## 2 Dynamic Modeling of Spalling Faults at Non-Central Positions of the Inner-Ring Raceway
### Time-Varying Energy of Rolling Elements in Spalling Area  
According to the law of conservation of energy, for a rolling bearing with spalling defects on the raceway, the energy of the rolling elements will change when passing through the defective area, causing energy fluctuations in the entire bearing system and thus triggering vibration responses. The contact deformation and movement speed are the main influencing factors of energy change, and the movement state of the rolling element when passing through the defect area is determined by its internal contour. Therefore, the length, width, and depth of the spalling defect are set as fixed values, and the defect slope and the time-varying energy of the rolling element when passing through are quantitatively studied.  
Circle center trajectory is shown in Figure 4, which is a sinusoidal trajectory. When the rolling element enters the defect area, the elastic deformation will be released, and the released elastic deformation is equal to the offset of the circle center relative to the pitch diameter of the bearing. According to the geometric relationship between the rolling element and the defect contour and the judgment of the vector direction, when the rolling element enters the defect area, it only contacts the entry end. At this time, the deformation is gradually released, the elastic potential energy decreases, and the kinetic energy increases until it runs to the circle center trajectory of the defect and leaves the contact end. This entry of the circle trajectory can be approximated as a half-sine curve. Next, the circle center trajectory is completed around the leaving end. The elastic potential energy gradually increases, and the kinetic energy decreases. According to the geometric relationship, the formula for the release of elastic deformation is established as:  
\[\delta=\begin{cases}-H_{o}\cdot(1-\cos\left\{\frac{\pi[\theta_{i}-\omega t- \varphi_{\text{spall}}]}{2\phi_{d}}\right\}&AB\\ H_{o}\cdot(1-\cos\left\{\frac{\pi[\theta_{i}-\omega t-\varphi_{\text{spall}}]}{2 \phi_{d}}\right\}&BC\\ 0&elsewhere\end{cases} \tag{6}\]  
where, H\({}_{o}\) is the maximum submergence depth of the V-shaped trajectory, and the calculation formula is as follows:  
\[H_{o}=\tan\alpha\cdot\frac{L-2d_{b}\cdot\tan(\frac{\pi}{2})}{2} \tag{7}\]  
Establish an expression for the movement speed \(V_{y}\) in the y-direction of the rolling element with a circular-arc-type trajectory:  
\[V_{y}=\begin{cases}V_{x}\cdot(1-\cos\left\{\frac{\pi[\theta_{i}-\omega t- \varphi_{\text{spall}}]}{2\phi_{d}}\right\}&AB\\ V_{x}\cdot(1-\cos\left\{\frac{\pi[\theta_{i}-\omega t-\varphi_{\text{spall}}]}{2 \phi_{d}}\right\}&BC\\ 0&elsewhere\end{cases} \tag{8}\]  
The energy formula of the rolling element:  
\[E_{m}=E_{r}+E_{k} \tag{9}\]where, \(\rm E_{m}\) is the energy of the rolling element, \(\rm E_{r}\) is the potential energy, and the calculation formula is:  
\[E_{r}=m_{0}g\delta+\frac{1}{2}K\delta^{1.5} \tag{10}\]  
where, \(\rm m_{0}\) is the mass of the rolling element; \(\delta\) is the submergence distance of the rolling element's circle center; K is the elasticity coefficient; g is the gravitational coefficient, taken as 9.81 m/s; \(\rm E_{k}\) is the kinetic energy, and the calculation formula is as follows:  
\[E_{k}=\frac{1}{2}m_{0}\Big{(}v_{x}^{2}+v_{y}^{2}\Big{)} \tag{11}\]  
where, \(\rm v_{x}\) is the motion speed of the rolling element in the x-direction; \(\rm v_{y}\) is the speed of the rolling element in the y-direction.