## 5 Alignment Optimization
### Alignment Optimization Based on GPR Response Surfaces  
In view of the problems above, aiming at reducing the load of the rear stern bearing and improving the evenness of load distribution among bearings, the alignment optimization of the elastically supported submarine propulsion shafting was carried out based on the GPR response surfaces. First of all, the heights of the thirteen supporting points illustrated in Figure 2 were derived via numerical calculation, and the single equivalent supporting point of each bearing was obtained using the shear balance equation. The vertical displacements of the three bearings' supporting points \(\Delta y_{i}\) (\(j=1,2,3\)) under the equivalent line-surface method were selected as the optimization variables (\(\Delta y_{1}\in[-3.5,\,3.5]\) mm, \(\Delta y_{2}\in[-3.5,\,3.5]\) mm, and \(\Delta y_{3}\in[-1,\,1]\) mm). The reaction force of each bearing \(F_{i}\) can be expressed as:  
\[F_{i}=F_{0i}+\sum_{j=1}^{3}A_{ij}\Delta y_{j},\,i=1,\,2,3 \tag{17}\]  
where \(F_{0i}\) represents the reaction force of the \(i\)-th bearing in the straight-line alignment state and \(A_{ij}\) represents the bearing load influence number output by the response surface. The optimization objective function was defined as:  
\[ \begin{split}&\text{min}\varphi_{1}=F_{1}\\ &\varphi_{2}=\sqrt{\frac{1}{3}\sum_{i=1}^{3}\left(F_{i}-\overline {F}\right)^{2}}\end{split}  \tag{18}\]  
where \(\varphi_{1}\) represents the load index of the rear stern bearing; \(\varphi_{2}\) represents the evenness index expressed by the standard deviation of the loads of the three bearings; and \(\overline{F}\) is the average load value of the three bearings. In order to satisfy the alignment requirements of marine propulsion shafting, the constraints during the optimization were defined as:  
\[ \begin{split}&\text{s.t. }0.2G_{i}\leq F_{i}\leq F_{\text{max}}\\ &\varphi_{n}\leq[\varphi]\\ &\sigma_{\text{max}}\leq[\sigma]\end{split}  \tag{19}\]  
where \(F_{\text{max}}\) is the maximum allowable reaction force for the \(i\)-th bearing and \(G_{i}\) is the total weight of the two adjacent shaft segments that the \(i\)-th bearing spans; \(\phi_{n}\) is the maximum rotation angle at the connection of the \(n\)-th shaft segment and \([\phi]\) is the maximum allowable value; \(\sigma_{\text{max}}\) is the maximum stress of all the shaft segments; and \([\sigma]\) is the maximum allowable value. Combined with the built GPR response surfaces, the genetic algorithm [24] was used to realize the alignment optimization of the elastically supported shafting, and the optimization process is shown in Figure 11.  
In Figure 11, \(N\) is the genetic generation number in the algorithm. In the value ranges of \(\Delta y_{j}\), \(10^{3}\) points were sampled randomly, and the mapping relationship between \(\Delta y_{j}\) and the BLINs was obtained by the response surface method. The fitness function of the genetic algorithm was constructed according to Equation (9), which can be written as:  
\[\eta(x)=\frac{1}{\sigma_{f}^{2}\exp(-\frac{1}{2l^{2}}\left\|\Delta y_{i}- \Delta y_{j}\right\|^{2})+\sigma_{n}^{2}\delta_{ij}} \tag{20}\]  
The maximum generation number of the genetic algorithm was set to 300, and the crossover probability and mutation probability were set to 0.9 and 0.01, respectively. The final optimization results will be output when the iteration number exceeds the maximum generation number, or the optimization results satisfy the convergence error. The final optimization results are shown in Table 3. \begin{table}
\begin{tabular}{c c c c} \hline
**Bearing** & \(\Delta y_{i}\) **(mm)** & **Bearing Load (kN)** & **Rotation Angle (Rad)** \\ \hline Rear stem bearing & \(-0.57\) & \(14.95\) & \(7.86\times 10^{-4}\) \\ Front stern bearing & \(0.14\) & \(9.15\) & \(-3.25\times 10^{-4}\) \\ Thrust bearing & \(0.05\) & \(8.01\) & \(2.27\times 10^{-4}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Alignment optimization results of the elastically supported shafting.