## 4 Optimal Design of Passive Energy Dissipation Systems
### Viscous Dampers (VDs)  
Viscous damping is the dissipation of energy that occurs when a particle in a vibrating system is resisted by a force, the magnitude of which is a constant independent of displacement and velocity, and the direction of which is opposite to the direction of the velocity of the particle. Uniaxial force is a result of a pressure across the piston head. Since the fluid is nearly incompressible, a reduction in fluid volume results in a restoring force, which is prevented by the use of a rod make-up accumulator [12]. Previous research show an increase in temperature can be significant, particularly in long-durations or large-amplitude seismic motions. This temperature is compensated by a relatively small effect through mechanisms [43, 44]. An efficient mathematical model to describe VD behavior (linear or nonlinear) was proposed by [PERSON] and [PERSON] [45] based on experimental results. The force of the damper \(P(t)\) is calculated by the following Equation (6):  
\[P(t)=C_{d}\big{|}\dot{u}(t)\big{|}^{\alpha}\,sgn\big{[}\dot{u}(t)\big{]} \tag{6}\]  
where \(C_{d}\) is the damping coefficient, \(u(t)\) is the displacement across the damper, and \(\alpha\) is a coefficient, depending on the piston head design and viscosity properties of fluid. The coefficient \(\alpha\) is the first important parameter to verify which could be less or equal to 1. Figure 7a describes the force-velocity relationship for linear and nonlinear behavior, while Figure 7b shows the force-displacement hysteretic loops. For earthquake resistance structures, the \(\alpha\) coefficient has a value ranging from 0.3 to 1.0, in order to provide larger forces and to minimize shocks for high velocities with no degradation of performance [46]. In addition, the lowest value needed to maintain a high amount of energy absorbed per cycle of vibration is shown in Figure 7b, and minimizes at the same time the stress at adjacent structural members [47]. So, in our study, an \(\alpha\) value equal to 0.3 has been selected. Velocity is the second important parameter to fix. This is because the VD force varies with velocity, which is related to structural motion and depends on the structural fundamental period. The horizontal flexibility of the structure injects the full movement directly into the horizontal component of the damper, so VD is considered efficient for flexible rather than rigid structures [47]. In this study, a parametric study was done in order to select the correct velocity in accordance with previous studies [47]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**Longitudinal**} & \multicolumn{3}{c}{**Transversal**} \\ \cline{2-7}  & **Undamped** & **With TMD** & **With PTMD** & **Undamped** & **With TMD** & **With PTMD** \\ \hline Fundamental period [s] & 1.983 & 2.091 & 1.801 & 2.202 & 2.322 & 2.67 \\ Top roof displacement [cm] & 38.377 & 36.39 & 25.01 & 12.567 & 10.14 & 9.23 \\ Base shear [kN] & 19,306.7 & 17,476.06 & 16,974.11 & 5222.9 & 5331.14 & 4386.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Response of the high-rise building for the TMD and PTMD configuration. Figure 7: (**a**) Force-velocity behavior of VD and (**b**) Force-displacement hysteretic-loops. The damping coefficient (\(Cd\)) is the third important parameter to define, related to the desired effective damping \(\xi_{eff}\), and attributed to the structure. Design codes do not provide any substantial procedure for the distribution of the calculated damping coefficient over the whole building. These days, a large variety of methods have been proposed, classified between two categories: standard and advanced methods [48; 49].