## 4 Vibration Data Analysis: Definition of the RWEE Damage Index
### Relative Wavelet Entropy (RWE): RWEE Damage Index Definition  
Results show that entropy based on the wavelet transform is a good candidate to characterize damage because of its sensitivity to local structural damage [37]. That is, the wavelet entropy changes when the structure is damaged. The RWE is known to be associated with different frequency bands of signals obtained from the baseline or inspection phases, thereby allowing for detection of changes in the level of damage of the I-section. The relative wavelet energy entropy (RWEE) is based on this concept together with entropy, i.e.,  
\[RWEE=-\sum_{i=1}^{N}p_{i}\cdot log\left[\frac{p_{i}}{q_{i}}\right] \tag{9}\]  
where \(p_{i}\) and \(q_{i}\) are the vectors that relate the wavelet energy of each node with respect to the total energy of the considered nodes of the vibration signals recorded in the inspection phase and baseline phase, respectively. The vectors \(p_{i}\) and \(q_{i}\) are defined as  
\[p_{i}=\left(\frac{E_{i}}{E_{i}}\right)_{ inspected} \tag{10}\]  
\[q_{i}=\left(\frac{E_{i}}{E_{i}}\right)_{undamaged} \tag{11}\]  
If the evaluated I-section is undamaged or has a very low level of damage, the value of RWEE will be close to zero. If damage is substantial, the RWEE value will be farther from null.  
In order to calculate the RWEE damage index, the following steps are taken:  
1. Firstly, normalization and scaling must be applied to counteract different excitation levels and environmental conditions and to compensate the off-set and amplitude variability among signals. This normalization is carried out as follows: \[f(t)=\frac{f(t)_{r}-\overline{f}(t)_{r}}{\sigma(f(t)_{r})}\] (12) where \(f(t)\) is the normalized signal, \(f(t)_{r}\) is the recorded signal and \(\overline{f}(t)_{r}\) and \(\sigma(f(t)_{r})\) are the mean value and the standard deviation of the \(f(t)_{r}\), respectively.  
The normalized signal \(f(t)\) is decomposed into multiple sub-signals in various frequency bands using the WPT. Let \(f_{0}^{1}(t)\) denote the normalized response signal shown in the first step. Through trial and error tests, it was found that the optimum level of decomposition is 7; thus, 27 (128 nodes) is the number of sub-signals to be obtained in the proposed procedure.
3. The energy of each sub-signal is calculated according to the following equation \[E_{j}^{i}=\int\Big{|}f_{j}^{i}(t)\Big{|}^{2}dt\] (13) providing the energy \(E_{j}^{i0}\) for the undamaged I-section and energy \(E_{j}^{i}\) for the inspected I-section, where i is the number of the node and j is the level of decomposition. Steps 1, 2 and 3 must be carried out in both baseline and inspection phases.
4. Finally, the RWEE index is calculated using Equation (14), where \(E_{j}^{i}\) is the energy of each sub-signal on the level of decomposition \(j\), and \(E_{jt}\) is the sum of the energy of the all sub-signals of level \(j\), both calculated for the inspection phase. The same parameters calculated for the baseline inspection are \(E_{j}^{i0}\) and \(E_{jt}^{0}\), respectively. \(N\) is the total number of sub-signals, in the present study 128. \[RWEE=-\sum_{i=1}^{N}\left(\frac{E_{j}^{i}}{E_{jt}}\right)\cdot log\left(\frac{ E_{j}^{i}/E_{jt}}{E_{j}^{i0}/E_{jt}^{0}}\right)\] (14) The RWEE index is obtained for the whole signal, i.e., considering all frequencies of the signals.