## 4 Analysis and Verification of Rolling Bearing Fault Diagnosis Tests
### Analysis of the Experimental Process and Results  
In this study, we used a test dataset consisting of 2048 sampling points collected under normal operating conditions. Based on the parameter selection from the literature, we set K to 4 and \(\alpha\) to 2000, and the reasonableness of this parameter choice was verified through preliminary experiments [42]. Similarly, in the calculation of PE, we also need to determine the embedding dimension (m) and the delay time (\(\tau\)), which were set to 6 and 1, respectively, based on parameters derived from the literature [43]. The following Figure 6 displays the original vibration signal of the rolling bearing and the results of its VMD are displayed in Figure 7.  
Subsequently, all vibration signals from the rolling bearings were processed, encompassing 480 sets of data corresponding to various fault types (L0 to L3). Given the substantial volume of data, this paper presents only a selection of these results. Five random samples from each operational type were chosen, and the PE values were calculated and documented in the accompanying Table 3.  
The parameters of the penalty factor \(\alpha\) and the decomposition mode number K were optimized for VMD using the IGTO method. The envelope entropy corresponding to various parameter configurations was calculated. The values of \(\alpha\) and K were determined based on the principle of minimizing envelope entropy. The IGTO-VMD was performed on  
Figure 6: Timing diagram of the original vibration signal. Figure 7: Results of VMD. the same dataset as previously mentioned; the initial IGTO population was set to 32, and the maximum number of iterations was established at 20 [33]. The search range for K was defined as [3; 15], while the search range for \(\alpha\) was set between [200, 2000]. The minimum envelope entropy served as the fitness function to identify the optimal parameters for VMD. The relationship between the iterations of IGTO and the envelope entropy is illustrated in the accompanying Figure 8.  
The figure illustrates that the minimum envelope entropy of 3.20422 is attained during the 16 th iteration. The corresponding values for K and \(\alpha\) are [299, 7]. Subsequently, these optimal parameters are employed to decompose the vibration signal using VMD, resulting in the decomposition of each IMF component, as depicted in Figure 9 below. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Fault Label**} & \multirow{2}{*}{**Sample Number**} & \multicolumn{4}{c}{**PE of Different IMFs**} \\ \cline{3-6}  & & **PE1** & **PE2** & **PE3** & **PE4** \\ \hline \multirow{6}{*}{L0} & 1 & 0.299069 & 0.320392 & 0.493529 & 0.511928 \\  & 2 & 0.312674 & 0.334967 & 0.496648 & 0.514999 \\  & 3 & 0.28996 & 0.310634 & 0.487277 & 0.517158 \\  & 4 & 0.283549 & 0.303766 & 0.493788 & 0.518129 \\  & 5 & 0.29542 & 0.316484 & 0.489354 & 0.518198 \\ \hline \multirow{6}{*}{L1} & 1 & 0.434235 & 0.477213 & 0.564439 & 0.726385 \\  & 2 & 0.437768 & 0.481004 & 0.563366 & 0.726868 \\  & 3 & 0.438445 & 0.481748 & 0.556385 & 0.726981 \\  & 4 & 0.428374 & 0.470683 & 0.563448 & 0.727338 \\  & 5 & 0.433564 & 0.476385 & 0.567046 & 0.728214 \\ \hline \multirow{6}{*}{L2} & 1 & 0.426129 & 0.430338 & 0.560985 & 0.717812 \\  & 2 & 0.432561 & 0.436833 & 0.558578 & 0.717836 \\  & 3 & 0.433015 & 0.439311 & 0.557989 & 0.718189 \\  & 4 & 0.447779 & 0.452202 & 0.55926 & 0.718142 \\  & 5 & 0.435799 & 0.458281 & 0.562103 & 0.718434 \\ \hline \multirow{6}{*}{L3} & 1 & 0.439023 & 0.444415 & 0.545552 & 0.692361 \\  & 2 & 0.428514 & 0.433776 & 0.556694 & 0.694069 \\ \cline{1-1}  & 3 & 0.420156 & 0.425316 & 0.553378 & 0.695108 \\ \cline{1-1}  & 4 & 0.42091 & 0.426079 & 0.551172 & 0.695938 \\ \cline{1-1}  & 5 & 0.417063 & 0.422185 & 0.55144 & 0.695975 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: PE values of fault samples extracted by VMD. Figure 8: Curve of envelope entropy. Subsequently, the PE values of the IMFs were derived through the IGTO-VMD applied to four distinct types of sample signals, as illustrated in the accompanying table.