.75E–050.00E+003.90E–078.96E–08
Fz*Fz−7.88E–086.68E–091.72E–075.66E–090.00E+003.24E–09
My*My8.30E–07−1.21E–073.54E–06−1.63E–08−5.79E–070.00E+00
Fy*Mz2.22E–07−2.51E–081.36E–06−5.72E–081.52E–07−1.47E–08
Fy*Fx−2.24E–072.53E–088.94E–073.04E–09−1.84E–082.22E–09
Fy*Mx−2.60E–082.84E–07−4.76E–08−2.67E–072.19E–062.64E–07
Fy*Fz0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
Fy*My0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
Mz*Fx−1.29E–072.35E–07−3.91E–07−4.01E–083.38E–076.01E–10
Mz*Mx9.44E–07−2.05E–076.60E–07−1.23E–06−5.32E–06−6.91E–07
Mz*Fz0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
Mz*My0.00E+000.00E+000.00E+000.00E+000.00E+000.00E+00
Fx*Mx5.84E–075.66E–08−1.93E–071.25E–078.95E–07−2.51E–07
Fx*Fz−8.84E–08−2.87E–09−3.11E–08−2.98E–08−1.08E–07−3.95E–08
Fx*My7.48E–077.17E–08−4.62E–072.25E–071.35E–064.26E–07
Mx*Fz−6.13E–073.43E–071.62E–06−3.65E–073.36E–079.47E–08
Mx*My−3.00E–061.79E–073.14E–05−1.17E–07−2.56E–07−6.27E–07
Fz*My3.37E–07−2.93E–08−2.17E–063.60E–08−1.09E–06−1.62E–07
Fig. 4.

The SC setup for the investigated balance.

Table 4.

Comparison of sensitivity coefficients.

ComponentFyMzFxMxFzMy
VC [N/(mV/V) or N·m/(mV/V)]12236.951379.643506.152638.975350.37603.80
SC [N/(mV/V) or N·m/(mV/V)]11922.801347.853332.672568.625228.89589.48
Relative error [%]2.62.45.22.72.32.4
Fig. 5.

The schematic diagram of the strain gauge bonding position on the investigated balance.

Table 5.

Comparison of first-order linear terms between the VC and SC matrices.

ComponentFyMzFxMxFzMy
VC [% FS]0.100.4323.93−17.01−0.040.45
SC [% FS]0.66−0.7225.05−16.783.43−1.22
Differences [% FS]−0.561.15−1.12−0.23−3.471.67

[i] Note: FS - full scale

Fig. 6.

First-order linear interactions on the Fx component.

Table 6.

Comparison of quadratic interactions.

ComponentFyMzFxMxFzMy
VC [% FS]0.05−0.05−1.590.000.010.00
SC [% FS]−0.010.03−0.84−0.020.070.02
Differences [% FS]0.06−0.08−0.750.02−0.06−0.02
Fig. 7.

Quadratic interactions on the Fx component.

Table 7.

Comparison of interactions of cross-product terms.

ComponentFyMzFxMxFzMy
VC [% FS]−0.400.566.781.850.31−0.08
SC [% FS]−0.010.457.25−0.971.020.22
Differences [% FS]−0.390.11−0.472.82−0.71−0.3
Fig. 8.

Cross-product term interactions on the Fx component.

Table 8.

Comparison of percent interactions across all terms.

ComponentFyMzFxMxFzMy
VC [%FS]−0.240.9429.12−15.160.280.37
SC [%FS]0.64−0.2431.47−17.774.51−0.97
Differences [%FS]−0.881.18−2.352.61−4.231.34
DOI: https://doi.org/10.2478/msr-2026-0034 | Journal eISSN: 1335-8871
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Language: English
Page range: 278 - 286
Submitted on: Apr 6, 2025
Accepted on: Sep 10, 2026
Published on: Oct 18, 2026
Published by: Slovak Academy of Sciences, Institute of Measurement Science
In partnership with: Paradigm Publishing Services
Publication frequency: Volume open
Keywords:
strain gauge balance,
finite element analysis,
static calibration,
virtual calibration
Related subjects:
Engineering,
Electrical engineering,
Control engineering, metrology and testing

© 2026 Chao Wang, Tao Ma, Zhiwei Xu, Guangwei Xiang, Huangwei Zhang, Peng Mi, published by Slovak Academy of Sciences, Institute of Measurement Science
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 26 (2026): Issue 5 (October 2026)
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