## 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
Assuming the position of the equivalent support point is \(\overline{L}\), and the assumed position of the support point is \(L\), then the assumed support point deviation \(\Delta L\) can be expressed as:  
\[\Delta L=L-\overline{L} \tag{9}\]Assuming that the deviation of the support point position \(\Delta L\) follows a normal distribution, \(\Delta L{\sim}N(0,\sigma_{\Delta L}^{2})\) (mainly affected by journal misalignment), then, based on the principle of maximum likelihood, the PDF of the support forces \(F_{s_{i}}\) obtained through SGM can be expressed as:  
\[f_{s_{i}}(x)=\left(2\pi\sigma_{s_{i}}^{2}\right)^{-1/2}\exp(-\frac{(x-\mu_{s_{ i}})^{2}}{2\sigma_{s_{i}}^{2}}) \tag{10}\]  
These parameters can be calculated by:  
\[\left\{ \begin{array}{l}\mu_{s_{i}}=\frac{a_{i}M_{i,i}+b_{i}M_{i,2}+c_{i}M_{i,3}+d_{i}}{L_{i,3}}\\ \sigma_{s_{i}}^{2}=\sigma_{M}^{2}\frac{\left|aM_{i,1}^{2}\right|+\left|M_{i,2} ^{2}\right|+\left|cM_{i,3}^{2}\right|}{L_{i,3}^{2}}+\frac{\sigma_{\Delta L}^{ 2}F_{s_{i}}^{2}}{L_{i,3}^{2}}\end{array} \right. \tag{11}\]  
Using the probability density functions of the \(n\) support forces \(F_{s}\) obtained from SGM, data fusion based on confidence distance is carried out. The key to the fusion is the selection of a distance metric. Commonly used distance measures include the Jensen-Shannon Divergence and the Wasserstein Distance [24]. In comparison with the Jensen-Shannon Divergence, the Wasserstein Distance is recognized as a rigorous metric within mathematics, as it adheres to the principles of symmetry, non-negativity, non-degeneracy, and the triangle inequality. Conversely, the Jensen-Shannon Divergence does not adhere to the triangle inequality, which means that it is categorized as a divergence and not as a strict distance measure [25]. Consequently, for the merging of probability density functions, the Wasserstein Distance is considered the more suitable distance measure. Confidence distance fusion based on the Wasserstein distance can both integrate to the centroid of the probability density functions and ensure that the physical/geometric characteristics of each probability density function remain unchanged [20]. Assuming that \(\mu,\
u\) are the marginal distributions of the random vectors \(X,Y\), \(X{\sim}\mu\), \(Y{\sim}\
u\), according to the reference [26], the Wasserstein distance is defined as:  
\[W_{p}(\mu,\
u)=\inf_{ \begin{array}{l}X{\sim}\mu\\ Y{\sim}\
u\end{array} }E(\|X-Y\|^{p})^{1/p} \tag{12}\]  
Based on Equation (12) and the probability density functions of the support forces \(F_{s}\), the fused result is obtained through the following steps:  
The first step is to construct a confidence distance matrix \(d_{ij}\) based on the probability density functions of the support forces \(F_{s}\), and to obtain the confidence distances between each pair of distribution functions. \[d_{ij}=W(p_{i},p_{j}) \tag{13}\]  
where \(p\) is the PDF of the support reaction force, \(i,j=1,2,\cdots,n\). The second step is to define the mutual support degree \(s_{i}\) between each support reaction force. \[s_{i}=\frac{1}{\sum\limits_{j=1}^{n}d_{ij}} \tag{14}\]  
The third step is to derive the fusion weights \(w_{i}\) for each support reaction force from the mutual support degree \(s_{i}\). \[w_{i}=\frac{s_{i}}{\sum\limits_{i=1}^{n}s_{i}} \tag{15}\]The fourth step involves fusing the support reaction forces based on the fusion weights, and the result of the fusion is:  
\[P=\sum_{i=1}^{n}w_{i}p_{i} \tag{16}\]  
In the redundant strain design, the \(L_{1}\) of each strain section combination is located on both sides of the bearing, and the fusion result cancels out some of the result errors caused by the assumed support point deviation. Therefore, it is necessary to correct the variance of the probability density function of the fusion result.