## 3 Application and Results
### Fault Diagnosis of Bearing with Fault at the Outer Raceway
For the short-time signals, there is an error between the uncorrected fault frequency and the theoretical fault frequency. As shown in Figure 12a, the fault frequency 105.6 Hz has a difference of 2.5 Hz according to the theoretical fault frequency 103.1 Hz. However, the fault frequency 103.3 Hz, which is corrected, is almost equal to the theoretical fault frequency 103.1 Hz, as shown in Figure 12b. The comparison chart of the Hilbert envelope spectrum of \(PF_{1}(t)\) is shown in Figure 13, where the remaining three segments of data \(x_{out2}\), \(x_{out3}\), and \(x_{out4}\) are calculated. Without spectrum correction, the fault frequency 105.6 Hz has a clear error according to the theoretical fault frequency 103.1 Hz, as shown in Figure 13a. That is because the frequency resolution \(\Delta w\) is related to the sampling frequency \(F_{s}\) and the sampling number \(N\). The relationship between them is expressed by the formula of \(\Delta w=F_{s}/N\). When the sampling frequency is constant, the frequency resolution \(\Delta w\) is inversely proportional to the sampling number \(N\). When the number of the four segments is 2048, the frequency resolution decreases and the spectrum is distorted. However, with spectrum correction, the fault frequency 103.4 Hz is almost equal to the theoretical fault frequency 103.1 Hz, as shown in Figure 13b. The spectrum correction method can correct the spectral distortion of the short-time signals and restore the authenticity of the spectrum to some extent. Figure 11: Comparison chart of the Hilbert envelope spectrum of \(PF_{1}(t)\) which is from Figure 10a: (**a**) Without spectrum correction; (**b**) with spectrum correction. Figure 12: Comparison chart of the Hilbert envelope spectrum of \(PF_{1}(t)\) which is from Figure 10b: (**a**) Without spectrum correction; (**b**) with spectrum correction. Take 20 inner raceway fault samples to form the sample sets \(S_{in}\). The number of each sample of 10 samples is 12,000, and the number of each sample of the other 10 samples is 2048. Similarly, take 20 outer raceway fault samples to form the sample sets \(S_{out}\). The number of each sample of 10 samples is 12,000, and the number of each sample of the other 10 samples is 2048. Their IRF and ORF are shown in Table 1. The fault frequencies were extracted from \(S_{in}\) and \(S_{out}\), as shown in Figures 14 and 15. In the Figures 14 and 15, WOSC denotes the fault frequency without spectrum correction and WSC denotes the fault frequency with spectrum correction. As can be seen from Figures 14a and 15a, for the long-time signals, there is very small error between the corrected fault frequency or the uncorrected fault frequency and the theoretical fault frequency. The very small error does not affect the accurate extraction of fault frequency. As can be seen from Figures 14b and 15b, for the short-time signals, there is a large error between the uncorrected fault frequency and the theoretical fault frequency, or the corrected fault frequency. The large error may affect the accurate extraction of fault frequency. Figure 14: The sample sets \(S_{in}\): (**a**) The sampling number is 12,000; (**b**) the sampling number is 2048. Figure 13: Comparison chart of the Hilbert envelope spectrum of \(PF_{1}(t)\) after calculating the remaining three segments of data: (**a**) Without spectrum correction; (**b**) with spectrum correction.