## 2 Materials and Methods
### Power Spectral Density (PSD)  
Modal parameters, such as natural frequencies \((f_{n})\), enable the analysis of structural vibrations under various scenarios, including ambient noise, external force excitation, and structural damage. One common way to understand the behavior of vibration signals is to transform them from the time domain into the frequency domain using Fast Fourier Transform (FFT), which reveals their frequency content. Additionally, PSD estimates the power distribution of the signal in the frequency domain. Particularly, for random vibration signals, PSD is preferable over FFT because it normalizes the amplitude values by frequency resolution (frequency bin width), providing more accurate identification of harmonic components and spectral peaks.  
For non-parametric spectral estimation, a common method to estimate the power spectral density (PSD) is the periodogram. Considering \(x[n]\) as the discrete acceleration signal obtained by sampling the continuous-time signal \(x(t)\) at a sampling frequency \(fs=1/\Delta t\) results in a signal of length \(N\) samples. The one-sided periodogram estimate \((P_{xx})\) is defined by Equation (9):  
\[P_{xx}(f)=\frac{\Delta t}{N}\biggl{|}\sum_{n=0}^{N-1}w[n]x[n]e^{-2\pi f\Delta tn }\biggr{|}^{2},0\leqslant f\leqslant\frac{1}{2\Delta t} \tag{9}\]  
where:  
\begin{table}
\begin{tabular}{c c c c c c c} \hline
**AV** & **AVD** & **IH** & **ID** & \(f_{\Delta t}\) & \(f_{\Delta t2}\) & \(f_{\Delta t3}\) & **Total** \\ \hline
181 \(\times\) 4 & 36 \(\times\) 4 & 31 \(\times\) 4 & 34 \(\times\) 4 & 32 \(\times\) 4 & 33 \(\times\) 4 & 37 \(\times\) 4 & 384 \(\times\) 4 \\ \hline \end{tabular}
\end{table}
Table 1: Dataset specification for each experiment.  
Figure 12: Acceleration measurements with a shaker with frequencies \(f_{d1}=2.11\) Hz, \(f_{d2}=6.41\) Hz, and \(f_{d3}=9.68\) Hz.  
* \(\Delta t\) is the sampling interval.
* \(w[n]\) is a window function of length \(N\). In this study, the Hamming window was employed to mitigate spectral leakage.
* \(P_{xx}(f)\) represents the PSD with physical units of \(\frac{\text{m}/\text{s}^{2}}{\text{Hz}}\).  
Equation (9) corresponds to the squared magnitude of the Discrete Fourier Transform (DFT) of the windowed data, normalized by the factor \(\Delta t/N\). This normalization ensures that \(P_{xx}(f)\) describes the energy distribution of the acceleration signal over the measured frequency range [36; 37; 38]. In this article, the periodogram method was employed to quantify the relative power at each natural frequency \(f_{n}\) for the excitation cases previously described, using accelerometer sensors mounted on each mass (S2, S3, and S4). The one-sided periodogram \(P_{xx}(f)\) defined in Equation (9) was converted to decibels per hertz according to Equation (10).  
\[P_{xx,\text{dB}}(f)\ =\ 10\log_{10}(P_{xx}(f))\quad\text{[dB/Hz]} \tag{10}\]  
For each excitation case, the power spectral densities (PSDs) from all repetitions were analyzed. The median spectrum was computed across repetitions to identify the dominant frequency corresponding to the natural frequency of the structure while reducing the influence of outliers. This approach ensured robust identification of natural frequencies across different scenes.