## 2 Electric Propulsion Shaft System Model and Static Characteristic Analysis
### Static Analysis of Propulsion Shafting System  
#### 2.3.1 Static Mechanical Analysis  
The established three-dimensional model of the propulsion shaft system is imported into the finite element analysis software, combined with the theory of the ship's propulsion shaft system. The shaft segment from the propeller to the tail shaft tube seal, which is simplified, is treated with water coverage, and the buoyancy coefficient is taken as \(1025\) kg/m\({}^{3}\). According to the known model mass, the material properties corresponding to the corresponding shaft segment model are set. The density of unknown metal structures is processed at 7850 kg/m\({}^{3}\), the elastic modulus is \(2.0\times 10^{11}\) Pa, and Poisson's ratio is 0.3, resulting in a system mass of 33 tons for the propulsion shaft system. In terms of boundary condition definition, considering the rubber joint surface of the high-elasticity coupling with the flange disc and the flange surface with the external shaft segment, a binding connection is used and there is no relative displacement between the parts.  
In static calculation, it is necessary to impose corresponding constraints on the shaft system to meet the actual state of the shaft system. The spring element is used to simulate the bearing of the propulsion shaft system, assuming that the bearing has isotropic stiffness in the radial direction. To more accurately simulate the actual situation, in the cross-section of the bearing support point, vertical and horizontal spring elements are used, and corresponding stiffness is assigned for simulation calculation. In the post-processing stage, the deformation of the propulsion shaft system is visualized by adding displacement transformation, and the state parameters of the key cross-section or the entire system are extracted. To obtain the load conditions of each bearing, a spring probe is introduced to measure the load at the bearing support point position and the deflection of the spring. The final load of each bearing model is obtained as shown in Table 2, and the deflection curve of the propulsion shaft system is shown in Figure 5.  
From Figure 5, it can be observed that the bending deflection at various positions of the propulsion shaft system, as calculated through static mechanics, is delineated. The \(X\)-axis corresponds to the locations of cross-sections along the shaft from the propeller to the motor, while the \(Y\)-axis denotes the deflection values at each of these cross-sections. The deflection curve represents the deformation profile of the propulsion shaft system across different regions. At the vicinity of the \(X\)-axis at 0 mm, the deflection is recorded as 0.83 mm, predominantly due to the substantial mass of the propeller which results in a downward deflection. Within the shaft segment spanning from 2000 mm to 8500 mm, the section is supported by the stern bearing at the rear and the forward stern bearing. This  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **Name** & **Position (mm)** & **Load (kN)** \\ \hline
1 & after stern bearing & 2048 & 86.02 \\
2 & before stern bearing & 8299 & 13.95 \\
3 & thrust bearing & 11,727 & 43.72 \\
4 & the rear end of the motor bearing & 14,242 & 111.62 \\
5 & the front end of the motor bearing & 16,758 & 63.91 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Straight state bearing load table of the propulsion shaft system.  
Figure 4: Physical model diagram of the propulsion shaft system.  
segment, under the influence of the propeller's gravitational force, exhibits an upward deflection with a maximum deformation reaching 0.37 mm.