## 2 Relevant Theoretical Models  
### Bearing Dynamics Modeling  
The bearing model is composed of the inner ring, rolling elements and the inner ring. When constructing the bearing dynamics model, assumptions are made as follows:  
1. The outer ring is fastened to the bearing housing.
2. The inner ring rotates smoothly along with the shaft.
3. The elements are arranged at equal intervals on the raceway for pure rolling. At time \(t\), the \(j\)-th rolling element's angular position is \(\phi_{j}\), which can be expressed as \[\phi_{j}=\omega_{c}t+\frac{2\pi(j-1)}{N_{b}}+\phi_{0}\ j=1,2,\dots,N_{b}\] (1)  
When a bearing malfunctions, the contact force inside the bearing will change significantly, which impacts the bearing's motion state. The fault model assumes a rectangular notch on the outer circle. The position angle of the fault is \(\phi_{F}\), the width of the fault is \(W_{F}\), the corresponding angle of the fault is \(\Delta\phi_{F}\), and the depth is \(C_{d}\). The contact force between the rings and the rolling element varies as the rolling element moves through the fault point. Thereby generating periodic pulses. The contact state returns to normal after passing the fault. At this time, the total contact deformation is  
\[\delta_{j}=xsin\phi_{j}+ycos\phi_{j}-\delta_{0}-\beta_{j}C_{d} \tag{2}\]  
where \(x\) and \(y\) represent the bearing's displacements in the \(x\) and \(y\) directions, respectively, \(\delta_{0}\) is the radial clearance, and \(\beta_{j}\) is the judgment function, which can be described as  
\[\beta_{j}=\left\{\begin{aligned} & 1,\phi_{F}<\phi_{J}\leq\phi_{F}+ \Delta\phi_{F}\\ & 0,else\end{aligned}\right. \tag{3}\]  
The contact force of the bearing can be obtained from Harris contact theory as  
\[\left\{\begin{aligned} & F_{x}=K\Sigma_{j=1}^{N_{b}}\ R_{j}\delta_{j}{}^{n} cos\phi_{j}\\ & F_{y}=K\Sigma_{j=1}^{N_{b}}\ R_{j}\delta_{j}{}^{n}sin\phi_{j} \end{aligned}\right. \tag{4}\]  
where \(F_{x}\) and \(F_{y}\) represent the contact forces in the \(x\) and \(y\) directions, respectively, \(n\) is related to the contact mode, \(K\) represents the stiffness coefficient of the bearing contact, and \(R_{j}\) is the judgment function, which can be expressed as  
\[R_{j}=\left\{\begin{aligned} & 1,\delta_{j}>0\\ & 0,else\end{aligned}\right. \tag{5}\]  
The contact stiffness coefficient \(K\) can be expressed as  
\[K=\left[\frac{1}{(1/k_{i})^{1/n}+(1/k_{o})^{1/n}}\right]^{n} \tag{6}\]  
where \(k_{o}\) is the rolling element's contact stiffness coefficient with the outer raceway, and \(k_{i}\) is the rolling element's contact stiffness coefficient with the inner raceway. They can be expressed as  
\[\left\{\begin{aligned} & k_{i}=(\frac{n^{2}\sigma^{2}E^{2}\delta_{0}^{2}}{ 3C_{1}\Sigma_{1}\mu_{1}})\\ & k_{o}=(\frac{n^{2}\sigma^{2}E^{2}\delta_{0}^{2}}{3C_{1}\Sigma_{ 1}\mu_{1}})\end{aligned}\right. \tag{7}\]where \(\rho_{i}/\rho_{o}\) are the contact curvatures between the inner/outer rings, respectively, and the rolling elements; \(E\) is the equivalent elastic modulus; \(e\) is the ellipticity parameter; \(\xi_{1}\) and \(\xi_{2}\) are the first and second types of elliptic integrals, respectively.  
According to [PERSON]'s second law, the outer ring dynamic equation of the bearing can be obtained as  
\[\left\{\begin{array}{l}m_{o}\dot{x}+c_{o}\dot{x}+k_{o}x=F_{lhx}+F_{x}\\ m_{o}\ddot{y}+c_{o}\dot{y}+k_{o}y=F_{hy}+F_{y}-m_{o}g\end{array}\right. \tag{8}\]  
where \(m_{o}\) represents the outer ring mass, \(c_{o}\) represents the outer ring damping, and \(F_{h}\) represents the load.