# Research on a Nonlinear Dynamic Incipient Fault Detection Method for Rolling Bearings
## 4 Experimental Results and Analysis
### Simulation Results and Analysis of The Rolling Bearing Rolling Body Faults  
#### 4.2.1 The Rolling Body Fault Simulation Results  
The fault characteristics of rolling elements were more complicated, which was determined by the motion characteristics of rolling elements. When the rolling bearing is in operation, it must not only ensure the rotation with the cage, but also complete its own rotation [36]. When a rolling body fails, the modulation frequency of its high-frequency resonance signal will be consistent with the rotation frequency of the cage, and the signal in the frequency domain will have a larger amplitude spectral line. Theoretically, the sideband distance on both sides of the spectral line should be consistent with the rotation frequency of the cage. However, due to the interference of the rolling motion of the rolling body, the performance of the actual collected vibration signal had weak characteristics. Fault detection was difficult. This section tested the incipient failure of rolling elements to verify the accuracy of the proposed Deep ADKPCA method. The simulation results were as follows. The validity and accuracy of proposed methods were tested, and the results were also compared with other known methods such as ADKPCA, Deep KPCA, and Deep DKPCA. Figure 9 describes the simulation results of the Deep ADKPCA fault detection method for the rolling bearing rolling body. Figure 9a,b shows the simulation results of \(T_{01}^{2}\) and \(Q_{01}\) statistics of the original space \(X\) based on the Deep ADKPCA method, that is, the simulation results based on the ADKPCA method. Figure 9c-j shows the simulation results of \(T_{ij}^{2}\) and \(Q_{ij}\) statistics of the second-order subspace \(X_{21}\)-\(X_{24}\) based on the Deep ADKPCA method, respectively. It can be seen from Figure 9a,b that the failure detection rate of the ADKPCA method was about 50%, this method basically did not have incipient fault detection capabilities. Figure 9c shows the detection result of the fault by the \(T_{21}^{2}\) statistic. It can be seen from the image that the \(T_{21}^{2}\) statistic was less sensitive to the fault than other  
Figure 8: (a)The fault detection rate of rolling bearing outer ring fault and (b) the detection delay of rolling bearing outer ring fault. statistic. Figure 9g,i,h shows \(T_{23}^{2}\), \(T_{24}^{2}\), and \(Q_{23}\) statistics for the fault detection result. From the images we could obtain that \(T_{21}^{2}\), \(T_{24}^{2}\), and \(Q_{23}\) were more sensitive to this fault, and the fault detection rate was about 85%. So, the Deep ADKPCA method had a good incipient fault detection capability. Figure 9: Experimental results of the Deep ADKPCA fault detection method for the rolling bearing rolling body fault. (**a**) The result of \(T_{01}^{2}\), (**b**) The result of \(Q_{01}\), (**c**) The result of \(T_{21}^{2}\), (**d**) The result of \(Q_{21}\), (**e**) The result of \(T_{22}^{2}\), (**f**) The result of \(Q_{22}\), (**g**) The result of \(T_{23}^{2}\), (**h**) The result of \(Q_{23}\), (**i**) The result of \(T_{24}^{2}\) and (**j**) The result of \(Q_{24}\). Figure 10 describes the simulation results of the Deep KPCA fault detection method for the rolling bearing rolling body. Figure 10a-h describes the simulation results of \(T_{ij}^{2}\) and \(Q_{ij}\) statistics of the second-order subspace \(X_{21}\)-\(X_{24}\) based on the Deep KPCA method, respectively. Figure 10c,e shows the detection results of incipient failures of rolling bearing outer rings of \(T_{22}^{2}\) and \(T_{23}^{2}\), we could obtain that \(T_{22}^{2}\) and \(T_{23}^{2}\) were less affected by this fault. From Figure 10g, we could obtain that \(T_{24}^{2}\) was more sensitive to the fault. The Deep KPCA method had some incipient fault detection capabilities, but it still had some disadvantages compared to the Deep ADKPCA method. Figure 11 describes the simulation results of the Deep DKPCA fault detection method for the rolling bearing rolling body.