## 4 Conclusions  
Double interpolation PF optimization and the mutation interval reconstruction envelope are two independent improvements of LMD, and this paper combines the two improvements in one algorithm to form double interpolation and mutation interval reconstruction local mean decomposition (DIMIRLMD). 1. The main reason for the demodulation mutation phenomenon is found to be the intersection of upper and lower envelopes, and the extreme symmetry points are employed to reconstruct the envelopes. The mutation interval reconstruction algorithm effectively suppressed the demodulation mutation phenomenon. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{3}{c}{**Iteration**} & \multicolumn{3}{c}{**IO**} & \multirow{2}{*}{**IEC**} \\ \cline{2-2} \cline{4-7}  & **PF1** & **PF2** & **PF3** & **PF1** & **PF2** & **PF3** \\ \hline \multirow{2}{*}{LMD} & 7.121 & 6.487 & 5.846 & 0.315 & 0.346 & 0.415 & 0.811 \\  & \(\pm\)1.311 & \(\pm\)1.552 & \(\pm\)1.648 & \(\pm\)0.052 & \(\pm\)0.081 & \(\pm\)0.074 & \(\pm\)0.098 \\ \cline{2-7}  & 5.943 & 10.434 & 9.567 & 0.253 & 0.537 & 0.695 & 0.874 \\ \cline{2-7}  & \(\pm\)1.225 & \(\pm\)1.965 & \(\pm\)1.841 & \(\pm\)0.053 & \(\pm\)0.086 & \(\pm\)0.093 & \(\pm\)0.086 \\ \cline{2-7}  & 5.454 & 4.454 & 5.524 & 0.185 & 0.243 & 0.286 & 0.887 \\ \cline{2-7}  & \(\pm\)0.891 & \(\pm\)1.078 & \(\pm\)1.447 & \(\pm\)0.049 & \(\pm\)0.054 & \(\pm\)0.065 & \(\pm\)0.072 \\ \cline{2-7}  & 4.665 & 3.823 & 3.735 & 0.177 & 0.232 & 0.273 & 0.905 \\ \cline{2-7}  & \(\pm\)0.628 & \(\pm\)0.752 & \(\pm\)0.848 & \(\pm\)0.046 & \(\pm\)0.051 & \(\pm\)0.058 & \(\pm\)0.048 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of decomposition results of different LMD methods. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Index**} & \multicolumn{3}{c}{**Method**} \\ \cline{2-5}  & **Original** & **LMD** & \begin{tabular}{c} **CSI** \\ **LMD** \\ \end{tabular} & \
\begin{tabular}{c} **MPCHI** \\ **LMD** \\ \end{tabular} &
\begin{tabular}{c} **DIMIRLMD** \\ \end{tabular} \\ \hline Peak Value & 0.5432 & 0.5121 & 0.5073 & 0.5118 & 0.5336 \\ RMS Value & 0.1521 & 0.0824 & 0.0728 & 0.0695 & 0.0652 \\ Peak Index & 3.5713 & 6.2148 & 6.9684 & 7.3640 & 8.1840 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Comparison of index values of envelope frequency spectrum for various LMD methods. 2. Two typical interpolation envelope methods, CSI and MPCHI, are used to obtain PF components, and then the optimal PF component is selected based on the index of orthogonality (_IO_). Double interpolation PF optimization achieves a better hierarchical property for each order PF component. The orthogonality of decomposition results is improved, and the accuracy of the PF component is guaranteed. 3. A numerical simulation experiment indicated that DIMIRLMD outperforms the other three LMD methods in MSE and sifting iterations of PF components. The fault characteristics of the strong nonstationary vibration signal of the reciprocating compressor are more accurately quantified. 4. DIMIRLMD was applicated in feature extraction from the vibration signals of the reciprocating compressor in an oversized bearing clearance fault state; results show it is superior in the sifting iterations, _IO_, and IEC compared to the other three LMD methods. Furthermore, the significant peak index of the PF component envelope frequency spectrum verified the superiority of DIMIRLMD in increased reciprocating pressure compressor fault identification accuracy. Conceptualization, H.Z. ; software, X.L. ; validation, X.L., Z.L., H.W. and J.H. ; formal analysis, Z.L. ; investigation, H.W. ; writing--original draft preparation, H.Z. ; writing--review and editing, X.L. and Z.L. ; supervision, J.H. ; project administration, H.Z. All authors have read and agreed to the published version of the manuscript.