# Study of the Dynamic Response of a Rigid Runway with Different Void States during Aircraft Taxing
## 2 Research Methodology
### Wavelet Packet Energy Analysis Method  
The concept of wavelet transform was first proposed by geophysicist [PERSON] in 1981, and then, in 1984, [PERSON] and physicist [PERSON] invented the term wavelet [38]. The wavelet transform is able to perform local transformations of time and frequency; compared with the traditional signal processing method of Fourier analysis, it efficiently extracts information from the signal and analyzes the signal with multiscale refinement using operational functions such as scaling and translation [39]. In wavelet packet analysis, the low- and high-frequency parts of the signal are decomposed simultaneously; thus, there is also a high rate of separation of the high-frequency components of the signal. In wavelet packet analysis, the frequency band is divided into multiple homogeneous layers; then, the signal energy analysis is performed by focusing on the segments [40]. In the signal analysis field, wavelet packet analysis is more refined than wavelet analysis.  
If the vibration acceleration signal is a discrete time series \(\{x(i)\}\) of length \(N\), the wavelet packet energy ratio (WPER) and wavelet packet energy entropy (WPEE) can be calculated as described below.  
First, the \(j\)-th layer wavelet packet decomposition of sequence \(\{x(i)\}\) yields \(2^{j}\) nodes; \(C_{n,k}\) (\(n=0,1,2,\ldots,j\); \(k=0,1,2,\ldots,2^{j-1}\)) represents the \(k\)-th node of the \(n\)-th layer, and the coefficients of each node represent the signal characteristics of a frequency band.  
Then, wavelet packet coefficients \(D_{n,k}\) are extracted separately for each frequency band from low to high frequencies.  
The energy value, \(E_{n,k}\), of each frequency band in the \(n\)-th layer is calculated using Equation (19).  
\[E_{n,k}=\int{\big{|}D_{n,k}(t)\big{|}^{2}\mathrm{d}t}, \tag{23}\]  
where \(D_{n,k}\) (\(t\)) is the wavelet packet reconstruction coefficient, and \(t\) is the time of the corresponding frequency band signal. The total energy, \(E_{0}\), of the analyzed vibration acceleration signal can be expressed by Equation (20).  
\[E_{0}=\sum_{k=0}^{2^{j}-1}E_{n,k}. \tag{24}\]  
The ratio of the energy of the \(k\)-band of the \(n\)-th layer in the total energy of vibration signal \(P_{n,k}\) is obtained from Equation (21).  
\[P_{n,k}=\frac{E_{n,k}}{E_{0}}. \tag{25}\]  
Figure 5: Dynamic load of aircraft main landing gear.  
Finally, according to the definition of Shannon entropy, the energy entropy, \(H_{n,k}\), of each frequency band in the \(n\)-th layer is calculated using Equation (22).  
\[H_{n,k}=-\sum_{i=1}^{N}P_{n,k}\ln|P_{n,k}|. \tag{26}\]  
In this paper, we chose the Daubechies wavelet function to decompose the signal. The number of wavelet packet decomposition layers, \(n\), and the order of wavelet function were determined based on the \(\mathbb{P}\) norm entropy index [41]. The form of the norm \(\mathbb{P}\) (\(1\leq p\leq 2\)) is shown in Equation (23).  
\[S(E_{n})=-\sum_{k=0}^{2^{n}-1}\left|E_{n,k}\right|^{p}, \tag{27}\]  
where \(E_{n,k}\) represents the reconstructed signal energy of the \(k\)-th frequency band in the \(n\)th layer.