## 2 Theoretical Method and Physical Model
### Process of Modeling  
The structure of the simplified rotor-bearing system is shown in Figure 8. This model basically consists of two bearings including the ring SFD and a shaft, a disk 1, and a disk 2. Figure 9 shows that the ring SFD is composed of the floating ring, and the inner and outer oil films. Six holes are evenly spaced on the whole circumference of the floating ring. Based on the model proposed above, the lubricating oil flow channel between the inner and outer oil chambers is built. Then, the flow rate of lubricating oil between the inner and outer membranes through the oil holes and the pressure transmission characteristics of the oil film can be highlighted. Figure 8: The floating ring bearing-rotor system FEM model. Figure 7: The dimensionless 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. The connection between the rotor and the bearing is based on the \"multi-physics coupling characteristics\". The force transmission between the bearing and the rotor can be used to describe the behavior of the hydrodynamic pressure effect. In this simulation calculation, the stress and deformation of the floating ring are neglected. The centroids of the two disks are set to offset 6 \(\times\) 10\({}^{-7}\) m from the center to the \(z\)-axis direction. Due to factors such as gravity and errors of assembly, the position is set to be approximately equal to the mass unbalance and eccentricity of the rotor system. From the final calculation results, note that the system has reached a steady state before 0.05 s. Therefore, it is reasonable to set the analysis time as 0.2 s in the numerical simulation. Known from the result of the simulation, the rotor speed is 8000 rad/min while the time step is 5 \(\times\) 10\({}^{-4}\) s. Thereby, the backward difference formula solution method is selected. The floating ring bearing adopts a full-floating model. With the Sommerfeld hypothesis taken into account, the solution result can be compared with the Reynolds boundary conditions. Table 1 shows the geometric parameters, material parameters, and settings of the floating ring bearing-rotor system, and the attribute parameters of the floating ring are shown in Table 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter Name** & **Numerical Value** & **Parameter Name** & **Numerical Value** \\ \hline Young’s modulus & 205 GPa & Poisson’s ratio & 0.3 \\ Rotor material density & 7800 kg/m\({}^{3}\) & Length of the rotor, L & 0.15 m \\ Position of disk 1 & 0.1 L & Position of bearing 1 & 0.3 L \\ Position of disk 2 & 0.9 L & Position of bearing 2 & 0.7 L \\ Quality of disk 1 & 1.4 kg & Quality of disk 2 & 1.0 kg \\ Lateral moment of inertia of disk 1 & \(6.3\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) & Lateral moment of inertia of disk 2 & \(4.5\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ Polar moment of inertia of disk 1 & \(1.26\times 10^{-5}\) kg\(\cdot\)m\({}^{2}\) & Polar moment of inertia of disk 2 & \(9\times 10^{-4}\) kg\(\cdot\)m\({}^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model parameters of the rotor system. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter Name** & **Numerical Value** & **Parameter Name** & **Numerical Value** \\ \hline Quality of the floating ring & 0.02 kg & Outer gap & 0.08 mm \\ Outer radius of the floating ring & 9 mm & Inner gap & 0.02 mm \\ Inner radius of the floating ring & 6 mm & Lubricating oil viscosity & 0.06 Pa\(\cdot\)s \\ Length of the floating ring & 0.01 m & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Floating ring bearing parameters. Figure 9: Schematic diagram of the inner and outer oil film channels. The rotor model in this section is meshed with Timoshenko beam elements. The beam element is a one-dimensional line element in space, which can resist certain linear deformation and has high calculation efficiency.