## 2 Misalignment and Rub-Impact Coupling Dynamic Modeling
### Dynamic Model of the Rotor System  
The dynamic model of the rotor system is developed based on the following assumption: torsional and axial displacement are ignored and the transverse bending vibration of the rotor system is considered; since the operating temperature of the rotor system is not high enough, the influences of temperature on material properties and system behaviors  
Figure 2: Rub-impact model between the rotor and casing. are small and can be ignored; after estimation, it is found that the gyro effect has a weak influence on the vibration of the rotor. Accordingly, the gyro effect is not considered in this work; the power turbine rotor is the whole forging rotor, the impeller and the shaft are integral, and there is no interaction between the components in the rotor system; the hub and blade are simplified as a rigid disk considering the rotation effect; and the shaft is simplified as a massless elastic shaft. A schematic diagram of the offset rotor system is shown in Figure 3. The mass of the disk is \(m_{d}\), the eccentricity of the imbalanced disk is \(e\), and the viscous damping of the disk in all directions is \(c\); the clearance between the rotor and the casing is \(\delta\); the distance between the left support and the disk is \(a\), the distance between the right support and disk is \(b\), and the length of the shaft is \(l\). These offset parameters have an influence on the vibration characteristics of the system. The dynamic characteristics under the determination offset will be studied in this work; the supporting mass at the left and right ends is \(m_{a}\) and \(m_{b}\), the supporting stiffness is \(k_{a}\) and \(k_{b}\), and the supporting viscous damping is \(c_{a}\) and \(c_{b}\), respectively. In addition, the elastic modulus of the rotating shaft is \(E\), and the moment of inertia of the cross section of the rotating shaft is \(l\); the polar moment of inertia of the disk is \(l_{p}\), and the moment of inertia of the disk diameter is \(J_{d}\). The parameter values of the rotor system studied in this work are from a real rotor, as shown in Table 1.  
The equation of motion for the rotor system is established by using the Lumped Mass Method and the Lagrange Energy Equation. Lagrange equation is  
\[\frac{d}{d_{l}}\left(\frac{\partial L}{\dot{q}}\right)-\frac{\partial L}{ \partial q}+\frac{\partial\mathbf{\Phi}}{\partial\dot{q}}=F(t) \tag{17}\]  
where \(L=T-U\); \(q\) represents the displacement of each degree of freedom; and \(F(t)\) is the external load of the system, the unbalanced force is considered in this study; compared with the unbalanced force, the influence of the aerodynamic force is negligible. The yaw angles of the wheel around the \(x\) axis and the \(y\) axis are \(\theta_{x}\) and \(\theta_{y}\), respectively; the displacements \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Mass of disk, left and right support \(m_{d}\), \(m_{a}\), \(m_{b}\) (kg) & 34.6, 2, 2 \\ Damping of disk, left and right support \(c\), \(c_{a}\), \(c_{b}\) (N\(\cdot\)s/m) & 2100, 1050, 1050 \\ Support stiffness at left and right end \(k_{a}\), \(k_{b}\) (N/m) & \(1\times 10^{7}\), \(1\times 10^{7}\) \\ Shaft length and wheel position \(l\), \(a\), \(b\) (m) & 0.5, 0.5/3, 1/3 \\ Polar and diameter moment of inertia of the disk \(J_{p}\), \(J_{d}\) (kg\(\cdot\)m\({}^{2}\)) & 0.7, 0.35 \\ Wheel eccentricity \(e\) (m) & 3 \(\times\) 10\({}^{-5}\) \\ Elastic modulus of shaft \(E\) (kg\(\cdot\)m\({}^{2}\)) & 2.09 \(\times\) 10\({}^{11}\) \\ Moment of inertia of cross section of shaft \(l\) (m\({}^{4}\)) & 1.2566 \(\times\) 10\({}^{-7}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The related parameters of the rotor system. Figure 3: Schematic diagram of the offset rotor system. of the bearing at the left and right ends in the \(x\) and \(y\) directions are \(x_{a}\), \(y_{a}\) and \(x_{b}\), \(y_{b}\), respectively; the kinetic energy of the rotor system is  
\[