## 4 Optimal Design of Passive Energy Dissipation Systems
### Tuned Mass Damper
This is why the low-rise building is studied over the two following small mass ratio values 0.5% and 1%, the mid-rise building over 1% and 3%, while the high rise building will be studied for 2% and 3%. As for the installation location of TMD, choosing the best location is the most important factor to consider in the optimal design process to show excellent control performance for the controlling dynamic response [39]. It is important to note that there are not enough studies carried on the installation of TMD in a spatial structure, as well as a lack of data on the optimal installation [40]. The TMD is commonly installed at the center plan to avoid creating torsional effects. In their study on the performance and placement of one or more TMDs in buildings, [PERSON] et al. [41] concluded that the optimum location is near the geometric center of the plan, whether for symmetric or asymmetric buildings. In addition to that, TMD is a motion-based system, which means that the TMD efficiency in reducing structural response is gained by applying it at the story that will experience the most motion. In symmetric buildings, it is usually on the upper story level [42]. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Building Type** & \(\widehat{\mathbf{m}}\) & \(\mathbf{m_{d}}\) & \(\mathbf{k_{d}}\) & \(\mathbf{c_{d}}\) \\ \hline Low-rise & 0.50\% & 7.86 & 6561.47 & 54.63 \\  & 1\% & 15.72 & 12,955.58 & 134.49 \\  & 1\% & 46.27 & 1829.10 & 86.70 \\  & 3\% & 138.8 & 5238.61 & 373.86 \\  & 2\% & 171.2 & 1321.15 & 179.82 \\  & 3\% & 256.8 & 1937.40 & 309.24 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: TMD design parameter values for the three investigated buildings. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{3}{*}{**Building Type**} & \multirow{3}{*}{**Case**} & \multirow{3}{*}{**Direction**} & \multicolumn{2}{c}{**Fundamental Period**} & \multicolumn{2}{c}{**Top Roof Displacement**} & \multicolumn{2}{c}{**Base Shear**} \\ \cline{3-8}  & & & **Value** & **[cm]** & **Margin** & **Value** & **Value** & **Margin** \\ \hline \multirow{8}{*}{Low-rise} & Undamped & Longitudinal & 0.156 & & 0.8556 & & 9536.64 & \\  & Transversal & 0.216 & & 0.4248 & & 2772.68 & \\  & Damped with & Longitudinal & 0.158 & \(-1.28\%\) & 0.8243 & 3.66\% & 9530.28 & 0.07\% \\  & \(\overline{m}=0.5\%\) & Transversal & 0.218 & \(-0.93\%\) & 0.4183 & 1.53\% & 2751.67 & 0.76\% \\  & Damped with & Longitudinal & 0.157 & \(-0.64\%\) & 0.8237 & 3.73\% & 9287.71 & 2.61\% \\  & \(\overline{m}=1\%\) & Transversal & 0.216 & 0.00\% & 0.4164 & 1.98\% & 2693.44 & 2.86\% \\  & Undamped & Longitudinal & 0.697 & & 13.2068 & & 24,870.8 & \\  & Transversal & 0.985 & & 5.4141 & & 4652.36 & \\  & Damped with & Longitudinal & 0.71 & \(-1.87\%\) & 12.6213 & 4.43\% & 23,726.16 & 4.60\% \\  & \(\overline{m}=1\%\) & Transversal & 0.997 & \(-1.22\%\) & 5.3094 & 1.93\% & 4567.02 & 1.83\% \\  & Damped with & Longitudinal & 0.733 & \(-5.16\%\) & 12.5036 & 5.32\% & 22,507.32 & 9.50\% \\  & \(\overline{m}=3\%\) & Transversal & 1.031 & \(-4.67\%\) & 4.7931 & 11.47\% & 4559.55 & 1.99\% \\  & Undamped & Longitudinal & 1.983 & & 38.377 & & 19,306.7 & \\  & Transversal & 2.202 & & 12.5677 & & 5222.92 & \\  & Damped with & Longitudinal & 2.058 & \(-3.78\%\) & 35.9845 & 6.23\% & 18,077.8 & 6.37\% \\  & \(\overline{m}=2\%\) & Transversal & 2.284 & \(-3.72\%\) & 10.2574 & 18.38\% & 5112.19 & 2.12\% \\  & Damped with & Longitudinal & 2.091 & \(-5.45\%\) & 36.3981 & 5.16\% & 17,476.06 & 9.48\% \\  & \(\overline{m}=3\%\) & Transversal & 2.322 & \(-5.45\%\) & 10.1373 & 19.34\% & 5331.14 & \(-2.07\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Responses of the three investigated buildings for the two different mass ratio of TMD with margin from undamped case. The margins in Table 5 are calculated according to the undamped case, which means that a reduction is detected for positive values. The negative margins mean that there is an increase in parameter values after installing the TMD.