## 3 Numerical Assessment of the New Methodology Performance
It is important to mention that they only analyze the \(y\)-direction; hence, the other four natural frequencies of the \(x\)-direction are not considered. In addition, the obtained results allow to affirm that the damping ratios values estimated by the proposal are more accurate than the ones calculated by [PERSON] and [PERSON] [35] and [PERSON] et al. [11], since a maximum error for the proposal and the other two works are: 6%, 12% and 15%, respectively. In particular, the proposal outperforms the methodology of the former reference by achieving an error reduction of over 50%. This reduction can be achieved as the proposal uses the envelope values in a qualitative way, that is, how similar are the two envelopes, making them inherent more robust to any slightly spike that can be presented. On the contrary, the methodology of [PERSON] et al. [11] is more susceptible to these spikes, as the value is an integral part of the equations used to obtain the DR value. It can be seen that the approach used by [PERSON] and [PERSON] [35], show that the used methodology can successfully estimate the DR with a reasonable accuracy, indicating that their methodology has a better performance to estimate the aforementioned values. The main objective of [PERSON] et al. [34] is to find the best FE-based model; thus, the authors do not estimate the damping ratios of structure. From these works, it is evident that the ones that use multi-paradigm approaches lead to better estimations of the modal parameters. Further, it is worth noting that [PERSON] et al. [53] mentioned that for obtaining an accurate value, it should be present a maximum error of 2% and 20% for natural frequency and damping ratio, respectively. In this regard, the proposed methodology contributes to demonstrate the abovementioned statement, as the obtained results can be considered as accurate. In summary, the obtained results demonstrate that the proposal is an excellent tool to identify and quantify the NF and DR. This is proven as the errors between the obtained values and the theoretical counterparts do not exceed 0.5% and 0.6%, respectively. Moreover, the validity of the proposed objective function is also demonstrated, as the correlation value of the estimated modes compared to the original ones is higher than 0.9. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Mode (Direction)**} & \multicolumn{5}{c}{**Natural Frequency (Hz) (Error \%)**} \\ \cline{2-6}  & **FEA** & **[PERSON] et al. [34]** & **[PERSON] and [PERSON] [35]** & **[PERSON] et al. [11]** & **New Methodology** \\ \hline
1 (\(y\)) & 9.410 & 8.48 (9.88) & 10.62 (12.86) & 9.407 (0.03) & 9.39 (0.21) \\
2 (\(x\)) & 11.79 & 9.03 (23.41) & - & 11.82 (0.20) & 11.72 (0.59) \\
3 (\(y\)) & 25.54 & 23.07 (9.67) & 25.39 (0.59) & 25.55 (0.03) & 25.59 (0.20) \\
4 (\(x\)) & 32.01 & 25.45 (20.49) & - & 31.95 (0.20) & 32.03 (0.06) \\
5 (\(y\)) & 38.66 & 36.32 (6.05) & - & 38.61 (0.13) & 38.67 (0.03) \\
6 (\(y\)) & 48.01 & 41.81 (12.91) & 47.97 (0.08) & 47.97 (0.08) & 48.05 (0.08) \\
7 (\(x\)) & 48.44 & 46.57 (3.86) & - & 48.50 (0.10) & 48.65 (0.44) \\
8 (\(x\)) & 60.15 & 56.09 (6.75) & - & 60.35 (0.33) & 60.16 (0.02) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Identified modal parameters of the benchmark structure shown in Figure 4. (FEA: finite element analysis.)