## 3 Measurement of Elastic Displacement
### Equivalent Line-Surface Method  
To address the limitations mentioned above, this study proposes an equivalent line-surface method to measure the elastic deformation of the raft frame and the bearing displacement induced by it. The equivalent surface of the raft frame was obtained by the robust weighted total least squares method [23; 24] to measure the elastic deformation, as illustrated in Figure 5a, and the deflection state of the centerline of the bearing housings was fitted by the polynomial fitting method [25] to measure the bearings' displacement characteristics, as illustrated in Figure 5b.  
In the equivalent line-surface method, nodal coordinates on the raft frame surface are first extracted; then, the robust weighted total least squares method is used to derive the equivalent surface. The positional difference on the equivalent surface before and after displacement represents the rigid displacement of a certain node on the raft frame, and the shortest spatial distance between the nodes on the raft frame and the equivalent surface \(\Delta y\) represents the elastic deformation of the raft frame, as shown in Figure 5a. Taking the centerline of the bearing housings as the reference baseline, the coordinates of each point along the line were used by the linear regression method to fit another centerline that can measure the displacement changes of the bearings, namely the equivalent line of the raft frame surface, as shown in Figure 5b. The displacement difference between a bearing and this equivalent line represents the displacement change of this bearing. This method enables the transformation of the absolute displacement obtained from finite element calculation to the displacement variations of the equivalent supporting points of the bearings, thereby obtaining the accurate elastic displacements of the bearings. In practical applications, the fitting accuracy can be enhanced by expanding the number of sample points, which ensures the measurement precision of the elastic displacement. Figure 6 shows the deflection curves of the shafting and the raft frame surface in the initial installation state, as well as the fitted equivalent line.  
As shown in Figure 6, the deflection curve of the raft frame surface exhibits a shape approximating an arch, which means that the raft frame has produced significant elastic deformation. Based on the shafting's deflection curve, the vertical displacements of the bearings \(\Delta y_{j}\) (\(j\) = 1, 2, 3) following the equivalent line-surface method can be obtained  
Figure 5: Equivalent line-surface method. (**a**) Equivalent surface; (**b**) equivalent line.  
by calculating the distance between each bearing and the equivalent line. The change in reaction force of each bearing can be calculated by the following equation:  
\[\Delta F_{i}=\sum_{j=1}^{3}A_{ij}\Delta y_{j} \tag{2}\]  
where \(\Delta F_{i}\) represents the reaction force change of the \(i\)-th bearing. \(A_{ij}\) is the element in the BLINs matrix, which represents the load change of the \(i\)-th bearing induced by a unit length change in the \(j\)-th bearing's vertical displacement [26]. However, in practice, elastic deformation will make the BLINs change dynamically.