## 2 Dynamic Modeling of Spalling Faults at Non-Central Positions of the Inner-Ring Raceway
### Rolling Element Deformation Quantity  
In this work, the theoretical analysis of a three-dimensional structure and motion characteristics of rolling bearing is made. Taking the 6205 deep groove ball bearing as an example, the bearing parameters are shown in Table 1. Considering that when the spalling fault occurs in the non-center position of the raceway, the bearing under the radial view angle is not symmetrical, and the movement direction of the rolling element is affected, the axial position of the raceway is introduced into the establishment condition of the model. Extend a z-axis on the x/y plane to describe the axial position of the spalling defect in the raceway, thereby establishing a three-dimensional mathematical model of the deep groove ball bearing, and on this basis, analyze the forces between the rolling element and the groove under different sections. Analyze the rolling ball bearing with single-point pitting failure under three-dimensional conditions, and decompose it into two sectional plane views, as shown in Figure 3, plan view of x-axis and y -axis sections as shown in Figure 3a, plan view of x-axis and z-axis sections as shown in Figure 3b.  
Figure 1: Bearing inner ring raceway non-center spalling fault.  
Figure 2: Rolling bearing model diagram.  
The rolling elements within the load area are deformed by the load distributed at their positions, causing an offset between the circle centers of the inner and outer rings of the bearing. This offset is the sum of the deformation vectors of all the rolling elements. Among them, \(\delta_{\rm i}\) is the contact deformation of the i-th bearing rolling element, which is expressed as:  
\[\delta_{\rm i}=(x\cos\theta_{\rm i}+y\sin\theta_{\rm i}-0.5C_{r})\cos\tau+z\sin\tau \tag{1}\]  
where, x, y and z are the offsets of the bearing's center of mass decomposed in the x, y and z directions, respectively; C\({}_{\rm r}\) is the bearing clearance; \(\theta_{\rm i}\) is the position angle of the i-th rolling element relative to the \(x\) axis, and it can be expressed as:  
\[\theta_{\rm i}=\omega_{\rm c}t+\frac{2\pi(j-1)}{Z}+\theta_{0} \tag{2}\]  
where, \(\theta_{0}\) is the initial position angle of the first rolling element; z is the number of rolling elements; \(\omega_{\rm c}\) is the angular velocity of the rolling element around the bearing circle center, and it can be expressed as:  
\[\omega_{\rm c}=\frac{1}{2}(1-\frac{d_{b}}{D_{w}}\cos\beta)\omega_{\rm s} \tag{3}\]  
where, d\({}_{\rm b}\) is the diameter of the rolling element; D\({}_{\rm w}\) is the pitch diameter; \(\beta\) is the bearing contact angle; \(\omega_{\rm s}\) is the rotor angular velocity; and it can be expressed as:  
\[\omega_{s}=\frac{2\pi n}{60} \tag{4}\]  
\begin{table}
\begin{tabular}{c c} \hline
**Parameter** & **Numerical Value** \\ \hline The inner raceway diameter Di/mm & 25 \\ The outer raceway diameter Db/mm & 52 \\ Pitch circle diameter & 38.5 \\ D\({}_{\rm w}\)/mm & \\ Rolling element diameter & 7.9 \\ d\({}_{\rm b}\)/mm & 9 \\ Number of rolling elements & 1 \\ Rolling element mass & 2.14 \\ m\({}_{0}\)/g & 2.14 \\ \hline \end{tabular}
\end{table}
Table 1: 6205 Bearing parameter table.  
Figure 3: Three-dimensional load distribution of bearings.  
where, \(\tau\) is the position angle of the pitting on the raceway relative to the center of the rolling element, as shown in Figure 3b. \(\phi_{\text{spall}}\) is the radial position angle of the fault center point. \(\phi_{\text{d}}\) is the half of the circle center position angle of the spalling fault, and the calculation formula is:  
\[\varphi_{d}=\arcsin\left(\frac{L}{d_{b}}\right) \tag{5}\]  
where, L is the length of the spalling fault, and d\({}_{\text{b}}\) is the diameter of the rolling element.