## 4 Numerical Example
### Seismic Performance
Nevertheless, the reductions in the maximum displacements at each floor achieved by the devices designed with the remaining objective functions are notable as well, exhibiting reductions up to 39, 38, 38, and 40% for the cases of optimization OA2, J1, J2, and J3, respectively. Once more, using the Northridge earthquake, the smallest reductions obtained ranged from 12 to 17% at the higher story-levels of the structure. Besides, for the three remaining accelerograms, the reductions in the maximum floor displacement oscillate between 20 and 43%. Figure 8: Maximum floor displacement at each story level of the structure equipped with TMDs with \(\mu=0.05\). Furthermore, the dynamic response of displacements observed in Figures 7 and 8 denotes an S-shape trend. The curve shows a linear behavior between 1 st and 20 th story-levels; then, there is a setback in the curve between 27 th and 32 nd story-levels, where it becomes linear again, although with a different inclination from the first segment representing the greatest response reductions. This behavior is closely related to the whipping lash effect that has been previously observed in high rise buildings [66]. This behavior is largely caused by changes in the compressive strength in the concrete used in the column sections of the building, which causes changes in stiffness and the dynamic behavior of the structural system. On the other hand, Figures 9 and 10 present the behavior of the RMS values of the displacements at each level of the building controlled by TMDs with mass ratios \(\mu\) = 0.02 and \(\mu\) = 0.05, respectively. Figure 9: RMS displacement at each story level of the structure equipped with TMDs with \(\mu\) = 0.02. Figure 9 shows reductions fluctuating from 19% (Northridge excitation at 23 rd story-level) and 42% (Kobe excitation at 23 rd story-level). Besides, the reductions in the RMS response of displacement of the roof story-level are 12, 31, 24, and 17% for the El Centro, Kobe, Loma Prieta, and Northridge accelerograms, respectively. The analysis of the RMS response of displacement curves at each floor allows the establishment that the control devices designed through the OA2, J1, and J2 approaches present the best performances when the RMS of displacements is analyzed; accordingly, TMDs designed with objective functions OA1 and J4 achieve lower reductions. The best reductions in the RMS values of the displacements at each level of the structure are obtained when the building is equipped with TMDs with \(\mu\) = 0.05. These reductions are attained using the OA2 and J1 approaches: 20, 42, 33, and 26% for the El Centro, Kobe, Loma Prieta, and Northridge accelerograms, respectively. Among these reductions, the most outstanding is attained under Kobe excitation, since the reductions between 15 th to 32 nd story-levels ranged between 40% (0.286 m to 0.170 m at 15 th story-level) and 52% (0.334 m to 0.159 m at the 22 nd story-level). It can be noticed that the reductions in the RMS response of displacement at each floor of the structure exhibit a much more uniform and efficient behavior than the reductions of the maximum horizontal peak floor displacements, especially with \(\mu\) = 0.05 TMDs, and objective functions OA2, which focus on reducing the RMS displacements, and J1, which gives the greatest weight to this parameter. Using the Kobe and Northridge records, the  
Figure 10: RMS displacement at each story level of the structure equipped with TMDs with \(\mu\) = 0.05.  
results report the highest response magnitudes and, at the same time, the TMDs exhibit the greatest performance. This behavior suggests that the objective functions for the optimal design of TMDs should be focused on minimizing the RMS values of displacements, or, instead, a linear combination in which the greatest weight is given to this parameter should be used. To verify this statement, Figures 11 and 12 present a comparative analysis of the six objective functions and the attained PI.