## 4 Numerical Example
### Practical Design Recommendations  
Once the optimal design parameters have been determined for each benchmark record, an attempt is made to establish a single set of optimal design parameters to work properly under any acceleration record. In that sense, the methodology proposed by [PERSON] and [PERSON] in [PERSON] and [PERSON] (2018) is implemented for that purpose. This methodology establishes average design parameters according to the weighted response reduction percentages obtained previously. Therefore, the design parameters obtained from the objective function J1, which was the most effective to control the dynamic response of displacement, are replaced in Equations (23) and (24):  
\[\zeta_{d\text{ avg}}=\frac{\sum\limits_{i=1}^{4}\left(\zeta_{opt}*R_{\text{ mfd}}*R_{\text{RMS}}\right)}{\sum\limits_{i=1}^{4}\left(R_{\text{mfd}}*R_{ \text{RMS}}\right)} \tag{23}\]  
\[f_{avg}=\frac{\sum\limits_{i=1}^{4}\left(f_{opt}*R_{\text{mfd}}*R_{\text{RMS} }\right)}{\sum\limits_{i=1}^{4}\left(R_{\text{mfd}}*R_{\text{RMS}}\right)} \tag{24}\]  
where \(R_{\text{mfd}}\) and \(R_{\text{RMS}}\) denote, respectively, the reduction of the maximum floor displacement and the reduction of the maximum RMS displacement defined according to Equations (25) and (26). In addition, the counter \(i\) in the sum numbers the accelerograms used in the tuning process. More records in the optimization process will enhance the accuracy of the average design variables; however, for this example, only the numerical results computed from the records in Table 1 are considered. The computed set of average design parameters \(\zeta_{d\text{ avg}}\) and \(f_{avg}\) are reported in Table 5.  
\[R_{\text{mfd}}=\left(1-\frac{\text{max}(|x_{n}|)}{\text{max}\left(|x_{n}^{*}| \right)}\right) \tag{25}\]  
\[R_{\text{RMS}}=\left(1-\frac{\text{max}(\text{RMS}(x_{n}))}{\text{max}\left( \text{RMS}(x_{n}^{*})\right)}\right) \tag{26}\]  
Thereafter, the case-study is subjected to the action of the Petrolia and San Fernando acceleration records, whose details are presented in Table 6. The main purpose of using acceleration records different from the benchmark records used in the optimization process is to verify the performance of the structure and the robustness of the WOA methodology under any type of random excitation.  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(\mu\) & \(\zeta_{d\text{ avg}}\) & \(f_{avg}\) \\ \hline
0.02 & 0.103 & 0.967 \\
0.05 & 0.209 & 0.929 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Average design parameters for TMDs with \(\mu=0.02\) and \(\mu=0.05\).  
Reductions up to 6 and 30% are observed in horizontal peak displacements and RMS displacements, respectively, using the TMD designed with a unique set of parameters derived from the methodology described in [60]. Although the methodology was originally proposed for base isolation systems, it can be adapted for the tuning of linear mass dampers since, for both base isolation systems and TMDs, optimal design variables must be determined. At first glance, the reductions attained with the average design values are not substantial. However, in earthquake engineering practice, it is impossible to know the ground motion excitation that will affect the structural systems, and, accordingly, the Petrolia and San Fernando Earthquake present dynamic properties different from the records for which the TMD was originally tuned. Thus, although small, these results validate the robustness of the optimization based on WOA and demonstrated the efficiency of the algorithm over other metaheuristics, like DEM and conventional tuning methodologies based on closed-form expressions. Finally, to achieve a realistic tuning, other critical variables should be considered, and a more comprehensive analysis is required (e.g., analysis on the stochastic nature of the seismic site conditions).