## 5 Conclusions and Future Work  
In this study, a fault diagnosis framework for rotating machinery based on IPSO-VMD-PNN was established by the combination of numerical simulation and experimental verification, and the effectiveness of the framework was verified with the open data of gears and rolling bearings.  
* In view of the difficulty in selecting the two key parameters \(K\) and \(\alpha\) in VMD, the adaptive inertia weight strategy and compression factor method were used to improve the PSO algorithm, and the adaptive optimization of parameters was carried out by combining the fitness function, so as to avoid the instability of manual value taking and cumbersome operation.
* In the extracted multi-dimensional feature vectors, the sensitivity of different fault feature parameters varies, and some parameters may be irrelevant or redundant. Laplace score was used to select sensitive feature parameters from multi-domain feature parameters, and the fault sensitive feature vector was constructed and input into the PNN model to realize fault diagnosis.
* Through the validation of simulated and real signals, the proposed IPSO-VMD-PNN framework shows excellent robustness and adaptability, demonstrating its potential for application in practical engineering.  
Future research will focus on further optimizing the proposed method to enhance its fault diagnosis accuracy under more complex conditions. Additionally, the potential application of this method in intelligent diagnosis under varying rotational speeds will be explored.  
Conceptualization, Z.L.; methodology, C.Z.; software, C.Z.; validation, Z.L. and C.Z.; investigation, J.H.; data curation, S.X.; writing--original draft preparation, Z.L. and C.Z.; writing--review and editing, Z.L. and Z.C.; visualization, W.Z.; supervision, H.L.; funding acquisition, H.L. All authors have read and agreed to the published version of the manuscript.  
This research was supported by the National Natural Science Foundation of China (grant number 52275505), and the Key Research and Development project of Hubei Science and Technology Plan (grant number 2023 BEB013).  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Gears Status** & \begin{tabular}{c} **Feature Extraction** \\ **Algorithm** \\ \end{tabular} & \begin{tabular}{c} **Classification** \\ **Algorithm** \\ \end{tabular} & \begin{tabular}{c} **Number of** \\ **Training Sets** \\ \end{tabular} & \begin{tabular}{c} **Number of** \\ **Test Sets** \\ \end{tabular} & **Right** & **Misjudgment** & \begin{tabular}{c} **Accuracy** \\ **Rate** \\ \end{tabular} \\ \hline \multirow{4}{*}{Normal} & IPSO-VMD & PNN & 140 & 60 & 59 & 1 & 98.33\% \\  & PSO-VMD & PNN & 140 & 60 & 57 & 3 & 95.00\% \\  & PSO-VMD & ELM & 140 & 60 & 55 & 5 & 91.67\% \\  & PSO-VMD & SVM & 140 & 60 & 58 & 2 & 96.67\% \\  & EMD & PNN & 140 & 60 & 54 & 6 & 90.00\% \\ \hline \multirow{4}{*}{\begin{tabular}{c} Rolling \\ element \\ failure \\ \end{tabular} } & IPSO-VMD & PNN & 140 & 60 & 58 & 2 & 96.67\% \\  & PSO-VMD & PNN & 140 & 60 & 56 & 4 & 93.33\% \\  & PSO-VMD & ELM & 140 & 60 & 54 & 6 & 90.00\% \\  & PSO-VMD & SVM & 140 & 60 & 57 & 3 & 95.00\% \\  & EMD & PNN & 140 & 60 & 53 & 7 & 88.33\% \\ \hline \multirow{4}{*}{\
\begin{tabular}{c} Inner ring \\ Failure \\ \end{tabular} } & IPSO-VMD & PNN & 140 & 60 & 60 & 0 & 100\% \\  & PSO-VMD & PNN & 140 & 60 & 59 & 1 & 98.33\% \\  & PSO-VMD & ELM & 140 & 60 & 58 & 2 & 96.67\% \\  & PSO-VMD & SVM & 140 & 60 & 58 & 2 & 96.67\% \\  & EMD & PNN & 140 & 60 & 56 & 4 & 93.33\% \\ \hline \multirow{4}{*}{
\begin{tabular}{c} Outer ring \\ Failure \\ \end{tabular} } & IPSO-VMD & PNN & 140 & 60 & 59 & 1 & 98.33\% \\  & PSO-VMD & PNN & 140 & 60 & 58 & 2 & 96.67\% \\ \cline{1-1}  & PSO-VMD & ELM & 140 & 60 & 57 & 3 & 95.00\% \\ \cline{1-1}  & PSO-VMD & SVM & 140 & 60 & 58 & 2 & 96.67\% \\ \cline{1-1}  & EMD & PNN & 140 & 60 & 55 & 5 & 91.67\% \\ \hline \hline \end{tabular}
\end{table}
Table 7: Diagnostic results of rolling bearing fault samples.