## 4 Vibration Data Analysis: Definition of the RWEE Damage Index  
### Wavelet Packet Decomposition and Wavelet Energy  
Wavelet Packet Transform (WPT) is one extension of the Wavelet Transform (WT) to decompose a signal repeatedly into successive low and high frequency components [47; 48]. A wavelet packet is a family of scaling functions and wavelet functions constructed by following a binary tree of dilations/translations. The wavelet packet, \(\psi^{i}_{j,k}(t)\), is a function of three indices, where \(i,j\) and \(k\) are integers representing the modulation, the scale and the translation parameters, respectively. It can be written as  
\[\psi^{i}_{j,k}(t)=2^{j/2}\psi^{i}\Big{(}2^{j}t-k\Big{)},\;i=1,\;2,\;3\ldots \tag{1}\]  
where the wavelets \(\psi^{i}\) are obtained from the following recursive relationships  
\[\psi^{2i}(t)=\sqrt{2}\sum_{k=-\infty}^{\infty}h(k)\psi^{i}(2t-k) \tag{2}\]  
\[\psi^{2i+1}(t)=\sqrt{2}\sum_{k=-\infty}^{\infty}g(k)\psi^{i}(2t-k)\, \tag{3}\]  
\(h(k)\) and \(g(k)\) being quadrative mirror filters associated with the scaling function and the mother wavelet function, respectively [49; 50]. In the two above equations, \(\psi^{i}(t)\) is the mother wavelet function. There are quite a few mother wavelets reported in the literature, developed to satisfy some important properties such as invertibility or orthogonality [51; 52]. Daubechies [50] developed a family of mother wavelets based on the solution of a dilation equation. In the present work, among several alternatives, Daubechies functions of the order 5 (db5) were selected as mother wavelets on the basis of trial and error tests.  
Figure 3: Response signal obtained from the vibration tests.  
The analytical procedure starts by computing the wavelet packet decomposition of the monitored vibration signal at level N using a given wavelet. This means there are 2\({}^{\text{N}}\) components at the Nth level, providing flexibility for choosing the way to encode the original signal, so that the reconstruction error is minimal. A schematic representation of the WPT of a time-domain signal \(f(t)\) up to the 3 rd level of decomposition is presented in Figure 4; it can be obtained, for instance, by summing the signal components \(f_{1}^{1}(t)\), \(f_{3}^{5}(t)\), \(f_{3}^{6}(t)\) and \(f_{2}^{4}(t)\). The approximations correspond to the low scale (low frequency components) of the signal, while the details correspond to the high scale (high frequency components) of the signal.  
The wavelet packet component signal \(f_{j}^{i}(t)\) can be expressed by a linear combination of wavelet packet functions \(\psi_{j,k}^{i}(t)\) as follows  
\[f_{j}^{i}(t)=\sum_{k=-\infty}^{\infty}c_{j,k}^{i}\psi_{j,k}^{i}(t) \tag{4}\]  
and the wavelet packet coefficients \(c_{j,k}^{i}\) can be obtained from  
\[c_{j,k}^{i}=\int_{-\infty}^{\infty}f(t)\psi_{j,k}^{i}(t)dt \tag{5}\]  
Each component in the WPT tree can be viewed as the output of a filter tuned to a particular basis function, so that the whole tree can be regarded as a filter bank. At the top of the WPT tree (lower level), the WPT yields good resolution in the time domain but poor resolution in the frequency domain. At the bottom of the WPT tree (higher level), the WPT results are in good resolution in the frequency domain but in poor resolution in the time domain.  
The RWEE index is based on the energy evaluation of the WPT. The energy of each sub-signal is calculated according to the following equation  
\[E_{j}^{i}=\int\left|f_{j}^{i}(t)\right|^{2}dt \tag{6}\]  
where \(i\) is the number of the node and \(j\) is the level of decomposition.