# Low-Speed Bearing Fault Diagnosis Based on Permutation and Spectral Entropy Measures
## 5 Results
Overall, and considering as reference the values of HS, the average value for D1 decrement 8% and D2 almost 30%. For the median, D1 decrement almost 9% and D2 29%. \begin{table}
\begin{tabular}{l c c} \hline  & **Slope** & \(y\)**-intercept** \\ \hline healthy & \(-4.24\times 10^{-6}\) & 0.29 \\ damage 1 & \(-4.74\times 10^{-5}\) & 0.26 \\ damage 2 & \(-3.18\times 10^{-5}\) & 0.13 \\ \hline \end{tabular}
\end{table} 
Table 1: Linear regressions for all the scenarios at 5 rpm. \begin{table}
\begin{tabular}{l c c c} \hline  & \(\bar{x}\) & **Md** & \(\bar{s}\) \\ \hline healthy & 0.292 & 0.291 & 0.012 \\ damage 1 & 0.254 & 0.256 & 0.014 \\ damage 2 & 0.129 & 0.125 & 0.031 \\ \hline \end{tabular}
\end{table} 
Table 2: Statistical data for all the scenarios at 5 rpm. Figure 5: Results of the spectral entropy (SE) calculation for the PE signal from the raw vibration signal for all scenarios at 5 rpm. (**a**) Values for each rotation of the bearing. (**b**) Box plot for statistical analysis of the calculated values. The results in Figures 5 and 6 indicate that the calculated SE of the PE signal can discriminate between a healthy bearing and a bearing with incipient damage from the point of view of a classification problem. This might be since the energy released by a damaged bearing, which is related to the operating speed of the bearing, the size of the damage, and the magnitude of the harmonics. The quantity of the released energy (detected as vibration by the sensor) for low-speed bearings is reflected in the magnitude of the produced harmonics, which are in general small enough to be confused as noise in the signal. Therefore, as the released energy for incipient damage is not high enough, it cannot be distinguished from the noise in the signal. Nevertheless, the released energy produces harmonics. The proposed method hardly recognizes the trends from the energy release caused by the incipient damage, allowing slight detection exposed to results from both rotational speeds. For the case of a healthy bearing and established damage, an expected result would be a bigger difference among scenarios for higher than lower rpm (a higher rotational speed means higher energy release). However, results show a bigger difference between HS and D2 for data at 5 rpm in comparison to 10 rpm. As reviewed in the literature, not all indicators are useful for all rotational speed range. From this perspective, the proposed method might have a better behavior for rotational speed closer to 1, as the numerical difference between a healthy and established damage scenario is bigger for the lower rotational speed. Further studies could confirm this assumption. The energy quantity from the established damage is large enough to be distinguishable from healthy and incipient damage, as the results reveal. For a better awareness of the method and its behavior under several types of damage and rotational speed, so that a robust method for the diagnosis of bearing damage can be developed, a variety of damage types and rotational speed should be studied. \begin{table}
\begin{tabular}{l l l l} \hline  & \(\mathbf{\hat{x}}\) & **Md** & \(\mathbf{s}\) \\ \hline healthy & 0.245 & 0.246 & 0.016 \\ damage 1 & 0.225 & 0.225 & 0.022 \\ damage 2 & 0.173 & 0.175 & 0.018 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical data for all the scenarios at 10 rpm. Figure 6: Results of the SE calculation for the PE signal from the raw vibration signal for all scenarios at 10 rpm. (**a**) Values for each rotation of the bearing. (**b**) Box plot for statistical analysis of the calculated values. \begin{table}
\begin{tabular}{l l l} \hline  & **Slope** & \(\mathbf{y}\)**-intercept** \\ \hline healthy & \(-3.05\times 10^{-5}\) & 0.25 \\ damage 1 & \(-1.51\times 10^{-5}\) & 0.23 \\ damage 2 & \(-1.71\times 10^{-5}\) & 0.18 \\ \hline \end{tabular}
\end{table} 
Table 3: Linear regressions for all the scenarios at 10 rpm.