# 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 Outer Ring Faults  
#### 4.1.1 The Rolling Bearing Outer Ring Fault Simulation Results  
When there is a defect point on the outer surface of a rolling bearing, the rolling bearing passes the defect point during operation, which will generate a periodic pulse force to the system. Generally, because the outer ring is fixed, the pulse intensity of each cycle is considered as constant, and the periodic pulse signal presents the law of periodic equal amplitude vibration changes [35]. This section simulates the incipient failure of the rolling bearing outer ring. 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. The original data was divided into multiple subspaces according to the steps in Section 3.2, and calculate the \(T_{ij}^{2}\) and \(Q_{ij}\) control limits and statistics of each subspace. The results are shown in Figure 5. Figure 5a,b show 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 5c-j show 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. In the figures, the solid black line is the statistic, and the red dotted line is the control limit with 95% confidence. If the statistics exceeded the control  
Figure 4: Comprehensive mechanical failure test bench. limit, a fault occurred, otherwise it is normal. Since the model was updated every three sets of data when updating the model, the number of samples in the figure was about 340. From Figure 5a,b, \(T_{01}^{2}\) and \(Q_{01}\) were slightly affected by this fault. Figure 5d,f shows the detection results of incipient failures of rolling bearings outer rings of \(Q_{21}\) and \(Q_{22}\), we could obtain that \(Q_{21}\) and \(Q_{22}\) were not sensitive to this fault. Figure 5g,i show the results of the \(T_{23}^{2}\) and \(T_{24}^{2}\) statistics on the fault detection. From the image, it can be seen that \(T_{23}^{2}\) and \(T_{24}^{2}\) were more sensitive to the fault, the fault detection rate was around 90%. So, Deep ADKPCA the method had a good incipient fault detection capability. Figure 6a-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 6b,c,d shows the detection results of incipient failures of rolling bearing outer rings of \(T_{22}^{2},Q_{21}\), and \(Q_{22}\), we could obtain that \(T_{22}^{2},Q_{21}\), and \(Q_{22}\) were less affected by this fault. From Figure 6g, we could obtain that \(T_{24}^{2}\) was more sensitive to the fault. The Deep KPCA method has some incipient fault detection capabilities, but it still has some disadvantages compared to the Deep ADKPCA method. Figure 7a-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 DKPCA method, respectively. Figure 7a,g describes the detection results of incipient failures of rolling bearing outer rings of \(T_{21}^{2}\) and \(T_{24}^{2}\), the rolling bearing outer rings fault could be successfully detected by \(T_{21}^{2}\) and \(T_{24}^{2}\). From Figure 7c, we could obtain that this fault had no obvious influence on \(T_{22}^{2}\). Figure 5: Experimental results of Deep ADKPCA fault detection method for the rolling bearing outer ring fault. (**a**) The result of \(T^{2}_{01}\), (**b**)The result of \(Q_{01}\), (**c**) The result of \(T^{2}_{21}\), (**d**) The result of \(Q_{21}\), (**e**) The result of \(T^{2}_{22}\), (**f**) The result of \(Q_{22}\), (**g**) The result of \(T^{2}_{23}\),(**h**) The result of \(Q_{23}\), (**i**) The result of \(T^{2}_{24}\) and (**j**) The result of \(Q_{24}\).