# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 4 DDS Designs
The optimal solution attained by the GA solver for DC2 includes the allocation list  
\[\overline{\alpha}=[\overbrace{1,1,1,1,0}^{\hat{\mathbb{A}}^{(2)}},\;\overbrace {0,1,1,1,0}^{\hat{ \mathbb{A}}^{(4)}},\;\overbrace{0,0,1}^{\hat{\mathbb{A}}^{(1)}},\;\overbrace {0,0,1}^{\hat{\mathbb{A}}^{(2)}},\;\overbrace{0,0,1}^{\hat{ \mathbb{A}}^{(3)}},\;\overbrace{1,1,0}^{\hat{\mathbb{A}}^{(4)}}], \tag{76}\]  
and the damping coefficients presented in the second row of Table 1. \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Conf.** & \(\mathbf{d_{1}}\) & \(\mathbf{d_{2}}\) & \(\mathbf{d_{3}}\) & \(\mathbf{d_{4}}\) & \(\mathbf{d_{5}}\) & \(\mathbf{d_{6}}\) & \(\mathbf{d_{7}}\) & \(\mathbf{d_{8}}\) & \(\mathbf{d_{9}}\) & \(\mathbf{d_{10}}\) & \(\mathbf{d_{11}}\) & \(\mathbf{d_{12}}\) \\ \hline DC1 & 1.4480 & 1.5137 & 1.3414 & 2.0098 & 2.1014 & 1.9725 & 1.5460 & 1.3074 & 0.4334 & 0.3683 & 0.8023 & 0.1528 \\ DC2 & 1.8626 & 1.9882 & 1.7986 & 1.6611 & 2.1895 & 1.9489 & 1.6484 & 0.6311 & 0.4362 & 0.1519 & 0.0006 & 0.6828 \\ DC3 & 2.6792 & 2.7151 & 2.5843 & 2.1407 & 0.2790 & 0.9873 & 0.7835 & 0.0009 & 1.3963 & 0.9310 & 0.0007 & 0.5018 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of the damping coefficients corresponding to the linked damping configurations DC1, DC2 and DC3 (\(\times 10^{7}\) Ns/m). Figure 5: Damping configuration DC1. Full-linked distributed damping system (DDS) with three actuated buildings, \(\hat{n}=8\) interstory dampers and \(\hat{n}=4\) interbuilding dampers. Finally, we consider the third damping configuration (DC3) displayed in Figure 7, which allows placing interbuilding dampers at all interbuilding positions and restricts interstory damper implementation to building \(\mathcal{B}^{(1)}\). For that case, the schemes of allowed interstory and interbuilding damping positions are  
\[\vartheta^{(1)}=[1,2,3,4,5],\ \vartheta^{(2)}=[\ ],\ \vartheta^{(3)}=[\ ],\ \vartheta^{(4)}=[\ ],\ \vartheta^{(5)}=[\ ] \tag{77}\]  
and  
\[\vartheta^{(1)}=[1,2,3,4,5],\ \vartheta^{(2)}=[1,2,3,4,5],\ \vartheta^{(3)}=[1,2,3,4,5],\ \vartheta^{(4)}=[1,2,3,4,5], \tag{78}\]  
respectively, and the total number of optimization variables is \(37\) with \(\overline{\eta}=25\) binary damper-allocation variables and \(n_{d}=12\) continuous damping-coefficient variables. The optimal allocation list obtained for DC3 is  
\[\overline{\alpha}=[\overbrace{1,1,1,1,0}^{\hat{\alpha}^{(1)}},\ \overbrace{0,0,1,1,1}^{\hat{\alpha}^{(1)}},\ \overbrace{0,0,1,0,1}^{\hat{\alpha}^{(2)}},\ \overbrace{0,0,0,0,1}^{\hat{\alpha}^{(3)}},\ \overbrace{0,1,0,0,1}^{\hat{\alpha}^{(4)}}], \tag{79}\]  
and the corresponding damping coefficients are collected in the third row of Table 1.  
Figure 6: Damping configuration DC2. Full-linked DDS with two actuated buildings, \(\hat{n}=7\) interstory dampers and \(\hat{n}=5\) interbuilding dampers. Looking at the damping coefficient values in Table 1 and the structure of the optimal damper placements in Figures 5-7, the following facts can be observed: (i) damping coefficients of interbuilding dampers are about one order of magnitude lower than those obtained for interstory dampers, (ii) interstory dampers tend to be placed in low building levels, and (iii) interbuilding dampers are preferably allocated at upper interbuilding positions. These facts are consistent with the results obtained in preliminary works on DDSs for MBSs [22; 23]. It can also be observed in Table 1 that there are some dampers with particularly small damping coefficients. Specifically, damping coefficients \(d_{11}\) in DC2 and \(d_{8}\) and \(d_{11}\) in DC3 are remarkably small when compared with the coefficient values of all the other interstory and interbuilding dampers. It should be noted that those small coefficients correspond to dampers placed at particularly low interbuilding positions and can be interpreted as a numerical side-effect of constraining the overall number of dampers to exactly \(n_{d}=12\) elements.