## 3 Results
### Power Spectral Density (PSD)
Furthermore, the AV and AVD cases also show a difference in their spectral power, considering that the excitation is the same. Table 4 illustrates that the excitation frequencies \(f_{d1}\), \(f_{d2}\), and \(f_{d3}\) show the highest spectral peaks in comparison with the natural frequencies, as mentioned before. Moreover, the S3 sensor exhibits a major power contribution for the frequencies close to \(f_{d1}\) and \(f_{d3}\), whereas S2 shows a predominant contribution at frequency \(f_{d2}\). The values of the natural frequencies are inferior compared with the frequencies of the AV case. Additionally, both tables indicate that the frequency of 9.32 Hz appears in the AVD, ID, IH, \(f_{d1}\), \(f_{d2}\), and \(f_{d3}\) cases. Table 5 summarizes the relative error (RE) between the experimental and theoretical natural frequencies, \(f_{ne}\) and \(f_{n}\), respectively, for all cases. The AV case exhibits the lowest errors for \(f_{n1}\) and \(f_{n2}\), with \(f_{ne}=1.86\) Hz and \(f_{ne}=5.87\) Hz, respectively, which corresponds to a lower RE of less than 4%. On the other hand, the \(f_{d2}\) case shows the highest error for \(f_{n1}\), where \(f_{ne}=1.76\) Hz yields a RE of approximately 9%. In addition, for \(f_{n3}=8.86\) Hz, all experimental values exceed the theoretical frequency, indicating a consistent negative  
Figure 16: Power spectral density of the AV and AVD cases of the S1, S2, S3, and S4 sensors. Figure 17: Power spectral density of the ID and IH cases of the S1, S2, S3, and S4 sensors. bias. Nevertheless, comparing the AV case with the other cases, the REs remain below 6%, making it a suitable baseline for evaluating the effects of structural damage versus the undamaged condition. Moreover, the \(f_{we}\) values closest to \(f_{n2}\) exhibit the the lowest RE, with values below 3% in all the cases. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Frequency [Hz]** & **AV Case [dB/Hz]** & **AVD Case [dB/Hz]** & **ID Case [dB/Hz]** & **IH Case [dB/Hz]** \\ \hline \multirow{3}{*}{1.80} & - & \(-53.76\), (\(53\)) & \(11.39\), \((53)\) & \(5.14\), \((53)\) \\  & - & \(-55.20\), (\(54\)) & \(10.28\), \((54)\) & \(4.04\), \((54)\) \\  & - & \(-60.05\), \((52)\) & \(6.10\), \((52)\) & \(-0.14\), \((52)\) \\ \hline \multirow{3}{*}{1.86} & \(-55.79\), (\(53\)) & - & - & - \\  & \(-56.72\), (\(54\)) & - & - & - \\  & \(-62.76\), (\(52\)) & - & - & - \\ \hline \multirow{3}{*}{5.72} & - & \(-50.36\), (\(52\)) & \(-6.22\), (\(52\)) & \(-0.99\), (\(52\)) \\  & - & \(-55.18\), (\(54\)) & \(-11.02\), (\(54\)) & \(-5.80\), (\(54\)) \\  & - & \(-67.85\), (\(53\)) & \(-24.04\), (\(53\)) & \(-18.97\), (\(53\)) \\ \hline \multirow{3}{*}{5.87} & \(-60.94\), (\(52\)) & - & - & - \\  & \(-65.56\), (\(54\)) & - & - & - \\  & \(-74.86\), (\(53\)) & - & - & - \\ \hline \multirow{3}{*}{9.32} & - & \(-52.45\), (\(53\)) & \(-4.76\), (\(53\)) & \(6.95\), \((53)\) \\  & - & \(-53.50\), (\(54\)) & \(-6.36\), (\(54\)) & \(5.35\), (\(54\)) \\ \cline{1-1}  & - & \(-62.58\), (\(52\)) & \(-15.43\), (\(52\)) & \(-3.72\), (\(52\)) \\ \hline \multirow{3}{*}{9.33} & \(-48.56\), (\(53\)) & - & - & - \\ \cline{1-1}  & \(-50.37\), (\(54\)) & - & - & - \\ \cline{1-1}  & \(-61.50\), (\(52\)) & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Experimental natural frequencies of the power spectral density according to the method of excitation (AV, AVD, ID, and IH cases) of the 3-DoF structure. Figure 18: Power spectral density of shaker excitation with frequencies \(f_{d1}\) (Case 5), \(f_{d1}\) (Case 6), and \(f_{d3}\) (Case 7) of the S1, S2, S3, and S4 sensors.