## 2 Materials and Methods
### Theoretical Structure Modeling of Unbalanced Single-Side Cantilevered Rotor System  
A simplified model of the unbalanced single-side cantilevered rotor system is shown in Figure 4. The following assumptions were made during the analysis: (1) there is an elastic massless shaft; (2) regardless of the torsional vibration of the shaft, the angular velocity of the shaft is constant; (3) the axial length of the disc is neglected, and the imbalance of the disc is located on one side plane; and (4) the stiffness and damping of the integral squeeze film damper are isotropic [23; 24; 25; 26]. It is assumed that the lateral displacement the rotation angle of the geometric center of the cantilevered disk in the \(x\), \(y\) directions are \(x_{D},\ T_{D},\ y_{D},\ \text{and}\ U_{D}\), respectively. The lateral displacements of ISFBD A and ISFBD B in the \(x\), \(y\) directions are \(x_{A},\ y_{A},\ x_{B},\ \text{and}\ y_{B}\), respectively. The integral squeeze film bearing damper mass and cantilevered disk mass are \(m_{D}\) and \(m\), respectively. The polar moment of inertia and diameter moments of inertia of the cantilever disk are \(J_{pD}\) and \(J_{dD}\), respectively. The distance from damper A to the motor is \(l\). The distance between the two dampers is \(b\). The cantilevered distance is \(a\) and the angular velocity of rotation is \(\Omega\). The disk eccentricity is \(e_{uD}\), the initial phase angle is \(h_{uD}\), and the phase difference of the rotational dynamic deflection is \(J_{D}\). Therefore, the kinetic energy of the disc is:  
\[T_{rD}=\frac{1}{2}\Big{[}J_{dD}(\dot{\text{T}}_{D}^{2}+\dot{U}_{D}^{2})+J_{pD} k^{2}-2J_{pD}\Omega\dot{\text{T}}_{D}U_{D}\Big{]} \tag{18}\]  
The potential energy of the disc is:  
\[V_{D}=m_{D}gx_{D}. \tag{19}\]  
The dissipative energy of the disc is:  
\[H_{D}=\frac{1}{2}c_{tD}(\dot{x}_{D}^{2}+\dot{y}_{D}^{2})+\frac{1}{2}c_{rD}( \dot{\text{T}}_{D}^{2}+\dot{U}_{D}^{2}) \tag{20}\]  
where \(g\) is gravitational acceleration, and \(c_{tD}\) and \(c_{rD}\) are the lateral movement of the cantilever disk and the damping coefficient of the diameter rotation, respectively. Similarly, the kinetic, potential, and dissipative energy of integral squeeze film bearing dampers A and B are:  
\[T_{A}=\frac{1}{2}m(\dot{x}_{A}^{2}++\dot{y}_{A}^{2}) \tag{21}\]  
Figure 4: Dynamic model of single-side cantilevered rotor system. \[V_{A}=mgx_{A}+\frac{1}{2}kx_{A}^{2}+\frac{1}{2}ky_{A}^{2} \tag{22}\]  
\[H_{A}=\frac{1}{2}c(\dot{x}_{A}^{2}+\dot{y}_{A}^{2}) \tag{23}\]  
\[T_{B}=\frac{1}{2}m(\dot{x}_{B}^{2}++\dot{y}_{B}^{2}) \tag{24}\]  
\[V_{B}=mgx_{B}+\frac{1}{2}kx_{B}^{2}+\frac{1}{2}ky_{B}^{2} \tag{25}\]  
\[T_{rD}=\frac{1}{2}\Big{[}J_{dD}(\dot{\text{T}}_{D}^{2}+\dot{U}_{D}^{2})+J_{pD} k^{2}-2J_{pD}\Omega\dot{\text{T}}_{D}U_{D}\Big{]} \tag{18}\]0  
where \(k\) represents the stiffness of the damper in the \(x\) and \(y\) directions, and \(c\) represents the damping coefficient of the damper. The elastic potential energy of the shaft is:  
\[V_{R}=\frac{1}{2}\bigg{[} \begin{array}{c}x_{D}^{*}\\ T_{D}^{*}\end{array} \bigg{]}^{T}\bigg{[} \begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array} \bigg{[} \begin{array}{c}x_{D}^{*}\\ T_{D}^{*}\end{array} \bigg{]}+\frac{1}{2}\bigg{[} \begin{array}{cc}y_{D}^{*} \\ -U_{D}^{*}\end{array} \bigg{]}^{T}\bigg{[} \begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array} \bigg{[} \begin{array}{c}y_{D}^{*}\\ -U_{D}^{*}\end{array} \bigg{]} \tag{27}\]  
where \(x_{D}^{*},\ y_{D}^{*},\ T_{D}^{*},\ U_{D}^{*}\) are the actual deflections of the shaft at the center of the cantilevered disk. \(\left[ \begin{array}{cc}k_{11}&k_{12}\\ k_{21}&k_{22}\end{array} \right]\) is the stiffness influence coefficient matrix of the shaft.