## 2 Methodology  
The framework of the orthotropic plate damage identification method proposed in this study is shown in Figure 1.  
By applying a sinusoidal excitation load, the original vibration response can be obtained from sensors already arranged on a steel box girder or an orthotropic plate. However, these raw vibration signals should not be used directly, but should be pruned, as shown in Figure 1a. This is due to the fact that in signal processing and analysis, pruning and removing the initial portion of the data when sinusoidal excitation loads are applied to the structure helps to more centrally and accurately evaluate the dynamic behavior and integrity of the structure under sinusoidal excitation. Firstly, it helps remove transient effects, such as initial settling or response onset, ensuring that subsequent analysis focuses on the steady-state response of the structure under excitation. Secondly, by isolating the steady-state response, it enables a clearer analysis of structural behavior and extraction of relevant information, such as excitation frequency and resonant frequencies. Additionally, this process aids in noise reduction by eliminating noise or artifacts present in the initial portion of the data, thereby improving the signal-to-noise ratio and facilitating more accurate detection and analysis of structural responses. Then, the trimmed data are decomposed by Wavelet Packet Decomposition (WPD). In this process, WPD first uses a set of wavelet basis functions to represent the signal. These wavelets are localized in both time and frequency and are used to capture different features of the signal across various scales. WPD then decomposes the signal into a tree-like structure, allowing decomposition at scale and frequency bands, as shown in Figure 1b. At each layer of the decomposition tree, a filter bank is used to segment the signal into sub-bands or nodes. This segmentation process divides the signal into different frequency bands, with each node representing a specific frequency range. The decomposition process is recursively applied to each node of the tree, generating further sub-bands or nodes at deeper levels in the tree. This recursive decomposition continues until the desired level of detail or frequency resolution is reached. At the end of the decomposition process, the signal is represented as a set of wavelet packet coefficients that capture the amplitude and phase information of the signal at different frequency bands and scales. Analyzing these coefficients can extract meaningful features or features of the signal, such as dominant frequencies, transient components, or structural anomalies. If \(U_{b}^{a}\) is the signal of the a-_th_ level and the b-_th_ node, \(U_{2b}^{a+1}=\text{Lowpass}\left(U_{b}^{a}\right)\), and \(U_{2b+1}^{a+1}=\text{highpass}\left(U_{b}^{a}\right)\). The specific implementation of the entire decomposition process is as follows:  
\[W_{l}=U_{l-k}^{2k}\oplus U_{l-k}^{2^{k}+1}\oplus\cdots\oplus U_{l-k}^{2^{k+1} -1},\;l,\;k\in Z \tag{1}\]  
According to the theory of multi-resolution analysis, \(L^{2}(R)=\oplus W_{l}\), \(l\in Z\), where \(W_{l}\) is the \(l\)-th wavelet subspace, \(l\) is the current level of analysis, and \(k\) is the amount of step or level change in the current decomposition process. And \(L^{2}(R)\) is a real valued function space that can be squared integrally and can be decomposed into a direct sum of a series of wavelet subspaces \(W_{l}\). The \(W_{l}\) decomposition is further decomposed by wavelet packet decomposition, which gives a better division of the whole analysis frequency band, thus improving the frequency resolution. Wavelet packet decomposition can be expressed as follows:  
Figure 1: Framework of WPD-PE: (**a**) data pruning; (**b**) three-layer wavelet packet decomposition tree; (**c**) selection of the component with the highest energy; (**d**) signal reconstruction. Complexity is an important nonlinear feature to measure the degree of disorder of a system.