## 2 Numerical Models
### Governing Equation
\begin{array}{l}d_{ix}=\sum\limits_{k=1}^{i}\frac{F_{ix}z_{k}^{2}}{6 E_{I}^{2}}(3z_{i}-z_{k})+\sum\limits_{k=1}^{n}\frac{F_{ix}z_{k}^{2}}{6E_{I} ^{2}}(3z_{k}-z_{i})\\ d_{iy}=-\frac{Mz_{ix}^{2}}{6E_{I}^{2}}(3L-z_{i})+\sum\limits_{k=1}^{i}\frac{F _{ix}z_{k}^{2}}{6E_{I}^{2}}(3z_{i}-z_{k})+\sum\limits_{k=i+1}^{n}\frac{F_{ix} z_{k}^{2}}{6E_{I}^{2}}(3z_{k}-z_{i})\end{array} \right. \tag{5}\]  
where \(M\) is the self-weight of the propeller, \(L\) is the width of the bearing, \(E_{I}\) is the elastic modulus of the shaft, \(I\) is the inertia moment of the shaft section, and \(I=\pi D^{4}/64\), where \(D\) is the diameter of the shaft. The eccentricity, \(e_{i}\), between the point \(O_{i}\) and the point \(O^{\prime}\)\({}_{b}\), as shown in Figure 3, can be calculated as:  
\[e_{i}=d_{i}+e_{s} \tag{6}\]  
where the distance \(d_{i}\) can be calculated by \(d_{i}=\sqrt{d_{ix}^{2}+d_{iy}^{2}}\). Further, \(e_{s}\) is the eccentricity of the end face of the shaft. Subsequently, the \(\varepsilon_{i}\) in Equation (4) can be written as:  
\[\varepsilon_{i}=e_{i}/C \tag{7}\]  
The average geometric gap, \(h_{0}\), can be obtained by substituting Equations (5)-(7) into Equation (4). Figure 2: Schematic of water-lubricated rubber stern bearing. When asperity contact occurs, the total pressure, \(p\), acting on any nodal (\(\theta_{j}\), \(z_{k}\)) of the stern bearing inner face consists of two parts: the water film pressure, \(p_{h}\), and the contact pressure, \(p_{c}\). The latter can be determined using the Kogut-Etsion elastic-plastic contact model [33], which is given by:  
\[ \begin{array}{l}p_{c}^{*}=\frac{p_{c}}{H_{B}}=\frac{2}{3}\pi\sigma\beta DK \omega_{c}^{*}\left(\int_{h^{+}+\omega_{c}^{*}}^{h^{+}+\omega_{c}^{*}}I_{c}^{1.5}+1.03\int_{h^{+}+\omega_{c}^{*}}^{h^{+}+\delta\omega_{c}^{*}}I_{c}^{1.425} \right.\\ \left.+1.4\int_{h^{+}+\delta\omega_{c}^{*}}^{h^{+}+110\omega_{c}^{*}}I_{c}^{1.2 63}+\frac{3}{K}\int_{h^{+}+110\omega_{c}^{*}}^{\infty}I_{c}^{1}\right)\end{array}  \tag{8}\]  
where the superscript \"*\" denotes dimensionless parameters, \(H_{B}\) is the hardness of the bush, whose unit is Pa; \(\beta\) and \(D\) are the curvature radius and the density of the asperity, respectively; and \(I_{c}\) is the integral operator, which is calculated by:  
\[I_{c}^{b}=\left(\frac{y^{*}-h^{*}}{\omega_{c}^{*}}\right)^{b}\phi^{*}(y^{*}) \mathrm{d}z^{*} \tag{9}\]  
where \(\phi^{*}(y^{*})\) can be expressed as:  
\[\phi^{*}(y^{*})=\frac{1}{\sqrt{2\pi}}\frac{\sigma_{s}}{\sigma}\exp\left[-0.5 \left(\frac{\sigma_{s}}{\sigma}\right)^{2}(y^{*})^{2}\right] \tag{10}\]  
where \(\sigma_{s}\) is the surface roughness of the shaft, and the value of \(\sigma_{s}/\sigma\) is set to 1.0 in the present simulation. In Equation (8), the dimensionless parameters \(\omega_{c}^{*}\) and \(K\) can be calculated by the following expressions:  
\[\omega_{c}^{*}=\left(\frac{\pi KH_{B}}{2E_{c}}\right)^{2}\beta/\sigma,\ K=0.45 4+0.41\
u_{B} \tag{11}\]  
where \(E_{c}\) is the composite elastic modulus and \(\
u_{B}\) is the Poisson ratio of the bush. The deformation of the bearing, \(\delta_{R}\), can be expressed as the sum of the elastic deformation, \(\delta_{RE}\), and the thermal deformation, \(\delta_{RT}\). The elastic deformation of the journal, \(\delta_{IE}\), can be ignored because its elastic modulus is significantly larger than that of the rubber bushing. Additionally, the thermal deformation of the journal, \(\delta_{IT}\), is assumed to be a linear  
Figure 3: Schematic for the calculation of the deflection of the propeller shaft. thermal expansion so as to simplify the analysis. Consequently, the deformation equations of the journal and the rubber bushing are given as follows:  
\[\left\{