## 5 Experiment Data Analysis
### Bearing Vibration Signal Analysis and Bearing Fault Detection and Diagnosis
These frequencies are multiples of the inner ring  
Figure 14: Power spectrum analysis curve: (**a**) normal bearing power spectrum analysis curve; (**b**) the power spectrum analysis curve of faulty bearing vibration signal without Kalman filter noise reduction; and (**c**) the power spectrum analysis curve of faulty bearing vibration signal with Kalman filter noise reduction. fault characteristic frequency, respectively, are f\({}_{\text{IR}}\), 3f\({}_{\text{IR}}\), 4f\({}_{\text{IR}}\), 5f\({}_{\text{IR}}\), 7f\({}_{\text{IR}}\). It can be diagnosed that the bearing fault occurs in the inner ring. Among them, 96.55 Hz, 171.1 Hz are the interference frequencies, which have been marked with a red circle. Figure 15b is the envelope analysis of the signal after the Kalman filter. The frequency components included are 26.11 Hz, 75.79 Hz, 99.99 Hz, 121.6 Hz, 170.7 Hz. The above frequency components are the inner ring fault characteristic frequency f\({}_{\text{IR}}\), 3f\({}_{\text{IR}}\), 4f\({}_{\text{IR}}\), 5f\({}_{\text{IR}}\), 7f\({}_{\text{IR}}\). According to the analysis of Figure 15, the disadvantages of envelope analysis are as follows:  
1. If the working background of the bearing is noisy and the ambient noise is large, there will be more envelope components obtained through envelope analysis. It is easy to cause the fault signal envelope to be mixed with the noise envelope, and it is not easy to distinguish. 2. The envelope of the fault characteristic frequency is not obvious. The largest envelope in the envelope spectrum corresponds to the 7f\({}_{\text{IR}}\) frequency of the bearing fault characteristic frequency. The analysis of the envelope is required to determine the bearing fault location. Comparing Figure 15a,b, after Kalman filter noise reduction, the obtained envelope spectrum has the following advantages:  
1. There is no noise envelope in the envelope spectrum. The obtained envelopes are all multipliers of the fault characteristic frequency. 2. The envelope spectrum curve is smooth without any slight noise interference. The following is the analysis of autocorrelation envelope spectrum, as shown in Figure 10:0  
As visible in Figure 16a, in the improved autocorrelation envelope power spectrum, the bearing fault frequency and double frequency 25.63 Hz, 50.05 Hz, 75.07 Hz can be extracted, and the bearing rotation frequency 4.883 Hz can also be extracted. Compared with Figure 15b, the fault characteristic frequency is obvious, and the corresponding amplitude is the largest. The envelope spectrum is concise, does not contain more frequency components, and is easy to distinguish from other noise components. Figure 16b is the autocorrelation envelope power spectrum by the literature [31]. Compared with Figure 16a, Figure 16a can not only directly extract the bearing fault characteristic frequency, but also extract other frequency components. From this, the accuracy of the extracted fault frequency can be further proved. Figure 10:1 The envelope spectrum analysis curve: (**a**) the envelope spectrum analysis curve of faulty bearing vibration signal without Kalman filter noise reduction; and (**b**) the envelope spectrum analysis curve of faulty bearing vibration signal with Kalman filter noise reduction. The cepstral analysis method adopted by the literature [32] is shown in Figure 16c. In Figure 16c, the \(x\)-axis is the time domain, and the peak value appears every 0.04 s. From this, it can be determined that there is a periodic frequency, which is 25 Hz. The analyses in Figure 16a,c can both extract the characteristic frequencies of bearing faults. Comparing the two methods, the advantage of the improved autocorrelation envelope power spectrum analysis method is that the extracted fault frequencies are intuitive and do not require further analysis. The cepstrum contains too much noise and is easily disturbed by noise, and further analysis is required to obtain the fault frequency.