## 2 Friction-Induced Nonlinear System  
In this research, we consider that the rotor/stator system contains a flexible rotor and a frame. The equipment is physically described as a Jeffcott rotor with an interaction with a stator [13; 22] in Figure 1.  
The rotor is considered a massless shaft which has ideal bearings at two ends, and a disk with mass (\(m_{r}\)) mounted at the center of the shaft. The disk is characterized by mass eccentricity (\(e\)). \(\omega\) stands for the angular speed, \(k_{s}\) stands for the transverse stiffness, and \(c_{s}\) stands for the damping coefficient. For the consideration of the effects around the circumstance, we consider cross-coupling stiffness as \(Q_{s}\) in the rotor/stator system. As a pair with the rotor, we set the stator to be concentric with the rotor and paired with the disk. The stator is modeled as being supported by springs elastically with stiffness (\(k_{b}\)) in the radial direction. The mathematical equation of the rotor/stator system with friction is as follows:  
\[ \begin{array}{l}m\ddot{r}+c_{s}r+k_{s}r-mr\dot{\phi}^{2}+\Theta k_{b}(r-r_{0} )=me\omega^{2}\cos(\omega t-\phi)\\ mr\ddot{\phi}+c_{s}r\dot{\phi}+2m\dot{r}\dot{\phi}-Q_{s}r+\Theta\mu k_{b}(r-r_{0} )\cdot\mathrm{sgn}(v_{rel})=me\omega^{2}\sin(\omega t-\phi)\\ v_{rel}=\omega r_{d}+\dot{\phi}r\end{array}  \tag{1}\]  
where \(r_{d}\) is the radius of the disk; \(r_{0}\) is the clearance between the disk and stator; \(r\) is the vibration radius; \(\dot{\phi}\) is the vibration frequency of the disk. \(\Theta\) is defined as follows:  
\[\Theta=\left\{ \begin{array}{ll}0.&r<r_{0}\\ 1.&r\geq r_{0}\end{array} \right.\]  
\(v_{rel}\) stands for the relative velocity between the speed, \(\omega r_{d}\), and \(\dot{\phi}r\) of the disk. We define the direction of friction (\(F_{I}\)) as the variation with the direction of the velocity (\(v_{rel}\)) (for instance, \(\mathrm{sgn}(v_{rel})\)). O and O\({}_{1}\) are the geometry centers of the stator and the disk, respectively. \(\phi\) is the whirling motion angle. \(F_{n}\) represents normal force from the stator to the disk. \(F_{t}\) is the Coulomb friction force with a coefficient (\(\mu\)). Gravity was ignored in the research. When the rotor/stator is in contact, Equation (1) has the following form:  
\[ \begin{array}{l}R^{\prime\prime}+2\zeta R^{\prime}+\beta R-R\phi^{\prime 2}+ \Theta(R-R_{0})=\Omega^{2}\cos(\Omega\tau-\phi)\\ R\phi^{\prime\prime}+2\zeta R\phi^{\prime}+2R^{\prime}\phi^{\prime}-\gamma R+ \Theta\mu(R-R_{0})\cdot\mathrm{sgn}(V_{rel})=\Omega^{2}\sin(\Omega\tau-\phi) \\ V_{rel}=\Omega R_{d}+\phi^{\prime}R\end{array}  \tag{2}\]  
Figure 1: Rotor/stator model with clearance. (**a**) Schematic representation. (**b**) Side view at the position of the disk during backward rubbing. where \(R^{\prime}=dR/D\tau\), \(\phi^{\prime}=d\phi/D\tau\), \(R=r/e\), \(R_{0}=r_{0}/e\), \(R_{d}=r_{d}/e\), \(2\zeta=c_{s}/\sqrt{mk_{b}}\), \(\beta=k_{s}/k_{b}\), \(\gamma=Q_{s}/k_{b}\), \(\omega_{2}=\sqrt{k_{b}/m}\), \(\Omega=\omega/\omega_{2}\), \(V_{rel}\omega_{2}=v_{rel}/e\), and \(\tau=\omega_{2}t\). \(\omega_{2}\) is considered as the natural frequency of the rotor/stator system when the clearance is zero. Consider the following representation:  
\[x_{1}=R,\ x_{2}=R^{\prime},\ x_{3}=\phi,\ x_{4}=\phi^{\prime}\ \text{and}\ \theta=\text{mod}(\Omega\tau,2\pi)\]  
Equation (2) is rewritten as follows:  
\[x^{\prime}= \begin{cases}F_{1}(x),&H(x)>0\\ F_{2}(x),&H(x)<0\end{cases}  \tag{3}\]  
where \(x=\left(x_{1},\ x_{2},\ x_{3},\ x_{4},\ \theta\right)^{\text{T}}\in\mathbb{R}^{5}\) and \(H(x)=\Omega R_{d}+x_{1}x_{4}\). \(F_{1}(x)\) and \(F_{2}(x)\) are as follows:  
\[F_{1}(x)=\left( \begin{array}{l}x_{2}\\ \Omega^{2}\cos(\theta-x_{3})-2\zeta x_{2}-(\beta+1)x_{1}+x_{1}{x_{4}}^{2}+R_{0 }\\ x_{4}\\ \frac{\Omega^{2}\sin(\theta-x_{3})}{x_{1}}-2\zeta x_{4}-2\frac{x_{2}x_{4}}{x_ {1}}+\gamma-\mu(1-\frac{R_{0}}{x_{1}})\\ \Omega\end{array} \right)\]  
\[F_{2}(x)=\left(