## 2 Bearing Load Testing Method Based on Redundant Strain and Jacking Data Fusion
### Data Fusion Processing
#### 2.3.1 Redundant Strain Data Fusion Based on Confidence Distance  
To achieve the fusion of redundant strain data, firstly, based on SGM, multiple sets of measurement cross-sections should be arranged, combined, and screened to obtain \(n\) effective combinations. For the strain data of each combination, the supporting force corresponding to the pivot position \(F_{s_{i}}\) can be obtained through SGM. Let the section bending moment be \(M\), the shaft length be \(L_{1}\), \(L_{2}\), and \(L_{3}\), the uniformly distributed load be \(q\), and the supporting force be \(F\). According to reference [11], the following equation can be obtained:  
\[F=\frac{1}{L_{1}}(aM_{1}+bM_{2}+cM_{3}+d) \tag{1}\]  
Figure 5: Data processing flowchart. These parameters can be calculated by:  
\[\left\{ \begin{array}{l}a=-1\\ b=\frac{(L_{1}+L_{2}+L_{3})}{L_{3}}\\ c=-\frac{(L_{1}+L_{2})}{L_{3}}\\ d=\frac{q}{2}(L_{1}+L_{2})(L_{1}+L_{2}+L_{3})\\ \end{array} \right. \tag{2}\]  
The section bending moment in Equation (1) is obtained through the following formula:  
\[M=\frac{1}{2}\varepsilon EW \tag{3}\]  
where \(\varepsilon\) is the measured bending strain on the shaft cross-section; \(E\) is the elastic modulus of the material of the shaft section; \(W\) is the section modulus of resistance to bending. Assuming the true support reaction force is \(\overline{F}\), and the true section bending moment is \(\overline{M}\), according to Equation (1), the following can be derived:  
\[\overline{F}=\frac{1}{L_{1}-\Delta L}(a(M_{1}-\Delta M_{1})+b(M_{2}-\Delta M _{2})+c(M_{3}-\Delta M_{3})+d) \tag{4}\]  
Based on Equations (1) and (4), the error of the SGM results can be obtained as:  
\[\Delta F=F-\overline{F}=\frac{a\Delta M_{1}+b\Delta M_{2}+c\Delta M_{3}}{ \overline{L}_{1}}-F\frac{\Delta L}{\overline{L}_{1}} \tag{5}\]  
Taking into account the presence of measurement errors, the measurement equation for the sectional strain \(\varepsilon_{i}\) can be expressed as:  
\[\varepsilon_{i}=\overline{\varepsilon}_{i}+k_{\varepsilon}\overline{ \varepsilon}_{i}+\omega_{\varepsilon}\approx\overline{\varepsilon}_{i}+k_{ \varepsilon}\varepsilon_{i}+\omega_{\varepsilon} \tag{6}\]  
Assuming that the systematic error coefficient \(k_{\varepsilon}\) and the random error \(\omega_{\varepsilon}\) for the sectional strain \(\varepsilon_{i}\) follows a normal distribution, \(k_{\varepsilon}{\sim}N(0,\sigma_{k_{\varepsilon}}^{2})\) (mainly influenced by the quality of strain gauge bonding) and \(\omega_{\varepsilon}{\sim}N(0,\sigma_{\omega_{\varepsilon}}^{2})\) (mainly influenced by human factors). According to the principle of maximum likelihood, the PDF of the sectional bending moment \(M_{i}\) is:  
\[f_{M_{i}}(x)=\left(2\pi\sigma_{M_{i}}^{2}\right)^{-1/2}\exp(-\frac{(x-\mu_{M_{ i}})^{2}}{2\sigma_{M_{i}}^{2}}) \tag{7}\]  
Based on Equation (3), these parameters can be calculated by:  
\[\left\{ \begin{array}{l}\mu_{M_{i}}=\frac{1}{2}\varepsilon_{i}EW\\ \sigma_{M_{i}}^{2}=\sigma_{k_{\varepsilon}}^{2}\varepsilon_{i}^{2}+\sigma_{ \omega_{\varepsilon}}^{2}\end{array} \right. \tag{8}\]  
In the actual calculation process of SGM, due to journal misalignment [23], there is a difference between the equivalent bearing support point and the assumed bearing support point. SGM usually simplifies bearings into equivalent bearing support points, and the fulcrum position is generally selected based on experience. The fulcrum of the intermediate bearing is generally taken as the midpoint of the intermediate bearing [21].