## 4 Simulation Signal Analysis  
To validate the superiority of ISSD in processing rotor fault signals, a Jeffcott rotor rub-impact model was established to construct simulation signal firstly. Then the signal is analyzed by EMD, SSD and ISSD respectively for comparison. The rotor rubbing model is shown in Figure 4a. A single disc rotor, the mass of which is \(m\), is fixed in the middle of a simply supported shaft with the angular speed of \(\omega\). The mass of the shaft is neglected. It is assumed that the linear bending stiffness of the rotating shaft is \(k_{1}\) and the non-linear bending stiffness of the rotating shaft is \(k_{2}\). \(c\) and \(k_{\rm c}\) are the damping coefficient and the radial stiffness of the stator, respectively. Figure 4b shows a schematic diagram of the rotor rubbing force. Assuming that the rubbing force between the rotor and the stator conforms to [PERSON]'s law and \(r=\sqrt{x^{2}+y^{2}}\) is the radial displacement of the geometric center of the rotor, the normal contact force \(P_{n}\) and tangential rubbing forces \(P_{t}\) can be calculated as follows [34]:  
\[\left\{ \begin{array}{l}P_{n}=(r-\delta)k_{c}\\ P_{t}=(f+\alpha v)P_{n}\end{array} \right., \tag{20}\]  
where, \(\delta\) is the minimum gap between rotor and stator, \(v=(\dot{x}^{2}+\dot{y}^{2})^{1/2}\) is the relative sliding speed between the moving and stationary parts, \(f\) is the friction coefficient and \(\alpha\) is the speed influence factor. If the angle between the point of occurrence of rubbing and the \(x\)-axis is \(\varphi\), the normal and tangential rubbing forces can be expressed in the X-Y coordinate system as follows:  
\[\left\{ \begin{array}{l}P_{x}\\ P_{y}\end{array} \right.=\left[ \begin{array}{cc}-\cos\phi&\sin\phi\\ -\sin\phi&-\cos\phi\end{array} \right]\left\{ \begin{array}{l}P_{n}\\ P_{t}\end{array} \right\}=-\frac{(r-\delta)k_{c}}{r}\left[ \begin{array}{cc}1& -(f+\alpha v)\\ f+\alpha v&1\end{array} \right]\left\{ \begin{array}{l}x\\ y\end{array} \right\}. \tag{21}\]  
Then, the equation of motion of the rotor can be established as Equation (22) shows:  
\[\left[ \begin{array}{cc}m&\\ m&\end{array} \right]\left\{ \begin{array}{l}\dot{x}\\ \dot{y}\end{array} \right\}+\left[ \begin{array}{c}c\\ c\end{array} \right]\left\{ \begin{array}{l}\dot{x}\\ \dot{y}\end{array} \right\}+\left[ \begin{array}{c}k_{1}\\ k_{1}\end{array} \right]\left\{ \begin{array}{l}x\\ y\end{array} \right\}+\left[ \begin{array}{c}k_{2}\\ k_{2}\end{array} \right]\left\{ \begin{array}{l}\dot{x}^{3}\\ y^{3}\end{array} \right\}=\left\{ \begin{array}{l}P_{r}\\ P_{y}\end{array} \right\}+\mu\mu\omega^{2}\left\{ \begin{array}{l}\cos\omega t \\ \sin\omega t\end{array} \right\}+\left\{ \begin{array}{l}0\\ -mg\end{array} \right\}, \tag{22}\]  
Figure 4: (**a**) Jeffcott rotor model; (**b**) rubbing force model. where, \(\mu\) represents eccentricity of the rotor. With the transformation of \(x=\frac{x}{\delta}\), \(y=\frac{y}{\delta}\), \(\varpi=\frac{\omega}{\omega_{0}}\), \(\tau=\omega t\), \(\frac{c}{m}=2\sigma\omega_{0}\), \(\beta=\frac{k_{c}}{k_{1}}\), \(\overline{\mu}=\frac{\mu}{\delta}\), the differential equations of motion for the rotor are obtained:  
\[\left\{ \begin{array}{c}\ddot{x}+2\sigma\dot{x}+x+\lambda x^{3}+\beta(1-\frac{ 1}{r})\Big{[}x-fy-\alpha(\dot{x}^{2}+\dot{y}^{2})^{1/2}\Big{]}=m\varpi\cos \varpi t\\ \ddot{y}+2\sigma\dot{y}+y+\lambda y^{3}+\beta(1-\frac{1}{r})\Big{[}fx+y+\alpha (\dot{x}^{2}+\dot{y}^{2})^{1/2}\Big{]}=\overline{\mu}\varpi^{2}\sin\varpi t- mg\end{array} \right.. \tag{23}\]  
Using the four order Runge-Kutta method to solve Equation (23), we can get the vibration response of the signal in the direction of X and Y. For the simulated signal in this paper, the specific parameters were initialized to: \(m=2.8\) kg, \(\sigma=0.4\), \(\omega=3\), \(\lambda=0.45\), \(\beta=1.65\), \(f=0.1\), \(\alpha=0.3\) and \(\overline{\mu}=2\). Figure 5a,b shows the waveform and frequency spectrum of the simulated rubbing signal in the Y direction, respectively.