## 2 Methodology
The measurement of the complexity of vibration signals of orthotropic steel bridge panels is beneficial to excavate the damage state of orthotropic steel bridge panels from the complex vibration signals and judge the damage location effectively. This study focuses on the energy of the signal after decomposition. By observing the proportion of energy in each frequency band, the frequency band with the highest proportion of energy is selected, as shown in Figure 1c. The inverse wavelet transform is used to reconstruct the signal, as shown in Figure 1d. The specific calculation process of permutation entropy algorithm adopted in this study is as follows:  
For a time series of length \(N\): \(\{x(i),i=1,2,\ldots,N\}\), reconstruct any element \(x(i)\) according to the phase-space delay coordinate method, take \(m\) consecutive sample points of each sampling point, and obtain the \(m\)-dimension reconstruction vector of point \(x(i)\):  
\[\mathbf{X(i)}=\{x(i),x(i+f),\cdots,x(i+(m-1)f)\} \tag{2}\]  
where \(m\geq 2\) is the embedding dimension, \(f\) is the time delay, \(i=1\), \(2\), \(\ldots\), \(N\), and the \(m\) reconstructed components in \(X(i)\) are arranged in the order from smallest to largest:  
\[\{x(i+(j_{1}-1)f)\leq x(i+(j_{2}-1)f)\leq\cdots\leq x(i+(j_{m}-1)f)\} \tag{3}\]  
If \(\mathbf{X(i)}\) has the same element--i.e., \(x(i+(j_{p}-1)f)=x(i+(j_{q}-1)f)\)--then it is sorted according to the size of \(i\)--i.e., when \(p\leq q\), the arrangement is \(x(i+(j_{p}-1)\theta)\leq x(i+(j_{q}-1)f)\). Therefore, any vector \(\mathbf{X(i)}\) can obtain a set of symbol sequences:  
\[S(g)=(j_{1},j_{2},\cdots j_{m}) \tag{4}\]  
Among them, \(j=1\), \(2\), \(\ldots\), \(m\) (\(m\in N\)). According to the theorem of permutation and combination, there are a total of m! different permutations of symbol sequences that are different from the \(m\)-dimensional phase space mapping, and \(S(g)=(j_{1},j_{2},\ldots,j_{m})\) is one of these m! permutations. Assuming the probabilities of each occurrence of \(S(g)\) are \(P_{1}\), \(P_{2}\), \(\ldots\), \(P_{k}\), then according to [PERSON]'s entropy theorem, the permutation entropy of the sequence \(\{x(k),k=1,2,\ldots,N\}\) can be defined as:  
\[PE(m)=-\sum_{g=1}^{k}P_{g}lnP_{g} \tag{5}\]  
when \(P_{g}=1/m!\), PE(m) reaches its maximum value \(ln(m!)\). For convenience, \(ln(m! )\) is usually used to normalize \(PE(m)\). \[0\leq H_{P}=H_{P}/ln(m!)\leq 1 \tag{6}\]  
The randomness of the acceleration vibration signal is measured by the magnitude of the PE value. The smaller the PE value, the more regular the time series is; on the contrary, the larger the PE value, the higher the degree of disorder and signal complexity of the time series. The permutation entropy difference (PED) of under different states can be expressed as:  
\[PED=\left|PE^{u}-PE^{d}\right| \tag{7}\]  
where PE\({}^{u}\) and PE\({}^{d}\) represent the sample entropy under nondestructive and damaged conditions, respectively. For the selection of embedding dimension, when the embedding dimension is too small (m < 4), the range of change in permutation entropy is too small to effectively distinguish damaged vibration signals from normal vibration signals. However, when the embedding dimension is too large (m > 8), the operation time of permutation entropy becomes longer, and it will lead to a decrease in the range of change in permutation entropy, making it difficult to accurately measure signal amplitude [14]. This article selects an embedding dimension value of m = 6 to determine the damage location of the orthotropic steel bridge deck by calculating the PED value.