## 5 Experimental Results and Analysis
### Experiment 2: Analysis of Cylindrical Roller Bearing Faults  
To further verify the proposed method, the vibration signals collected from a cylindrical roller bearing under the motor speed of 1440 rpm were taken for analysis. The inner-race, outer-race and rolling element defects were introduced to the test bearings using electric spark machining. Defect size of the inner-race and outer-race is 0.2 mm in width and 0.1 mm in depth. The defect size of rolling element is 0.8 mm in diameter and 1 mm in depth. Figure 14 shows the three fault conditions of the faulty bearings. The parameters of the faulty bearings were shown in Table 4. The theoretical defect frequencies of the test bearing were given in Table 5. Figure 15 depicts the time waveform of four states of the test bearing. Figure 15a shows the time waveform of normal state. From Figure 15b,d, it can be seen that the vibration signals of inner-fault, outer-fault and rolling element fault have obvious impacts. Figure 14: Three fault conditions of the faulty bearings: (**a**) Inner-race fault; (**b**) outer-race fault; (**c**) rolling element fault. Figure 13: The classification results of (**a**) ELM; (**b**) SVM. The vibration signal of each state was divided into 40 subset signals. The improved fast spectral correlation was adopted to acquire the fault feature vectors of each subset as shown in Figure 16. It can be observed that the curve of the same fault pattern has the same trend and the curves of different fault patterns are quite different. It indicates that the feature vectors obtained using the improved fast spectral correlation can effectively express the fault characteristics, which is very useful for pattern recognition. Table 6 gives the specific setting of the fault feature vectors. As shown in Table 6, each state has 40 sets of fault feature vectors, which include 10 sets of training samples and 30 sets of test samples. Both RF and PSO-RF were used for testing and classifying the fault feature vectors. Figure 17a,b show the classification results of RF and PSO-RF, respectively. The accuracy of RF is 97.5% and the classification accuracy of PSO-RF is 100%. It indicates that the PSO-RF is superior to RF in classifying bearing faults. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Number of** & **Speed of Training/Test** & **Label of Class** \\
**Training/Test Sets** & **Fault Type** & **Samples (rpm)** & **Label of Class** \\ \hline
10/30 & Normal & 1440/1440 & 0 \\
10/30 & Inner-race fault & 1440/1440 & 1 \\
10/30 & Outer-race fault & 1440/1440 & 2 \\
10/30 & Rolling element fault & 1440/1440 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The description of the datasets. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Bearing Type** & **Roller Diameter** & **Pitch Diameter** & **Number of Roller** & **Contact Angle** \\ \hline N205 & 7.5 mm & 39 mm & 12 & 0\({}^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The defect frequencies of cylindrical roller bearing at 1440 rpm. Figure 15: The waveform of (**a**) the normal; (**b**) inner-race fault; (**c**) outer-race fault; (**d**) rolling element fault. Figure 16: The fault feature vectors of: (**a**) the normal; (**b**) inner-race fault; (**c**) outer-race fault; (**d**) rolling element fault. Figure 17: Classification results of: (**a**) RF; (**b**) PSO-RF. To further validate the effectiveness of the improved fast spectral correlation, two fault extraction methods, i.e., EMD-SVD and MPE were used for comparison. For an equitable comparision, the classifier of PSO-RF was used to identify the fault feature vectors generated using EMD-SVD and MPE. Figure 18 describes the state classification results of EMD-SVD and MPE. We can see that the classification accuracy of EMD-SVD and MPE are 94.1667% and 97.5%, respectively. The improved fast spectral correlation reflects a higher accuracy. Similar to experiment 1, the classifier PSO-RF was compared with ELM and SVM.