## 2 Materials and Methods
### FEM of Integral Squeeze Film Bearing Damper  
From Section 2.2, the unbalanced single-side cantilevered rotor system was modeled as shown in Figure 5. The model simulates the installation of the integral squeeze film bearing damper by modifying the stiffness and damping of the support. The following 4 working conditions were designed to investigate the influence of the position and quantity of the integral squeeze film bearing dampers on the stability and energy distribution of the system: (1) the rigid support is installed in the bearing housing (working condition 1); (2) damper is installed in the bearing housing near the motor end, and the bearing housing away from the motor end is still rigidly supported (working condition 2); (3) rigid support is installed in the bearing housing near the motor end, and an bearing damper in installed in the bearing housing away from the motor end (working condition 3); and (4) two dampers are installed in both bearing housings (working condition 4); The purpose of working conditions 1, 2, and 3 is to investigate the effect of the installation position on the unbalanced single-side cantilevered rotor system. The purpose of working conditions 1, 2, 3, and 4 is to investigate the effect of the number of installations on the rotor system. The model was established to study the effect of support stiffness on the first order critical speed, as shown in Figure 6 and Table 1. It can be found that when the support stiffness exceeds \(1.5\times 10^{7}\) N/m, the first order critical speed of the rotor system does not increase with the increasing stiffness. Therefore, it can be known that the stiffness of the integral squeeze film bearing damper needs to be designed to be less than \(1.5\times 10^{7}\) N/m to effectively reduce the first order critical speed of the system. When using Equation (29), in its first formula and the second formula, all of the damping parameters are set to 0. The finite element model calculates the corresponding critical speed by setting different support stiffness levels, and then connects all the points of the critical speeds corresponding to each stiffness level to obtain the curve shown in Figure 6.  
In this paper, considering that the support stiffness is too low, the damper can quickly achieve fatigue failure. Therefore, the design of the ISFBD used ANSYS for static analysis, as shown in Figure 7. \begin{table}
\begin{tabular}{c c} \hline \hline
**Support Stiffness (N/m)** & **Critical Speed (r/min)** \\ \hline \(1.66\times 10^{6}\) & 2573.71 \\ \(2.94\times 10^{6}\) & 2601.43 \\ \(3.55\times 10^{6}\) & 2611.81 \\ \(5.66\times 10^{6}\) & 2620.33 \\ \(7.54\times 10^{6}\) & 2627.47 \\ \(9.11\times 10^{6}\) & 2638.18 \\ \(1.09\times 10^{6}\) & 2648.26 \\ \(1.32\times 10^{6}\) & 2655.52 \\ \(1.49\times 10^{7}\) & 2667.54 \\ \(1.93\times 10^{7}\) & 2670.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Different support stiffness levels corresponding to critical speed. Figure 5: Rotor dynamics model. Figure 6: Influence of support stiffness on first order critical speed. First, we should calculate the stiffness of the shaft, as shown in Figure 7a. The shaft has a stiffness of \(1.0\times 10^{8}\) N/m. According to the machinery vibration and rotordynamics theory, the stiffness design range of ISFBD is one or two orders of magnitude lower than the shaft stiffness. So the ISFBD has a stiffness of \(5.6\times 10^{6}\) N/m. It can be seen from the displacement contour that the main deformation of the displacement is concentrated on the S-shaped elastic body. Stress analysis was performed on the novel integral squeeze film bearing damper, and the analysis type was von-Mises stress, as shown in Figure 8. It can be found that the damper has no obvious stress concentration, uniform stress distribution and good protection in terms of strength. Through the analysis of the above displacement contour and stress contour, the novel integral squeeze film bearing damper can ensure a safe and healthy service life.