# Legendre Multiwavelet Transform and Its Application in Bearing Fault Detection
## 4 Conclusions  
This paper proposes a novel bearing fault detection method based on LMWT, which calculates the relative energy ratio to select the most sensitive components for fault features. The LMWT method is applied to three different cases, and the results show that it can more effectively extract various component fault frequencies compared to the EWT method. Specifically, in simulated environments, the LMWT method outperforms the EWT method in high-noise conditions. In the single fault experiment, the LMWT method yielded relative energy ratios of 0.528, 0.176, and 0.049 for three different faults, while the EWT method only achieved values of 0.275, 0.083, and 0.000, respectively. This indicates that the LMWT method exhibits a 92% increase in sensitivity for outer race faults and a 112% increase in sensitivity for inner race faults compared to EWT. However, for ball faults, although the LMWT method significantly outperformed the EWT method in relative energy ratios, its fault feature frequencies still remained mixed with background noise. In the compound \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & **1** & **2** & **3** & **4** & **5** & **6** & **Average** \\ \hline \multirow{2}{*}{LMWT} & Inner race & 0.042 & 0.108 & 0.121 & 0.023 & 0.083 & 0.017 & 0.065 \\  & Outer race & 0.386 & 0.332 & 0.189 & 0.392 & 0.127 & 0.232 & 0.276 \\ \hline \multirow{2}{*}{EWT} & Inner race & 0.000 & 0.000 & 0.000 & 0.000 & 0.000 & 0.002 & 0.000 \\  & Outer race & 0.008 & 0.001 & 0.000 & 0.001 & 0.006 & 0.016 & 0.004 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Relative energy ratios of LMWT and EWT in different fault. Figure 19: The envelope spectrums of **(a)** BPFI, **(b)** BFFO on the empirical modes to the different parts of the compound fault signal in the Paderborn bearing dataset. fault experiment, due to the rich regularity and orthogonality of the LMWT method, it achieved relative energy ratios of 0.276 and 0.065 for outer and inner race faults, respectively, while the corresponding ratios in the EWT method were 0.004 and 0.000. Clearly, the LMWT method holds a considerable advantage over the EWT method. However, in the spectrum of the inner race fault, the fault feature frequencies are not distinct and are still entangled with background noise, lacking complete separation. Therefore, the LMWT method can effectively improve the accuracy and reliability of bearing fault diagnosis, and has excellent compound fault detection ability. In the future, combining deep learning methods with the Legendre multiwavelet theory will be developed. Conceptualization, X.Z. and Z.L. ; methodology, X.Z. ; software, Z.L. ; validation, Z.L. ; formal analysis, X.Z. ; investigation, Z.F. ; resources, L.C. ; data curation, Z.L. ; writing--original draft preparation, Z.L. ; writing--review and editing, Z.L. ; visualization, Z.L. ; supervision, X.Z. ; project administration, X.Z. ; funding acquisition, X.Z. All authors have read and agreed to the published version of the manuscript. This work is funded by Fundamental and Advanced Research Project of Chongqing CSTC of China, the project No. are cstc2019 jcyj-msxmX0386 and cstc2020 jcyj-msxmX0232. Not applicable. In this article, two publicly available bearing datasets were used, namely Case Western Reserve University (CWRU) bearing data and Paderborn University (PU) bearing dataset. CWRU bearing dataset can be found in [[https://engineering.case.edu/bearingdatacenter](https://engineering.case.edu/bearingdatacenter)]([https://engineering.case.edu/bearingdatacenter](https://engineering.case.edu/bearingdatacenter)) (accessed on 23 March 2023), and PU bearing dataset can be found in [[https://mb.uni-paderborn.de/kat/forschung/kat-datacenter/bearing-datacenter/](https://mb.uni-paderborn.de/kat/forschung/kat-datacenter/bearing-datacenter/)]([https://mb.uni-paderborn.de/kat/forschung/kat-datacenter/bearing-datacenter/](https://mb.uni-paderborn.de/kat/forschung/kat-datacenter/bearing-datacenter/)) (accessed on 3 April 2023). The authors declare no conflicts of interest.