## 2 Basic Theory of the TPOD Method and the Rotor Model with Coupling Faults
### The Rotor Model with Coupling Faults  
The high-pressure rotor system supported by a pair of sliding bearings with both crack and rub-impact faults is modeled as a 24-DOFs system by the [PERSON]'s second law, as shown in Figure 1.  
Here we assume that the axial and torsional vibrations of the system and the gyroscopic moment are neglectful. \(O_{i}(i=1\ldots 12)\) are the geometric centers of the discs. \(m_{i}(i=1\ldots 12)\) are the equivalent lumped masses. \(c_{i}(i=1\ldots 12)\) are the equivalent damping coefficients at the position of the lumped masses. \(k_{i}(i=1\ldots 11)\) are the equivalent stiffness of the corresponding discs. Assume that the crack occurs on the shaft between the first and second disc. The rotor model is supported by a pair of liquid-film bearings on both ends. The schematic diagram of the cracked section is shown in Figure 2a. \(\alpha\) is the depth of crack, \(\beta\) is the intersection angle between the unbalance and the crack normal vector. Law of open and close describes the effects of crack intersection angle to the shaft stiffness. There are three classic models that  
Figure 1: Rotor model with coupling faults (crack and rub-impact). characterizing the time-varying property of a crack when the rotor is operating: the cosine model [53], square wave model and hybrid one [54]. In Figure 2a, the initial intersection angle of the cracked normal vector and \(x\)-axis is 0. We consider the case of gravity dominance, the open and close function of crack \(f(\varphi)\) can be expressed as the function of rotating angle \(\omega t\). \(k\) is the cracked stiffness of rotating shaft, \(\Delta k_{\xi}\) and \(\Delta k_{\eta}\) are the stiffness variations of cracked normal and tangential vectors. The stiffness of rotating shaft with crack is expressed as:  
\[\left[ \begin{array}{cc}k_{x}&k_{xy}\\ k_{yx}&k_{y}\end{array} \right]=\left[ \begin{array}{cc}k&0\\ 0&k\end{array} \right]-f(\varphi)\left[ \begin{array}{cc}\Delta k_{\xi}\cos^{ 2}(\omega t)+\Delta k_{\eta}\sin^{2}(\omega t)&\left(\Delta k_{\xi}-\Delta k_{ \eta}\right)\sin(\omega t)\cos(\omega t)\\ \left(\Delta k_{\xi}-\Delta k_{\eta}\right)\sin(\omega t)\cos(\omega t)& \Delta k_{\xi}\sin^{2}(\omega t)+\Delta k_{\eta}\cos^{2}(\omega t)\end{array}  \right] \tag{5}\]  
The expression of \(f(\varphi)\) is shown in Formula (6), this model can select the square wave and cosine model based on the depth of the crack respectively. \[f(\varphi)=\left\{ \begin{array}{cc}\frac{1}{2}+\frac{2}{\pi}\cos(\varphi)- \frac{2}{3\pi}\cos(3\varphi)+\frac{2}{5\pi}(5\varphi)-\ldots&a/\mathrm{R}\leq 0.5\\ (1+\cos\varphi)/2&a/\mathrm{R}>0.5\end{array} \right. \tag{6}\]  
The rub-impact model is shown in Figure 2b, it is combined with linear contact force and Coulomb friction [55], the formula is expressed as:  
\[\left[ \begin{array}{c}P_{x}\\ P_{y}\end{array} \right]=\left\{ \begin{array}{cc}0&\mathrm{r}\leq r_{0}\\ k_{b}\Big{(}1-\frac{r_{0}}{r}\Big{)}\left[\begin{array}{c}x-\mu_{0}y\\ \mu_{0}x+y\end{array}\right]&\mathrm{r}>r_{0}\end{array} \right. \tag{7}\]  
where \(\mathrm{r}\) is radial displacement, \(r_{0}\) is clearance between rotor and stator, \(k_{b}\) is rub-impact stiffness, \(\mu_{0}\) is frictional coefficient, \(P_{x}\) and \(P_{y}\) represent the rub-impact forces along \(x\) and \(y\) directions. The dynamical equation is shown in Formula (8) and the dimensionless process is expressed as follows:  
Figure 2: Schematic diagram of the cracked section and rubbing model. (**a**) The cracked model. (**b**) The rub-impact model. \[ \begin{array}{c}\tau=\omega t,\;x_{i}=\frac{X_{i}}{c},\;y_{i}=\frac{Y_{i}}{c}, \dot{x}_{i}=\frac{dx_{i}}{d\tau},\dot{y}_{i}=\frac{dy_{i}}{d\tau},\;\ddot{x}_{i} =\frac{d\dot{x}_{i}}{d\tau},\;\ddot{y}_{i}=\frac{d\dot{y}_{i}}{d\tau},M_{1}= \frac{m_{1}c\omega^{2}}{sP},\;M_{12}=\frac{m_{12}c\omega^{2}}{sP}\\ \end{array} \]  
\(X_{i}(i=1,\ldots 12)\) and \(Y_{i}(i=1,\ldots 12)\) represent the vibration displacement of x and y directions. The dimensionless equation is expressed as Formula (9).