## 4 Experimental Verification
### Case 1  
Gears are the core components of rotating machinery. The quality of gears can affect the performance of rotating machinery. In this case, the gear failure is analyzed by spur gear failures from the University of Connecticut gear public data set as an example [37]. This data set contains a full set of gearbox vibration data. The experimental bench used is composed of drive motor, reducer, brake, and speed controller. As shown in Figure 8, the transmission diagram of the gearbox in the University of Connecticut data test bench is a two-stage reduction structure. The gear box contains input shaft, intermediate shaft, and output shaft. ER-10K rolling bearings are used on both sides of each stage shaft to connect with the reducer box. The number of teeth of gear 1 is 32, gear 2 is 80, gear 3 is 48, and gear 4 is 64. Therefore, the first stage reduction ratio \(i_{1}=2.5\), the second stage reduction ratio \(i_{2}=4/3\), and the detailed parameters of data acquisition are shown in Table 1.  
Figure 7: Basic structure of PNN. In this experiment, the information of the experiment is that only gear fault data collected from this data set are used, bearing #1-bearing #6 are normal, input and output shaft are normal, gear fault includes flaking and pitting, and fault occurs on gear 1 and 3, respectively. Fault label and status information are shown in Table 2.  
In the experiment, 100 samples are collected for each status, with each sample consisting of 3600 data points covering three periods of experimental data. As shown in Figure 9, the time-domain diagram of three different state data after down-sampling is shown. IPSO adaptive search is performed on three types of fault data and the best parameter combination \(\left[\hat{K},\hat{\alpha}\right]\) is shown in Table 3. The best combination of parameters obtained by optimizing label 1 data is \([10,2000]\), label 2 is \([10,3150]\), and label 3 is \([8,4950]\). The data of the three labels are processed by IPSO-VMD and each IMF component is calculated. The correlation coefficient between each IMF component and the original signal is shown in Figure 10. It can be observed from the diagram that the correlation coefficient between all IMF components and the original signal under the three labels is greater than the set threshold \(\sigma=0.1\). It is concluded that all IMF components from the three different fault data of the gear wheel are effective components. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Acquisition** & **Processing Board** & \
\begin{tabular}{c} **Sampling** \\ **Frequency** \\ \end{tabular} &
\begin{tabular}{c} **Sampling** \\ **Time** \\ \end{tabular} & **Speed** \\ \hline dSPACE & DS1006 & 20 kHz & 10 s & 2820 rpm \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Detailed parameters of experiment. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Label** & **Gears 1** & **Gears 2** & **Gears 3** & **Gears 4** \\ \hline
1 & Normal & Normal & Normal & Normal \\
2 & Flake & Normal & Normal & Normal \\
3 & Pit & Normal & Normal & Normal \\ \hline \hline \end{tabular}
\end{table} 
Table 2: PHM gear fault label and status information. Figure 8: Gearbox transmission diagram. Further, the energy ratio feature of the effective components is shown in Figure 11. It can be observed that the energy ratio of the IMF3 component of label 1 data is the largest, accounting for 0.325, while the IMF5 component is the smallest, accounting for 0.06. IMF1 and IMF3 are the first two components in the proportion of label 2 data, 0.376 and 0.4039, respectively, and the other three components are close to each other. The proportion of label 3 data in IMF1 and IMF2 is the first two, 0.3259 and 0.2035, respectively. The proportion of other components decreases as they move forward. Figure 10: IMF component correlation coefficient of gear data. Figure 9: Time-domain diagram of gear data in three different conditions.