*4.1. Benchmark Records*

3

4

Four widely known accelerograms in the literature of recorded earthquakes were downloaded from the Pacific Earthquake Engineering Research (PEER) Centre database [64] to simulate the seismic action in the optimization process. The four records are labeled and described in Table 1.

> > 90 DEG

> > > S49W

 **Duration [s]**

 0.40

 0.84  53.73

 30.03

 39.98

 14.97


 Capitolia

 Rinaldi

**Table 1.** Characterization of the benchmark records.

 Loma Prieta

 Northridge

The four accelerograms listed in Table 1 correspond to historical ground motions from different locations and, certainly, all four records present very different dynamic characteristics. Moreover, it is impossible to know a priori future ground motions that will affect the structural system. Therefore, it should be clarified that the intention of this research is not to reproduce a realistic tuning process for TMDs subjected to earthquake loads; on the contrary, and as with other recently proposed methodologies [44–46,51,58,59], this paper's purpose is to prove the efficiency of the WOA to find the best-fit design variables for TMDs to improve the dynamic response of multi-story structures using actual accelerograms as input excitations, and examine extensively the dynamic behavior of the structure under the action of such accelerograms.

#### *4.2. Calibration of the Algorithm*

The main challenge of adapting the WOA methodology is to find a balance between the algorithm performance and the processing time. In that sense, multiple numerical simulations were carried out for the case study, using the objective function J1 and the Kobe excitation. Alternatives of 10, 30, 50, 100, 200, and 500 whales combined with 5, 10, 30, and 50 iterations were considered. Figure 3 depicts the performance index (PI), defined as the ratio between the maximum controlled and uncontrolled response of the system, and the processing time for every simulation.

**Figure 3.** Calibration process of the optimization algorithm.

Furthermore, Table 2 reports the design parameters calculated for each alternative, as a complement to the information detailed in Figure 3.


**Table 2.** Complementary information on the calibration process.

Based on the results presented in Figure 3 and Table 2, it is possible to state that, in terms of performance and time, the most attractive optimization alternatives are those that use five generations and 50 whales (Alternative A), and 10 generations and 30 whales (Alternative B). Comparing the performance of alternatives A and B with the one that presents the best PI reduction, which corresponds to 50 generations and 500 whales (alternative C), diminutions in the processing time of 98.98% and 98.82%, respectively, are obtained.

In the same way, the differences in the PI values of alternatives A and B compared with alternative C are 0.0119% and 0.0124%, respectively. Considering these data and taking into account the stability of the methodology by employing a larger number of generations, alternative B is adopted in this work for the determination of optimal design parameters of TMDs.
