## 2 Theoretical Method and Physical Model
### Numerical Results
#### 2.2.2 Oil Film Bearing Capacity and Deflection Angle Calculation  
The bearing capacity of the lubricating oil film versus for the journal and the floating ring provides lubrication and damping effects, with less friction and vibration. According to the oil film pressure distribution, note that the bearing capacity of the oil film inside and outside of the floating ring bearing is derived from the vector sum of the oil film force of each micro-element area of the bearing working surface. Based on the analysis of the working principle of the floating ring, the line of action of the bearing capacity of the inner oil film passes through the center of the journal and the center of the floating ring.  
Figure 4: Oil film pressure distributions of the inner oil film. (**a**) \(e=0.1\), \(ratio=0.5\); (**b**) \(e=0.3\), \(ratio=0.5\); (**c**) \(e=0.1\), \(ratio=0.3\); (**d**) \(e=0.3\), \(ratio=0.3\).  
After the dimensionless oil film pressure is measured in the oil cavity, the dimensionless bearing capacity \(W\) of the oil film can be obtained by integrating the oil film force on each micro-unit area of the bearing working surface. The calculation formula is written as  
\[W_{r,\theta}=\int_{-1}^{1}\int_{0}^{2\pi}Pd\varphi d\lambda \tag{17}\]  
\[\left\{\begin{aligned} W_{x}&=\int_{-1}^{1}\int_{0}^{2 \pi}P\cos\varphi d\varphi d\lambda=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m}P _{i,j}\cos\varphi_{i}\Delta\varphi\Delta\lambda\\ W_{y}&=\int_{-1}^{1}\int_{0}^{2\pi}P\sin\varphi d \varphi d\lambda=\sum\limits_{j=1}^{n}\sum\limits_{i=1}^{m}P_{i,j}\sin\varphi _{i}\Delta\varphi\Delta\lambda\end{aligned}\right. \tag{18}\]  
where \(W_{r,\theta}\) is the bearing capacity under the cartesian coordinates and \(W_{x,y}\) is the bearing capacity under the polar coordinate.  
The dimensionless oil film bearing capacity is defined as  
\[W=\sqrt{W_{x}{}^{2}+W_{y}{}^{2}} \tag{19}\]  
The deflection angle of the sliding bearing can be obtained as  
\[\theta=arcg\frac{W_{x}}{W_{y}} \tag{20}\]  
Figure 6 depicts the results of the maximum oil film pressure, bearing capacity, and deflection angle with eccentricity. The evolution of the maximum oil film pressure is close to linear growth in the case of a small eccentricity, as shown in Figure 6a. When the eccentricity exceeds 0.5, the maximum oil film pressure increases sharply with the  
increase in eccentricity, showing a non-linear increase. Also shown in Figure 6b is that the dimensionless bearing capacity of the oil film increases linearly in the case of small eccentricity. In comparison with Figure 6a, the bearing capacity increases sharply and grows nonlinearly in this instance of the eccentricity exceeding 0.5. The oil film squeezing effect is significantly enhanced, and it will show strong nonlinear characteristics. In the case of large eccentricity, the oil film may rupture, and the journal and the floating ring may be in direct contact at some locations. Figure 6c depicts that the deflection angle decreases linearly with the increase in eccentricity.  
The dimensionless maximum oil film pressure, bearing capacity, and deflection angle curves during one period are presented in Figure 7. Note that both the maximum oil film pressure and bearing capacity are closed due to the actions of the width-to-diameter. The wider the bearing, the greater the oil film pressure at the center of the wedge-shaped convergence zone. On the contrary, the deflection angle changes slightly with the increase in the width-to-diameter ratio.  
Figure 6: The maximum oil film pressure, bearing capacity, and deflection angle. (**a**) The maximum oil film pressure. (**b**) The oil film bearing capacity. (**c**) The deflection angle.