# Research on Multi-Directional Spalling Evolution Analysis Method for Angular Ball Bearing
## 4 Multi-Direction Spalling Expansion Analysis  
The analysis results from the EHL model indicate that the maximum stress point on the bearing's outer ring is situated at the lowest point of the outer ring, with a contact angle of 30.2469\({}^{\circ}\). Around this maximum stress point, an initial square spalling region with a depth of 0.03 mm is defined, encompassing an azimuth angle range of \(\pm 1^{\circ}\) and a contact angle range of \(\pm 10^{\circ}\). Using the established multi-directional crack FEA model, the propagation of spalling in four directions within this region is examined. The analysis continues with a scenario involving a crack of 0.02 mm depth at an initial angle of 90\({}^{\circ}\) relative to the vertical direction. The multi-directional crack FEA model facilitates the examination of spalling expansion in multiple directions by modifying the initial position and direction of motion of the roller. The progression of the crack, labeled as F, is analyzed as the roller moves from left to right. Conversely, the propagation of the crack labeled as B can be analyzed when the roller moves from right to left. In the case where the radial load \(F_{\rm z}=2000\) N, the axial load \(F_{\rm x}=10,000\) N, and the rotating speed is 7000 rpm, the crack propagation results for cracks labeled F and B are presented in Tables 2 and 3, respectively. For cracks labeled as L and R, a static analysis is required based on the initial spalling region width. When the roller is located at the point of maximum stress, the left and right edges of the spalling region are subjected to symmetrical stress when the effect of its self-rotation is ignored. When the roller rolls through the center of the spalling region, the forces are symmetrical in the L and R directions. At this time, the model's left and right translation degrees of freedom are restricted, and \(Q_{\rm e}\) is changed to half of its original value for analysis. Figure 14 shows a schematic diagram of the roller position under symmetrical conditions, and Table 4 shows the crack propagation simulation results under symmetrical conditions. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Iteration Number**} & **Required Time** & **Crack Deflection** & **Crack Propagation** \\  & **/s** & **Angle** & **Rate** \\  & & \(\varphi\varphi^{\circ}\) & \(v_{0}\)/mm\(\cdot\)s\({}^{-1}\) \\ \hline
1 & 714.7517 & 55.9226 & \(4.7856\times 10^{-8}\) \\
2 & 2872.4959 & 64.1817 & \(1.1908\times 10^{-8}\) \\
3 & 1219.1836 & \(-61.2824\) & \(2.8056\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Crack propagation results for cracks labeled F. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Iteration Number**} & **Required Time** & **Crack Deflection** & **Crack Propagation** \\  & **/s** & **Angle** & **Rate** \\  & **/s** & \(\varphi\varphi^{\circ}\) & \(v_{0}\)/mm\(\cdot\)s\({}^{-1}\) \\ \hline
1 & 564.3204 & 56.2685 & \(6.0613\times 10^{-8}\) \\
2 & 3603.2636 & 64.3281 & \(9.4928\times 10^{-9}\) \\
3 & 709.7066 & \(-63.2660\) & \(4.8196\times 10^{-8}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Crack propagation results for cracks labeled B. The use of the simplified 2D model for the analysis of spalling cracks results in higher computational efficiency. This simplification method is feasible when the spalling region coincides with the weak points of the bearing. By analyzing the crack propagation in four directions, it was found that cracks labeled as F and B are more likely to propagate. Therefore, the crack B was analyzed to investigate the relationship between the propagation time and the initial crack angle. Table 5 summarizes the relationship between the initial crack angle and the time required for spalling to propagate 1 mm. In the numerical simulation, when the initial angle exceeds \(90^{\circ}\), the crack will propagate far away from the raceway surface, making it difficult for spalling to propagate.