## 2 Materials and Methods
### Synchronization Validation
To ensure the robustness and reliability of the findings, each data recording session lasted approximately 15 min to ensure an adequate experimental period. This collection of 3 DOF structural responses from both sensor placement schemes allows for a comparative analysis to evaluate the quality of the collected data in terms of modal analysis results. Such analysis enables an assessment of the effectiveness and reliability of the collected data in capturing the structural response and modal characteristics of the system. Figure 6 presents the schematic diagram of FDD process applied to raw low-cost acceleration data. The FDD scheme consists of five steps: (1) reading the raw data, (2) detrending and resampling the data, (3) selecting a window size that contains the impulse response, (4) refining the alignment using adaptive sampling DTW, and (5) performing the FDD analysis. Achieving perfect time alignment of low-cost sensors is often unattainable. Nevertheless, the starting times provided by our proposed method serve as a valuable aid in the initial alignment. To overcome the challenge posed by imperfect temporal alignment, DTW is employed for time series alignment. The optimal warping path obtained through DTW can be regarded as the optimal temporal axis for point-wise alignment of the time series. The time series signal from the \(j\)-th low-cost sensor is denoted as \(Y_{j}=(Y_{j}[1],\ldots,Y_{j}[N])\), with \(N\) representing the number of data points for sensor \(Y_{j}\). The processing of unevenly sampled acceleration data involves the two primary steps, i.e., detrending and resampling. To perform detrending, the mean value of the \(j\)-th sensor, denoted as \(\mu_{j}\), is calculated as  
\[\mu_{j}=\frac{1}{N}\sum_{i=1}^{N}Y_{j}[i], \tag{1}\]  
where \(Y_{j}[i]\) is the \(i\)-th data point of the \(j\)-th sensor. Subsequently, this mean value \(\mu_{j}\) is subtracted from each data point of the \(j\)-th sensor, yielding the detrended discrete signal \(Y_{j,\text{detrended}}[k]\), given by  
\[Y_{j,\text{detrended}}[k]=Y_{j}[k]-\mu_{j}. \tag{2}\]  
where \(Y_{j}[k]\) represents the original signal from the \(j\)-th sensor and \(k\) denotes the index of the data point. This process removes any systematic trend or bias from the original data, resulting in a detrended signal suitable for further analysis or processing. The second step involves the resampling of the data to a frequency of 100 Hz. In practical scenarios of this study, the sampling frequency of the low-cost sensor is typically set to 100 Hz. However, the actual time sampling may not be consistently stable as desired. Therefore, the original sampling rate, denoted as \(f_{\text{original},i}\), is examined first before proceeding as follows:  
Figure 6: Schematic diagram of the Frequency Domain Decomposition (FDD) process applied to raw low-cost acceleration data. The diagram illustrates the five steps involved: reading raw data, detrending and resampling, selecting a window size, refining alignment using adaptive sampling DTW, and performing the FDD analysis. * If \(f_{\text{original},j}\) is greater than 100 Hz, the detrended signal \(Y_{j,\text{detrended}}[k]\) is downsampled using the decimation method as \[Y_{j,\text{downsampled}}[k]=\text{Decimate}(Y_{j,\text{detrended}}[k],f_{ \text{original},j}\,100).\] (3)
* If \(f_{\text{original},j}\) is less than 100 Hz, the detrended signal \(Y_{j,\text{detrended}}\) is upsampled using the interpolation method as \[Y_{j,\text{upsampled}}[k]=\text{Interpolate}(Y_{j,\text{detrended}}[k],f_{ \text{original},j},100).\] (4)  
The resulting signal, either the upsampled signal \(Y_{j,\text{upsampled}}[k]\) or the downsampled signal \(Y_{j,\text{downsampled}}[k]\), is represented by \(Y_{j,\text{resampled}}[k]\). Consequently, a windowing approach is employed to focus on the segments surrounding the peak response.