## 4 Optimal Design of Passive Energy Dissipation Systems
### Tuned Mass Damper
This configuration type can be represented by an equivalent SDOF system attached to the story, as shown in Figure 5.  
With the pendulum configuration [36], the equivalent stiffness is given by \(k_{eq}=m_{d}\cdot g/L\), the natural frequency is expressed by \(\omega_{d}=\sqrt{k_{eq}/m_{d}}=\sqrt{g/L}\) and the natural period is set by \(T_{d}=2\cdot\pi\cdot\sqrt{L/g}\). The tuning parameters of PTMD are the mass m\({}_{d}\) and the length \(L\). This configuration is advantageous over the conventional TMD, especially for high-rise buildings, because its frequency can be retuned easily by modifying the cable length [37]. Figure 4: Dynamic amplification factor as a function of \(\lambda\). The first important parameter in the optimal design process of the TMD system is the mass ratio \(\overline{\mu}\). It is well known that the structural response decreases as \(\overline{\mu}\) increases, but this ratio has a limit in practice which must not exceed 10% [34] based on geometrical and economical constraints. Then, optimized absorber parameters are calculated. All real systems contain some inherent damping, meaning that an absorber is added to a lightly damped system. The effect of the inherent damping in the real system is an important design consideration on the optimum tuning parameters of TMD. Table 2 shows the equations used in the present study for \(v\) and \(\xi_{d}\), which includes the structural damping ratio of the primary structural system \(\xi\), the derived rigidity \(k_{d}\), and damping coefficient \(c_{d}\) of the TMD, respectively [38]. Also, Table 3 shows the structure and TMD optimal parameter values taken for the three investigated buildings, noting that \(T\) is the fundamental period of the structure in the transversal direction in which the TMD is applied. In order to choose the most appropriate mass ratio, two analyses will be carried out for two mass ratios per building (Table 4). The structural response will be compared to the undamped case in Table 5. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{5}{c}{**Expressions**} \\ \hline Optimal tuning parameters of TMD given in [38] & \(v=\frac{1}{1+\overline{m}}\left[1-\xi\cdot\sqrt{\frac{\overline{m}}{1+\overline {m}}}\right]\) & (2) \\  & \(\xi_{d}=\frac{\xi}{1+\overline{m}}+\sqrt{\frac{\overline{m}}{1+\overline{m}}}\) & (3) \\ \hline Optimized absorber parameter & \(k_{d}=\omega_{d}{}^{2}\cdot m_{d}=v^{2}\cdot\omega^{2}\cdot\overline{m}\cdot m\) & (4) \\  & \(c_{d}=2\cdot\xi_{d}\cdot\omega_{d}\cdot m_{d}=2\cdot\xi_{d}\cdot\upsilon\cdot \omega\cdot\overline{m}\cdot m\) & (5) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: TMD optimized parameters expressions. Figure 5: PTMD schematic representation. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Building Type**} & \multicolumn{2}{c}{**Structure**} & \multicolumn{2}{c}{**TMD**} \\ \cline{2-6}  & **Parameter** & **Value** & **Parameter** & **Values** \\ \hline Low-rise & \(T\) [s] & 0.216 & \(\overline{m}=0.5\%\) & \(v=0.9915\) & \(\xi_{d}=12.03\%\) \\  & \(m\) [t] & 1572.29 & \(\overline{m}=1\%\) & \(v=0.9852\) & \(\xi_{d}=14.90\%\) \\ Mid-rise & \(T\) [s] & 0.985 & \(\overline{m}=1\%\) & \(v=0.9852\) & \(\xi_{d}=14.90\%\) \\ Mid-rise & \(m\) [t] & 4626.92 & \(\overline{m}=3\%\) & \(v=0.9626\) & \(\xi_{d}=21.92\%\) \\ High-rise & \(T\) [s] & 2.202 & \(\overline{m}=2\%\) & \(v=0.9735\) & \(\xi_{d}=18.90\%\) \\  & \(m\) [t] & 8560.39 & \(\overline{m}=3\%\) & \(v=0.9626\) & \(\xi_{d}=21.92\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Structure-TMD optimal parameter values. Flexible buildings undergo larger horizontal displacements, which may result in significant damages. In this case, it is preferable to choose a relatively big mass ratio, unlike rigid buildings, for which it is advisable to adopt a relatively small ratio, because increasing its value does not provide any additional damping effect.