## 4 Optimal Design of Passive Energy Dissipation Systems
### Viscous Dampers (VDs)
In the present study, the damping coefficient is distributed along the height of the building, based on the proportionality respective of the story shear force (Equation (7)); the effective damping \(\xi_{eff}\) is a sum of the structural inherent damping ratio (\(\xi_{0}\)) and the damping ratio of the viscous dampers (\(\xi_{d}\)), according to [50] recommendations (see Equation (8)). \[C_{d,i}=\frac{V_{i}}{\sum V_{i}}\sum C_{j} \tag{7}\]  
\[\xi_{eff}=\xi_{0}+\xi_{d}=\xi_{0}+\frac{\sum\lambda C_{j}\phi_{rj}^{1+\alpha} \cos^{1+\alpha}\theta_{j}}{2\pi A^{1-\alpha}\omega^{2-\alpha}\sum M_{i}\phi_{i }{}^{2}} \tag{8}\]  
\[\lambda=2^{2+\alpha}\frac{\Gamma^{2}\left(1+\frac{\alpha}{2}\right)}{\Gamma \left(1+\alpha\right)} \tag{9}\]  
where \(A\) is the amplitude, \(\phi_{rj}\) is the relative horizontal displacement of the damper, \(\theta_{j}\) is the inclined angle of the damper \(j\), \(\omega\) is the loading frequency supposed equal to the natural structural frequency, \(M_{i}\) is the vibrating mass of the story \(i\), \(\phi_{i}\) is the modal displacement at story i, and \(\lambda\) is a parameter calculated by Equation (9) [51]. [PERSON] [52] indicates that in order to establish the optimal effective damping, nonstructural elements must be taken into account. To have an essential damping ratio-repair cost relationship, the range of optimal effective damping is identified as \(30-40\%\) to minimize mean economic losses. However, the optimal damping amount also depends on the building's properties, such as the fundamental period of structure. Table 8 shows the selected, effective damping and velocities values, as well as the calculated damping coefficient. The design optimization of VD is not limited only on the mechanical parameters of VDs, but also on the position of the dampers in the plan-view of the building. It is important to ensure that the dampers are located in a configuration that does not introduce eccentricity to the structure; this is why the most efficient placement would be equivalently about the building's center of mass, for example, along the perimeter of typical structures. The main reason is to be able to control any torsional motion of the building [47]. In our study, the VDs are positioned in steel diagonal braces, half of them working under compression, and the other under tension. Different configurations of VD's placement were studied, while two of them are illustrated in the Figure 8. At least two dampers were positioned in each direction and on each side of the building's center mass at every story, even though it is not required. It could be terminated before the top levels or alternated at different story levels. Moreover, to limit damper force output, more than two dampers per direction could be used, especially for buildings with large footprints. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Building Type** & **Direction** & \begin{tabular}{c} **Fundamental** \\ **Period [s]** \\ \end{tabular} & \begin{tabular}{c} **Structural Rigidity** \\ **Description** \\ \end{tabular} & \begin{tabular}{c} **Suggested** \\ **Velocity [m/s]** \\ \end{tabular} & \
\begin{tabular}{c} **Suggestedeffective** \\ **Damping \(\xi_{eff}\)** \\ \end{tabular} &
\begin{tabular}{c} \(\Sigma C_{j}\) \\ **[kN-(s/m)]** \\ \end{tabular} \\ \hline \multirow{2}{*}{Low-rise} & Longitudinal & 0.156 & \multirow{2}{*}{Rigid} & 0.127 & 30\% & 71,537.07 \\  & Transversal & 0.216 & & 0.127 & 30\% & 59,405.58 \\ \multirow{2}{*}{Mid-rise} & Longitudinal & 0.697 & \multirow{2}{*}{Semi-rigid} & 0.254 & 35\% & 56,981.16 \\  & Transversal & 0.985 & & 0.254 & 35\% & 39,325.16 \\ \multirow{2}{*}{High-rise} & Longitudinal & 1.983 & \multirow{2}{*}{Flexible} & 0.381 & 40\% & 12,658.71 \\  & Transversal & 2.202 & & 0.381 & 40\% & 12,095.19 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Effective damping and calculated damping coefficient for the three investigated buildings (\(\alpha=0.3\)).