## 4 Optimal Design of Passive Energy Dissipation Systems
### Friction Dampers (FDs)  
FD is a displacement-based system which dissipates energy through friction across the surfaces between two solid elements [1; 3]. The dissipative mechanism generates heat through dry sliding friction with a stable hysteretic behavior [53]. A simple model for defining the behavior of the damper is given by the idealized Coulomb model of friction. The theory is based on the following hypotheses, which are experimentally validated [19]:  
* Force independent of the apparent contact surface  
Figure 8: (**a**) Alternative 1 of dampers placement and (**b**) Alternative 2 of dampers placement. Figure 9: Modeling configuration of VDs. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Building Type**} & \multirow{2}{*}{**Direction**} & \multicolumn{2}{c}{**Fundamental Period**} & \multicolumn{2}{c}{**Top Roof Displacement**} & \multicolumn{2}{c}{**Base Shear**} \\  & & **[s]** & **[cm]** & **[km]** & **[km]** \\  & & **Altern. 1** & **Altern. 2** & **Altern. 1** & **Altern. 2** & **Altern. 1** & **Altern. 2** \\ \hline \multirow{3}{*}{Low-rise} & Longitudinal & 0.112 & 0.112 & 0.167 & 0.176 & 1973.2 & 1905.6 \\  & Transversal & 0.148 & 0.147 & 0.218 & 0.345 & 2308.6 & 2070.0 \\ \cline{1-1}  & Longitudinal & 0.387 & 0.387 & 3.121 & 3.763 & 1888.1 & 7567.0 \\ \cline{1-1}  & Transversal & 0.775 & 0.775 & 2.997 & 1.760 & 4991.5 & 83.3 \\ \cline{1-1}  & Longitudinal & 0.543 & 0.543 & 5.812 & 6.821 & 10.828 & 322.5 \\ \cline{1-1}  & Transversal & 0.989 & 0.989 & 4.604 & 4.803 & 178.6 & 14,737.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Responses of the three investigated buildings for the two alternatives of VDs placement. * Force proportional to the total normal force acting through the interface
* Force independent of speed even with a slip at low speed  
As a result, the force can be written using the following expression \(F_{t}=\mu\cdot F_{n}\), where \(F_{t}\) and \(F_{n}\) represent the frictional and normal forces, respectively, and \(\mu\) the coefficient of friction which depends on the selection of sliding materials and present conditions of the sliding interface. \(F_{n}\) and \(\mu\) are maintained at constant values over extended durations of time, which is difficult to achieve in practice [3]. The damper hysteresis loop is rectangular, showing a great amount of energy dissipated per cycle of motion, and the cyclic behavior of FD is strongly nonlinear, as shown in Figure 10a. When the friction force is overcome, FD adds initial stiffness to the structural system. It is important to note that if no restoring force is provided, permanent structural deformation may exist after an earthquake [3]. As shown in Figure 10b, the response of the structure is highly affected by FD slip force, and a small variation of FD optimum slip load has a minimum effect on structure's response. The selected slip force must be high enough to prevent the damper from slipping under a small applied lateral loads value, and should be low enough to achieve slip before the yielding of the main structural elements [54]. A simple method used in the present study consists of taking a portion from the applied shear force, so the load at each story is estimated by the Equation (10)  
\[F_{t,\;optimal}=\frac{1}{3}[V_{i}/n_{i}] \tag{10}\]  
where \(F_{t,opt}\) is the optimal slip force or frictional force, \(V_{i}\) is the shear load, and \(n_{i}\) is the number of dampers per direction in the story \(i\). The shear load was calculated from the results of the Fast Nonlinear Analysis (FNA). For the diagonal configuration to the damper-brace assembly, it is clear that the device and the brace are connected in series.