## 4 Numerical Examples  
A ten-story shear building [36] was investigated for an optimum ATMD at the top. The structure has the same properties for each floor. The design constants (structural properties), the ranges of design variables, and the algorithm parameters are listed with values in Table 1. In addition, constant ATMD parameters are included in Table 1[14]. The mass of TMD was searched in a range with a minimum 1% and maximum 5% of the total mass of the structure. The range of T\({}_{\text{atmd}}\) is defined as 0.5 and 1.5 times the critical period of the uncontrolled structure. As seen from the ranges of the PID parameter, any specific range must not be found before the optimization due to modification done in Matlab Simulink, as explained in Section 2. The optimization process is done for three different st_max values such as 2, 3 and 4. Figure 4: The flowchart of the optimization process. During the optimization process, 22 earthquake records were used. These records have 44 excitations (two in the longitudinal direction for each record), and these records are grouped as far-field ground motion records in FEMA P-695: Quantification of Building Seismic Performance Factors [37]. These records were also considered in various optimum structural control methodologies [38; 39; 40; 41]. The information of excitations is listed in Table 2. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Definition** & **Value** & **Unit** \\ \hline m\({}_{\text{i}}\) & 360 & tons \\ k\({}_{\text{i}}\) & 650 & MN/m \\ c\({}_{\text{i}}\) & 6.2 & MNs/m \\ m\({}_{\text{d}}\) & 36–180 & tons \\ T\({}_{\text{atmd}}\) & 0.495–1.484 & s \\ \(\xi_{\text{d}}\) & 1–50 & \% \\ K\({}_{\text{p}}\) & (\(-\)10,000)–(10,000) & Ns/m \\ T\({}_{\text{d}}\) & (\(-\)10,000)–(10,000) & s \\ T\({}_{\text{i}}\) & (\(-\)10,000)–(10,000) & s \\ st\({}_{\text{max}}\) & 2–3–4 & - \\ HMS & 10 & - \\ HMCR\({}_{\text{in}}\) & 0.5 & - \\ PAR\({}_{\text{in}}\) & 0.05 & - \\ BSCR & 0.3 & - \\ MI & 5000 & - \\ R & 4.2 & \(\Omega\) \\ K\({}_{\text{f}}\) & 2 & N/A \\ K\({}_{\text{e}}\) & 2 & V \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The numerical data for structure case. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Earthquake** & **Date** & **Name** & **Component 1** & **Component 2** \\ Number & & & \\ \hline
1 & 1994 & Northridge & NORTHR/MUL009 & NORTHR/MUL279 \\
2 & 1994 & Northridge & NORTHR/LOS000 & NORTHR/LOS270 \\
3 & 1999 & Duzce, Turkey & DUZCE/BOL000 & DUZCE/BOL090 \\
4 & 1999 & Hector Mine & HECTOR/HEC000 & HECTOR/HEC090 \\
5 & 1979 & Imperial Valley & IMPVALL/H-DLT262 & IMPVALL/H-DLT352 \\
6 & 1979 & Imperial Valley & IMPVALL/H-E11140 & IMPVALL/H-E11230 \\
7 & 1995 & Kobe, Japan & KOBE/NIS000 & KOBE/NIS090 \\
8 & 1995 & Kobe, Japan & KOBE/H1000 & KOBE/SHI090 \\
9 & 1999 & Kocaeli, Turkey & KOCAEIL/DZC180 & KOCAEIL/DZC270 \\
10 & 1999 & Kocaeli, Turkey & KOCAEIL/ARC000 & KOCAEIL/ARC090 \\
11 & 1992 & Landers & LANDERS/VER270 & LANDERS/YER360 \\
12 & 1992 & Landers & LANDERS/CLW-LN & LANDERS/CLW-TR \\
13 & 1989 & Loma Prieta & LOMAP/CAP000 & LOMAP/CAP090 \\
14 & 1989 & Loma Prieta & LOMAP/G03000 & LOMAP/G03090 \\
15 & 1990 & Manjil, Iran & MANJIL/ABBAR-L & MANJIL/ABBAR-T \\
16 & 1987 & Superstition Hills & SUPERST/B-ICC000 & SUPERST/B-ICC090 \\
17 & 1987 & Superstition Hills & SUPERST/B-POE270 & SUPERST/B-POE360 \\
18 & 1992 & Cape Mendocino & CAPEMEND/RIO270 & CAPEMEND/RIO360 \\
19 & 1999 & Chi-Chi, Taiwan & CHICHI/CHY101-E & CHICHI/CHY101-N \\
20 & 1999 & Chi-Chi, Taiwan & CHICHI/TCU045-E & CHICHI/TCU045-N \\
21 & 1971 & San Fernando & SFERN/PEL090 & SFERN/PEL180 \\
22 & 1976 & Friuli, Italy & FRIULI/A-TMZ000 & FRIULI/A-TMZ270 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: FEMA P-695 far-field ground motion records [33]. The optimum values of design variables, objective function and design constraints are given as Table 3. In Table 3, the optimum results of passive TMD are also given.