# Low-Speed Bearing Fault Diagnosis Based on Permutation and Spectral Entropy Measures
## 6 Conclusions  
In this paper, PE and SE are introduced for the analysis of vibration signals belonging to scenarios with very low-speed bearings. The current study demonstrates the existence of a relationship between the calculated entropy-based indicator and the presence of damage on a bearing at 5 rpm and 10 rpm without the need of previous noise filtering. Specifically, the values of the entropy-based indicator are reduced as the damage size increases. The decrement of the values calculated from a damaged scenario can be up to 57% when the median value is compared to a healthy scenario and 55% for the average value.  
To apply the proposed entropy-based indicator in a real scenario, several conditions should be taken into account. In this study, the proposed entropy-based indicator is applied to a controlled scenario, where the speed and the forces on the bearing are always continuous. A subsequent study can consider variable forces to the bearing, as the wind speed through the day. The same condition can be implemented for the speed with different speed values to observe the capability of the proposed indicator near to 1 rpm. The use of a back and forth rotation condition to replicate the movement of a real pitch bearing, hence a more realistic scenario, can also be considered in further studies.  
The investigated scenarios included only two damage sizes located in one race of each bearing. Thus, the possible scenarios that could be utilized to study the performance of the proposed entropy-based indicator were limited. Further studies can focus on several types and combinations of damage, such as ball, or cage damage, to address the fault identification problem. The inclusion of several damage sizes (smaller and larger than the ones used) in race, ball or cage can also contribute to dealing with the quantification problem. A different scenario to consider is a naturally induced damage to the bearing, which increases in size over time. The mentioned scenarios could help researchers to identify the properties and limitations of the methodology presented and its use in CBM. Finally, the proposed entropy-based indicator can be used for the design of diagnosis models based on artificial intelligence for a subsequent stage in real CBM systems.  
Study conception and design was made by [PERSON], [PERSON], [PERSON], and [PERSON] Data collection, data analysis, data interpretation, and manuscript drafting were performed by [PERSON] Critical revisions of the manuscript were performed by [PERSON], [PERSON], and [PERSON] All authors have read and agreed to the published version of the manuscript.  
This study was partially funded by the Spanish Agencia Estatal de Investigacion (AEI)--Ministerio de Economia, Industria y Competitividad (MINECO), the Fondo Europeo de Desarrollo Regional (FEDER) through the research projects DPI2017-82930-C2-1-R and DPI2017-82930-C2-2-R, and by the Generalitat de Catalunya through the research project 2017 SGR 388.  
Any opinions, findings, and conclusions expressed in this article are those of the authors and do not necessarily reflect the views of funding agencies.  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} AE & acoustic emission \\ CBM & condition based maintenance \\ D1 & damage 1 scenario \\ D2 & damage 2 scenario \\ EEMD & ensemble empirical mode decomposition \\ EMD & empirical mode decomposition \\ HS & healthy scenario \\ O\&M & operation and maintenance \\ PE & permutation entropy \\ PSD & power spectral density \\ RMS & root mean square \\ rpm & revolutions per minute \\ SE & spectral entropy \\ \end{tabular}