## 4 Numerical Example
### Seismic Performance  
Now, the seismic performance of the building is evaluated under the action of the four seismic records used in the optimization process. It is worth noting that the scope of this investigation is limited to the linear behavior of the structural system. Thus, the numerical model does not take into account any type of damage or yielding process affecting the structure. Furthermore, the computed displacements are presumable within the elastic behavior of the system. Figures 5 and 6 show the history of displacements at the 32 nd story of the building, for \(\mu=0.02\) and \(0.05\), respectively. Even though both TMD options (\(\mu=0.02\), and \(0.05\)) can effectively reduce the dynamic response of displacements, it is visibly clear that TMDs  
Figure 4: Scatter graphs for TMD design parameters: (**a**) DE; (**b**) ES; (**c**) WOA. with \(\mu=0.05\) reach greater reductions in lateral displacements at the upper level of the structural system, regardless of the objective function. Figures 7 and 8 present the comparison of the peak displacement for \(\mu=0.02\) and \(0.05\), taking into account the six proposed objective functions. It is shown that the greatest reductions in the peak displacement at each story-level are achieved using the devices with the highest mass ratio, as reported in the literature [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50]. Moreover, the greatest reductions are observed at the higher story-levels of the structure. This behavior can  
Figure 5: Displacement of the 32 nd story of the structure equipped with TMDs with \(\mu=0.02\). Figure 6: Displacement of the 32 nd story of the structure equipped with TMDs with \(\mu=0.05\). be better understood by analyzing the proposed objective functions, which are mainly focused on diminishing the dynamic response at the 32 nd story of the building. Hence, the optimization process influences indirectly the response parameters of the floors near the upper level. The numerical results show that, using \(\mu=0.02\) TMD, the reduction of the maximum displacements of each floor of the structure is more effective using Kobe and Loma Prieta accelerators than El Centro and Northridge cases. Moreover, it may be affirmed that the greatest reduction in displacement is approximately 32.13%, reached at the 25 th story-level of the structure when it is controlled by the TMD designed by the J4 approach and using the Loma Prieta earthquake excitation. This reduction shows consistency, since the J4 approach gives the greatest possible weight to the reduction of the displacement peak over the RMS displacement values. Figure 7: Maximum floor displacement at each story level of the structure equipped with TMDs with \(\mu=0.02\). The greatest reductions in the horizontal peak displacement at the top-story level are attained using TMDs designed through the objective functions OA1 and J4. Using the El Centro earthquake, the horizontal peak displacement is reduced from 0.336 m to 0.280 m (16.60% reduction) using the function OA1 and J4 indistinctly; similar results are observed using the Kobe earthquake, where the peak displacement decreases from 1.243 m to 1.032 m (17% reduction). Now, under the Loma Prieta accelerograms, the maximum displacement reduces from 0.222 m to 0.162 m (27% reduction) using the objective function J4, while, for the Northridge excitation, the maximum displacement is decreased from 1.051 m to 0.980 m (7% reduction) with the OA1 approach. The controlled response of the structure using optimally designed TMDs with \(\mu=0.05\) is remarkable. The best performances are achieved with devices optimized through OA1 and J4 approaches, which are mainly focused on reducing the peak displacements, showing reductions of up to 43 and 42%, respectively.