## 5 Experiment Data Analysis
### Bearing Vibration Signal Analysis and Bearing Fault Detection and Diagnosis  
The normal bearing vibration signal and the faulty bearing vibration signal are shown in Figure 11. The vibration signal of the normal bearing is relatively stable, and the vibration signal of the faulty bearing has obvious fluctuations. It is difficult to directly determine the location of the bearing fault from the faulty bearing vibration signal. Because the bearing will produce different degrees of vibration when it is running, the vibration signal usually contains noise components and other interference information, resulting in the bearing fault signal shock characteristic's being not obvious, and affected by external interference. In the case of excessive external interference, the fault feature is often submerged. In addition, when an incipient fault occurs in the bearing, the amplitude of the fault component relative to the frequency of rotation and other components irrelevant to the fault is smaller. In order to accurately diagnose the fault of the bearing, it is necessary to perform noise reduction pretreatment on the bearing vibration signal. Figure 12 shows the state of the bearing after Kalman filter. Figure 10: Autocorrelation function curve. Figure 9: Model parameter identification curves. Analysis of Figures 11b and 12 shows that the vibration signal after Kalman filter noise reduction maintains the state of the original vibration signal. The signal after Kalman filter noise reduction has an obvious free damping oscillation process. In order to clearly observe the attenuated oscillation signal, the time domain signal of the first 10,000 points is selected in Figure 13. It is clear from the analysis that the weak vibration generated by the bearing fault and the fiber Bragg grating resonant sensor has generalized resonance, and a generalized resonance wave is generated. Figure 11: Bearing vibration signal: (**a**) normal bearing vibration signal curve; (**b**) faulty bearing vibration signal curve. Figure 12: State estimation after Kalman filter. Figure 13: Free damped oscillation curve of the signal after Kalman filter noise reduction. The following analyses are the power spectrum analysis of the vibration signals of the normal bearing and the faulty bearing. If there is a high frequency component around 6325 Hz in the power spectrum, it means that there is a fault in the bearing. The faulty bearing resonates with the sensor, releasing energy at the sensor's natural frequency [30]. The power spectrum analysis of the normal bearing's vibration signal, the faulty bearing's vibration signal and the vibration signal after noise reduction by the Kalman filter are shown in Figure 14. The three power spectrums in Figure 14a-c contain low frequency components of 1003 Hz, 1079 Hz and 1079 Hz respectively. These low frequency values do not affect the diagnosis of bearing faults. When the frequency value in the power spectrum is close to the natural frequency of the fiber Bragg grating resonance sensor at 6325 Hz, it indicates that the bearing is faulty, and the fault is in resonance with the sensor. In Figure 14b,c, the power spectrum of fault bearing signal includes a 6157 Hz high-frequency component, which is close to the natural frequency of the sensor. From this frequency, it can be determined that the bearing is faulty. Additionally, the extracted components of the filtered power spectrum are the same, without spectral distortion. Figure 14a There is no high-frequency component in the normal bearing signal power spectrum. The following is the envelope spectrum analysis of the faulty bearing vibration signal, as shown in Figure 15. Combined with the theoretical value of the bearing fault characteristic frequency in Table 1, \(\textsl{f}_{\text{IR}}\) = 25 Hz. In Figure 15a, the frequency components are 25.97 Hz, 77.24 Hz, 102.5 Hz, 121.8 Hz, 177.1 Hz.