## 6 Conclusions  
In this article, to solve the shortcoming of the subjective parameter selection and end effect phenomenon existing in the original VMD, a novel time-frequency analysis tool (i.e., IAVMD) consisting of waveform matching extension, GWO-based parameter optimization, and TEO demodulation is presented, which can automatically determine the inside parameters (\(\hat{\alpha}\), \(K\)) of VMD and separate a non-stationary compound signal into a superposition of modes. Several simulated data and experimental examples are provided to validate the effectiveness of our approach. Studies confirm that our approach can effectively identify fault features related to rotor early rub-impact, oil-whirl, and oil-whip. Besides, our proposed method outperforms four available approaches (i.e., OVMD, LMD, EMD, and WT) in analyzing the non-stationary compound signal with multi-frequency contents. The novelties and attractions of this paper are as follows:  
(1) The concept of waveform matching extension is introduced to alleviate the end effect problem in the traditional VMD method.  
(2) A novel time-frequency analysis algorithm termed as IAVMD is presented, which can avoid the effective manual parameter selection and improve diagnostic accuracy.  
(3) The efficacy of our method is demonstrated through the simulated and experimental data.  
Our future direction will focus on extending the proposed method to analyze multi-component signals in practical applications. Moreover, in our future work, fault diagnosis analyses under variable speeds will also be investigated by using the presented approach. Finally, in the follow-up work, the availability of our presented approach will be discussed in detail when strong noise and more interference are appended to the analyzed vibration signal.  
Methodology, Validation, Writing--original draft, X.Y.; Funding acquisition, Y.L.; Software, W.Z.; Supervision, M.J.; Conceptualization; X.W. All authors have read and agreed to the published version of the manuscript. This work was funded in part by the National Natural Science Foundation of China (Grant 51675098), in part by Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant KYCX170059), in part by Jiangsu Provincial Key Research and Development Program (Grant BE2019030637) and in part by Jiangsu Agricultural Science and Technology Independent Innovation Fund(Grant 2018325-SCX(18)2049). The authors would like to thank NCEPU for their kind permission to use their rotor fault data, and thank the editor and the anonymous reviewers for their helpful comments. The authors declare no conflicts of interest.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & \multicolumn{2}{c}{**Memory of Simulation (KByte)**} & \multicolumn{2}{c}{**Memory of Experiment (KByte)**} & \multirow{2}{*}{**Average Time (s)**} \\ \cline{2-2} \cline{4-7}  & **Case 1** & **Case 2** & **Case 1** & **Case 2** & **Case 3** \\ \hline IAVMD & 8.85 & 8.87 & 8.91 & 8.92 & 8.91 & 36.76\(\rightarrow\)High \\ OVMD & 5.46 & 5.48 & 5.04 & 5.03 & 5.03 & 28.67\(\rightarrow\)High \\ LMD & 1.93 & 1.96 & 1.98 & 1.97 & 1.98 & 3.04\(\rightarrow\)Low \\ EMD & 1.71 & 1.73 & 1.75 & 1.76 & 1.76 & 2.27\(\rightarrow\)Low \\ WT & 1.35 & 1.38 & 1.41 & 1.41 & 1.42 & 1.92\(\rightarrow\)Low \\ \hline \hline \end{tabular}
\end{table}
Table 7: Assessment of the time and memory cost of different approaches.