## 5 Conclusions  
This paper considers the vibration response of the bearing system when the spalling fault occurs at non-center positions on the raceway. A force analysis was conducted on bearings subjected to radial loads from a three-dimensional perspective, and the Hertz contact deformation area between the rolling elements and the raceway was calculated and described the position conditions of the spalling fault causing axial impact in the axial direction of the raceway. And the impact characteristics of spalling faults at different axial positions of the inner raceway were studied, thus proposing a three-degree-of-freedom dynamic model considering axial impact. Through simulation and experimental comparison analysis, the results show that the simulation and experimental results were similar, which verifies the effectiveness and accuracy of the model. When the spalling fault occurs in the non-central area of the raceway, it will generate axial impact and reduce the vibration signal intensity in the x direction, while the vibration signal in the y direction is not affected.  
The research made some achievements in the analysis of bearing peeling failure dynamics, but some aspects need to be improved. First, the time-varying stiffness of the rolling body through the peeling defect region. The existing time-varying stiffness is the contact stiffness change studied in the two-dimensional plane. In fact, both the rolling body and the inner coil rolling road are three-dimensional, so the time-varying stiffness is more in line with the actual situation with the spatial variation in deformation. Secondly, three-degrees of freedom dynamics can only simulate whether the axial impact is generated, and further development needs to be performed to realize the predicted axial impact amplitude.  
Conceptualization, S.L. and W.M.; methodology, S.L.; software, L.W.; validation, L.W., S.L. and W.M.; investigation, J.W.; data curation, J.W.; writing--original draft preparation, L.W. and Z.G.; writing--review and editing, S.L. and W.M.; visualization, W.M.; supervision, S.L.; project administration, S.L. and W.M.; funding acquisition, W.M. All authors have read and agreed to the published version of the manuscript.  
This research was funded by: National Natural Science Foundation of China (Grant No. 52205163), Youth Projects of Basic Scientific Research Projects in Colleges and Universities for Liaoning Provincial Education Department (JYQN2023383), National Defense Key Laboratory Open Foundation of Aerospace Manufacturing Process of Shenyang Aerospace University (No. SHSYS202408).  
Not applicable.  
The original contributions presented in this study are included in the article. Further inquiries can be directed to the corresponding author.  
The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Vibration** & **Axial Position** & **Axial Position \(\tau=15^{\circ}\)** & **Year-on-Year** \\
**Direction** & \(\tau=0^{\circ}\) & **Axial Position \(\tau=15^{\circ}\)** & **Growth** \\ \hline x & 0.2677 & 0.2077 & \(-22.4\%\) \\ y & 0.1259 & 0.1274 & \(+1.2\%\) \\ z & 0.0073 & 0.0847 & \(++\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Root mean square comparison of measured vibration signals.