## 6 Conclusions and Discussion  
In this paper, we proposed a vibration signal-based hybrid method for realizing gearbox fault diagnosis and carried out experiments to verify its effectiveness. The main contribution of the proposed method is the combination of signal processing, machine learning, and optimization techniques to effectively eliminate noise contained in vibration signals and to achieve high diagnostic accuracy. This hybrid method includes GWO-VMD-based vibration signal decomposition, correlation-based signal reconstruction, fault feature extraction, and DE-KELM-based gearbox fault classification. To eliminate as much noise as possible from the vibration signal and to alleviate the mode mixing problem during signal decomposition, VMD was adopted for vibration signal decomposition and GWO was used to optimize its [\(K\), \(\alpha\)] by minimizing the objective function shown in Equation (6); after IMFs were obtained, the correlation coefficient of each IMF was calculated to select highly correlated components for signal reconstruction. Method validation experiments were conducted and reported in Section 3 to demonstrate that the reconstructed signal by the proposed method eliminates more noise and has better Kurtosis, signal-noise ratio, RMSE, and permutation entropy values than those by the basic VMD method. With the rebuilt vibration signals, 18 feature parameters were calculated for each sample to form fault feature vectors. Then, KELM was used to classify the features with its parameters optimized by the DE algorithm for higher diagnostic accuracy. Method validation experiments were conducted and reported in Section 4 to demonstrate that the proposed DE-KELM can achieve higher classification accuracy than basic KELM. Finally, the gearbox vibration signals in eight different operating conditions were obtained and contrast experiments were conducted to verify that the proposed GWO-VMD-and DE-KELM-based gearbox fault diagnosis method can achieve higher diagnostic accuracy than basic VMD and KELM. Compared with other available hybrid fault diagnosis methods in the literature survey, for instance, the two methods achieved interesting research results in [25; 26]; the method proposed in this paper adopts the parameter-optimized time-frequency analysis technique (GWO-VMD), which can reflect more characteristics of the non-stationary vibration signal in the signal preprocessing and feature extraction stage. This method can also alleviate the \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Fault Classification Methods**} & **Classification Accuracy** & **Classification Accuracy** \\  & **of the Training Set** & **of the Testing Set** \\ \hline DE-KELM & 100\% & 98.125\% \\ KELM & 97.9808\% & 97.5\% \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Fault classification accuracy of training and testing sets by DE-KELM. Figure 23: Fault classification results by KELM on the feature matrix calculated through GWO-VMD-based decomposition method. mode mixing problem included in EMD and its varieties. In the fault classification stage, the proposed DE-KELM method used KELM as a fault classifier, which can achieve faster training speed than SVM-based methods. Although the effectiveness of the proposed method was demonstrated with the experiments in Sections 3-5, there are also limitations in the proposed hybrid diagnosis method--the introduction of optimization techniques to VMD and KELM leads to longer overall signal processing and fault classification time. In the experiments of Sections 3 and 5, it took 5-10 min for DE to search optimal values for \([K,\alpha]\). This deficiency makes the proposed hybrid method not applicable to real-time diagnosis. Therefore, improving the real-time capability of the proposed hybrid method will be one of the future research goals of the authors.