# Influence of Cavitation and Shaft Deformation in the Analysis of Lubrication of the Stern Bearing
## 4 Oil Film Thickness Equation Considering Shaft Deflection  
Due to the suspension of the ship stern shaft and the gravity of the propeller, the stern shaft is bent, which affects the pressure distribution of the bearing-lubricating oil film.  
Figure 4: Fluid element on the boundary of oil film reformation.  
The detailed expression of oil film thickness considering journal tilt can be analyzed, and the influence of stern shaft tilt caused by propeller-concentrated force on oil film thickness and the oil film pressure distribution of the stern bearing can be quantitatively calculated. However, because the oil film force and the asperity contact force act on the stern shaft at the same time, the stern shaft is also bent. In order to accurately consider the interaction between deflection and oil film force, asperity contact force, and external load, the displacement superposition method is used to calculate the shaft deflection.  
As shown in Figure 5, the tail end of the shaft is bent by the gravity of the propeller. When the shaft rotates, due to the dynamic pressure lubrication, there is oil film distribution force at the bearing. If the ratio of oil film thickness to the comprehensive roughness of the lubricating surface is less than 4, the asperity also bears the load. In order to accurately solve the shaft deflection, the shaft is divided into \(n\) sections along the axial direction in the bearing support area. The \(z\)-direction coordinate of each point is \(Z_{i}\), and the displacement of each point is \(v_{ix}\), \(v_{iy}\). The oil film thickness corresponding to each section is \(h_{i\theta}\); then:  
\[h_{i\theta}=c+e_{i}\cos(\theta-\phi) \tag{38}\]  
where  
\[e_{i}=\sqrt{\left(v_{ix}+e_{ix}\right)^{2}+\left(v_{iy}+e_{iy}\right)^{2}}\]  
\(e_{i}\) is the eccentricity of section \(i\) without considering shaft deformation; \(e_{ix}\) and \(e_{iy}\) are the components of eccentricity in the \(x\) and \(y\) directions, respectively, without considering shaft deformation respectively; \(\phi\) is the offset angle; and \(\theta\) is circumferential angle.  
According to the above mathematical model, the bearing lubrication simulation program is written by FORTRAN. The program flow is shown in Figure 6:  
1. The basic parameters of the bearing are input, such as bearing diameter, bearing radius clearance, bearing width, external load, lubricating oil viscosity, etc. The surface roughness of the bearing and the shaft and the pressure of the cavitation zone are input. The initial value of the eccentricity and the deviation angle are given, and the switching function (\(g\)) is set to 1;
2. The general Reynolds Equation (37) is solved to obtain the distribution of \(\alpha\) and update the value of \(g\). The finite difference method and the successive over-relaxation algorithm are applied for the calculation. The contact force of the asperity is solved. Then, the oil film force and the contact force of the asperity are substituted into the following formula to determine if the calculation is convergent: \[\left|\frac{F-F_{oil}-W_{A}}{F}\right|\leq\xi_{1},\left|\frac{F_{oilx}+W_{x}}{ F_{olly}+W_{y}}\right|\leq\xi_{2}\] (39) where \(\xi\), \(i=1,2\) is the convergence precision in the calculation. A typical value for the convergence criteria is 0.003. If the above convergence condition is not satisfied, the eccentricity, offset angle, and oil film thickness are adjusted until the convergence condition is satisfied;
3. From Equation (8) to Equation (11), the friction force is solved, and finally, the friction coefficient is solved according to the following formula: \(\mu_{j}=\frac{F}{P}\).  
Figure 5: The distribution of force acting on the bearing considering the shaft deflection.  
Figure 6: Program flow to calculate the bearing lubrication considering both cavitation and shaft deformation.