# EMD-Shannon Entropy-Based Methodology to Detect Incipient Damages in a Truss Structure
## 5 Results  
Following the proposed steps in the methodology, the measured vibration response of the bridge under different levels of damage is decomposed into its different frequency bands or IMFs using the three EMD-based methods. Figure 6 shows the first seven IMFs obtained for healthy and incipient damage conditions in Ax, respectively, using the EMD, EEMD, and CEEMD methods. From these figures, it is evident that there are not significant visual differences in the obtained IMFs using the three EMD-based methods; thus, it is necessary to perform a statistical analysis in order to determine both the method and the IMF that can provide the most discriminate information, since this condition will allow us to determine the structure's condition in the most accurate way. It is important to mention that the same situation (no visible differences) is found in the remaining axis (Ay and Az). To perform the statistical analysis, the SEI values of all IMFs for each structure condition are obtained; then, the one-way ANOVA is used to obtain the \(p\)-value, which is the probability that the there are no significant differences between two sets of data. In general terms, the lower the \(p\)-value, the better the differentiation between the two groups. In this way, the IMF 1 fused with SEI that has the lowest \(p\)-value can be considered as the one that is the most discriminant for determining the structure's condition. Table 1 summarizes the \(p\)-values computed by the three EMD methods for the first 7 IMFs of each axis. It is worth mentioning that the SEI values obtained by IMF 1 for all the three methods have the lowest \(p\)-value. Moreover, it is should be also noted that the three EMD methods have very closely \(p\)-values. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & \multicolumn{3}{c}{**Ax**} & \multicolumn{3}{c}{**Ay**} & \multicolumn{3}{c}{**Az**} \\ \cline{2-10}  & **EMD** & **EEMD** & **CEEMD** & **EMD** & **EEMD** & **CEEMD** & **EMD** & **CEEMD** \\ \hline IMF1 & \(8.63\times\) & \(1.32\times\) & \(6.29\times\) & \(2.65\times\) & \(2.17\times\) & \(4.38\times\) & \(1.47\times\) & \(1.43\times\) & \(3.55\times\) \\  & \(10^{-98}\) & \(10^{-97}\) & \(10^{-98}\) & \(10^{-91}\) & \(10^{-91}\) & \(10^{-91}\) & \(10^{-91}\) & \(10^{-91}\) & \(10^{-97}\) & \(10^{-98}\) \\ IMF2 & \(1.13\times\) & \(2.86\times\) & \(2.46\times\) & \(5.68\times\) & & & & & \(2.64\times\) & \(7.36\times\) & \(6.25\times\) \\  & \(10^{-13}\) & \(10^{-09}\) & \(10^{-05}\) & \(10^{-04}\) & & & & & \(10^{-06}\) & \(10^{-07}\) & \(10^{-07}\) \\ IMF3 & \(4.45\times\) & \(1.95\times\) & \(3.46\times\) & \(3.47\times\) & \(1.13\times\) & \(1.52\times\) & \(9.78\times\) & \(4.10\times\) & \(1.51\times\) \\  & \(10^{-61}\) & \(10^{-69}\) & \(10^{-65}\) & \(10^{-23}\) & \(10^{-40}\) & \(10^{-51}\) & \(10^{-57}\)  
Figure 7 shows graphically the SEI values for the healthy (H) and the incipient damage condition encountered in each bay using the IMF 1 for Ax, Ay, and Az, and the three EMD methods, respectively. It is possible to observe that the three EMD-based methods estimate similar results, as all the box plots follow the same pattern, thereby justifying that the EMD method is suitable for the decomposition of the two methods. Figure 6: The first seven IMFs obtained for the healthy and incipient damage conditions (damage located in first-bay) in Ax, respectively using (**a**) EMD, (**b**) EEMD and (**c**) CEEMD methods. of the signals. Figure 8 shows the distribution of the estimated SEI values of the healthy condition and incipient damage using the IMF 1 for Ax, Ay, and Az, and the three EMD methods, respectively; however, in this case the SEI values calculated by IMF 1 for each axis and each bay of the incipient damage have been joined in order to show that the EMD fused with SEI is capable of detecting damage in the bridge regardless of the damage localization.