# Bearing Anomaly Recognition Using an Intelligent Digital Twin Integrated with Machine Learning
## 3 Experimental Results and Discussion
### Residual Signals Test and Analysis  
After modeling and estimation, the bearing signals to design a digital twin and the residual signals are computed using the difference between the original (RAW) signals and the estimated signals. To test the effectiveness of the proposed digital twin estimation technique (MS-HRKF), this technique is compared with the following algorithms: (a) the MS-KF and (b) the MS-RKF. According to Equations (23) and (26), the residual signals for the MS-KF technique, \((R_{MS-KF}(k))\), and MS-RKF technique, \((R_{MS-RKF}(k))\), are computed as the following equations, respectively.  
\[R_{MS-KF}(k)=Z(k)-Z_{MS-KF}(k) \tag{40}\]  
\[R_{MS-RKF}(k)=Z(k)-Z_{MS-RKF}(k) \tag{41}\]  
Figure 11 shows the power of the MS-KF for the classification condition of the CWRUD bearing signals in diverse situations.  
Regarding the comparison between Figures 3 and 11, it can be seen that the MS-KF improves the classification. Moreover, Figure 12 shows the power of the MS-RKF for the classification condition of the bearing signals in different conditions. Regarding the comparison between Figures 3 and 12, it can be seen that the MS-RKF improves the classification. Regarding Figure 3, the CWRUD for the bearing has a significant challenge  
Figure 11: The residual signals of the bearing using the MS-KF algorithm when the motor torque loads vary from 0 to 3 hp; moreover, the crack sizes for abnormal conditions are 0.007, 0.014, and 0.021 inch.  
Figure 10: Impulse response estimated plot for modeled normal signal.  
in classification, especially when the crack sizes are 0.007 and 0.014 inch. The MS-RKF improves the classification's accuracy, especially when the crack size is 0.007 inch. Figure 13 illustrates the residual signals of the proposed digital twin (MS-HRKF) method. Regarding Figure 13, the overlapping and misclassification of the residual signals in the proposed digital twin (MS-HRKF) are reduced sharply.  
Figure 12: The residual signals of the bearing using the MS-RKF algorithm when the motor torque loads vary from 0 to 3 hp; moreover, the crack sizes for abnormal conditions are 0.007, 0.014, and 0.021 inch.  
Figure 13: The residual signals of the bearing using the proposed digital twin (MS-HRKF) algorithm when the motor torque loads vary from 0 to 3 hp; moreover, the crack sizes for abnormal conditions are 0.007, 0.014, and 0.021 inch: (**a**) all cases and (**b**) zoom view for NS and RFS.  
Regarding this figure, the proposed digital twin (MS-HRKF) method improves the detection and identification of the fault crack sizes sharply. According to the comparison between Figures 12 and 13, it is clear that the MS-RKF has a serious challenge for classification when the crack size is 0.014 inch. The proposed digital twin (MS-HRKF) solves this problem. Thus, based on Figures 11-13, the accuracy of the proposed digital twin (MS-HRKF) for the CWRUD for bearing signal classification is better than the MS-RKF and the MS-KF. In addition, the power spectral density estimation for NS, RFS, IFS, and OFS using proposed method is illustrated in Figure 14 to describe the distribution of power into the frequency components composing those signals. Regarding this figure, the distribution of the signal's power into the frequency for NS, RFS, IFS, and OFS are separable.  
Figure 15 shows the RMS residual signal using proposed digital twin (MS-HRKF) method. Based on this technique, it is clear that the signal is easy to classify.