## 2 Materials and Methods
### Synchronization Validation
The windowed segments of \(Y_{j,\text{resampled}}[k]\) and \(Y_{l,\text{resampled}}[k]\), denoted as \(Y_{j,\text{window}}[k]\) and \(Y_{l,\text{window}}[k]\) respectively, are defined as  
\[Y_{j,\text{window}}[k]= \begin{cases}Y_{j,\text{resampled}}[k],&\text{if }k\in[k_{\text{start}},k_{\text{end}}],\\ 0,&\text{otherwise},\end{cases}  \tag{5}\]  
\[Y_{l,\text{window}}[k]= \begin{cases}Y_{l,\text{resampled}}[k],&\text{if }k\in[k_{\text{start}},k_{\text{end}}],\\ 0,&\text{otherwise},\end{cases}  \tag{6}\]  
where \([k_{\text{start}},k_{\text{end}}]\) represents the time window centered around the peak response. By selecting the same time window for both signals, the focus is directed towards aligning the corresponding segments that encompass the peak responses. Within this window, the original resampled signal is retained, while outside the window, the signal is set to zero. This process isolates the segments of interest for further analysis, ensuring that both signals are aligned and comparable within the specified window. To align the windowed time series signals \(Y_{j,\text{window}}[k]\) and \(Y_{l,\text{window}}[k]\), the DTW algorithm is employed. DTW quantifies the dissimilarity between two sequences by accounting for temporal warping. The algorithm determines an optimal warping path denoted as \(X=(x_{1},x_{2},\dots,x_{p},\dots,x_{q})\). Each element \(x_{p}\) represents a pair of indices \((m,n)\), where \(m\) and \(n\) correspond to an index in time series \(Y_{j,\text{window}}[k]\) and \(Y_{l,\text{window}}[k]\), respectively. The DTW algorithm computes a distance matrix, often referred to as the two-dimensional cost matrix, denoted as \(\mathbf{D}\in\mathbb{R}^{(N_{j}+1)\times(N_{l}+1)}\), where \(\mathbf{D}[m,n]\) represents the cumulative distance between the \(m\)-th element of \(Y_{j,\text{window}}[k]\) and the \(n\)-th element of \(Y_{l,\text{window}}[k]\) given by  
\[\mathbf{D}[m,n]=d(Y_{j,\text{window}}[m],Y_{l,\text{window}}[n])+\text{min}( \mathbf{D}[m-1,n],\mathbf{D}[m,n-1],\mathbf{D}[m-1,n-1]), \tag{7}\]  
where \(d(Y_{j,\text{window}}[m],Y_{k,\text{window}}[n])\) represents the local distance or similarity measure between \(Y_{j,\text{window}}[m]\) and \(Y_{k,\text{window}}[n]\) and min is minimum function calculating the minimum value among the three values \(\mathbf{D}[m-1,n]\), \(\mathbf{D}[m,n-1]\), and \(\mathbf{D}[m-1,n-1]\). The alignment between \(Y_{j,\text{window}}[k]\) and \(Y_{l,\text{window}}[k]\) can be obtained by tracing back the optimal path through the distance matrix \(\mathbf{D}\). For more comprehensive information on DTW, refer to [70; 71]. After applying DTW, the aligned signals \(Y_{j,\text{aligned}}[k]\) and \(Y_{l,\text{aligned}}[k]\) can be obtained by warping the original signals based on the optimal alignment path. Implementing DTW allows for the alignment of the windowed time series signals, enabling further analysis of FDD for OMA applications. FDD is a widely employed method in the field of structural dynamics and vibration analysis for extracting modal parameters from measured data [72; 73]. FDD proves particularly valuable in scenarios where there is a lack of prior knowledge regarding excitation forces or precise boundary conditions. The theoretical validation of this method relies on the modal expansion of the structural response. In this study, the aligned signals \(Y_{j,\text{aligned}}[t]\) from the \(j\)-th sensor is used in the FDD analysis. Given a set of aligned signals \(Y_{j,\text{aligned}}[k]\), where \(j=1,2,\ldots,M\) represents the sensor index and \(k=1,\ldots,N\) represents the time (or discrete sample) index, and the aligned signals are stored in the matrix \(\mathbf{Y}\) as  
\[\mathbf{Y}= \begin{bmatrix}Y_{1,\text{aligned}}[1]&Y_{2, \text{aligned}}[1]&\ldots&Y_{M,\text{aligned}}[1]\\ Y_{1,\text{aligned}}[2]&Y_{2,\text{aligned}}[2]& \ldots&Y_{M,\text{aligned}}[2]\\ \vdots&\vdots&\ddots&\vdots\\ Y_{1,\text{aligned}}[N]&Y_{2,\text{aligned}}[N]& \ldots&Y_{M,\text{aligned}}[N]\end{bmatrix} .