## 3 Application and Results
### Fault Diagnosis of Bearing with Fault at the Inner Raceway  
The time-domain waveform of the fault signal \(x_{in}\) from the inner raceway is shown in Figure 4a, where the sampling number of \(x_{in}\) is 12000. Divide the signal \(x_{in}\) into four segments of \(x_{in1}\), \(x_{in2}\), \(x_{in3}\) \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data Code** & **RS** & **RF** & **IRF** & **ORF** \\ \hline \(x_{in}\) & 1721 r/min & 28.68 Hz & 155.3 Hz & \\ \(x_{out}\) & 1725 r/min & 28.75 Hz & & 103.1 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The fault frequency of inner raceway and outer raceway, where \(x_{in}\) is the inner raceway fault data, and \(x_{out}\) is the outer raceway fault data. Figure 3: The test stand for normal and faulty test data of ball bearings. Figure 2: The specific process of diagnostic method. and \(x_{in1}\). The number of each segment is 2048. Figure 4b displays the time-domain waveform of \(x_{in1}\). LMD is used to decompose \(x_{in}\) and \(x_{in1}\) separately into a series of PFs and \(u_{m}(t)\) is the residual component, as shown in Figure 5. The correlation coefficients \(\rho\) between the raw signal and its PFs were calculated, as shown in Table 2. It is obvious that both of \(PF_{1}(t)\) contain the richest fault information. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\rho\) & \(\mathbf{PF_{1}}(t)\) & \(\mathbf{PF_{2}}(t)\) & \(\mathbf{PF_{3}}(t)\) & \(\mathbf{PF_{4}}(t)\) & \(\mathbf{PF_{5}}(t)\) & \(\mathbf{PF_{6}}(t)\) \\ \hline \(\rho_{in}\) & 0.9966 & 0.18162 & 0.0202 & 0.0004 & \(-\)4.42E\(-\)5 & 2.66E\(-\)5 \\ \(\rho_{in1}\) & 0.9770 & 0.2096 & 0.0208 & 0.0019 & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The correlation coefficients \(\rho\): \(\rho_{in}\) are the correlation coefficients between \(x_{in}\) and its PFs; \(\rho_{in1}\) are the correlation coefficients between \(x_{in1}\) and its PFs. Figure 4: Time-domain waveforms of \(x_{in}\) and \(x_{in1}\) with different sampling number: (**a**) The sampling number of \(x_{in}\) is 12,000; (**b**) the number of first segment \(x_{in1}\) is 2048. Figure 5: PFs of \(x_{in}\) and \(x_{in1}\) with different sampling number: (**a**) The sampling number of \(x_{in}\) is 12000; (**b**) the number of first segment \(x_{in1}\) is 2048. To raise the effect of contrast, the comparison chart of the Hilbert envelope spectrum of \(\mathit{PF}_{1}(t)\) which is from Figure 5a is shown in Figure 6. The fault frequency 154.6 Hz, which is not corrected, is basically equal to the theoretical fault frequency 155.3 Hz, as shown in Figure 6a. The double frequency 309.9 Hz and the triple frequency 464.4 Hz exist obviously. Similarly, the fault frequency 154.9 Hz, which is corrected, is basically equal to the theoretical fault frequency 155.3 Hz, as shown in Figure 6b. The double frequency 309.8 Hz and the triple frequency 465.1 Hz exist also obviously. In this case, whether it is corrected or not, the fault frequency is almost equal to the theoretical fault frequency. In other words, for the long-time signals, it does not affect the accurate extraction of fault frequency whether the fault frequency is corrected or not. However, for the short-time signals, there is a difference between the fault frequency and the theoretical fault frequency. As shown in Figure 7a, there is a clear error of 2.8 Hz between the fault frequency 152.5 Hz and the theoretical fault frequency 155.3 Hz. The fault frequency 155 Hz, which is the value of spectrum correction, is almost equal to the theoretical fault frequency 155.3 Hz, as shown in Figure 7b. After calculating the remaining three segments of data \(x_{in2}\), \(x_{in3}\), and \(x_{in4}\), the comparison chart of the Hilbert envelope spectrum of \(\mathit{PF}_{1}(t)\) is shown in Figure 8. Without spectrum correction, the fault frequency 152.5 Hz has an error according to the theoretical fault frequency 155.3 Hz, as shown in Figure 8a.