# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 4 DDS Designs  
To illustrate the effectiveness and flexibility of the proposed design methodology, in this section we design three different DDSs for the seismic protection of a MBS formed by a row of \(m=5\) identical five-story buildings with story masses (\(\times 10^{5}\) kg) \(m_{1}=2.152\), \(m_{2}=2.092\), \(m_{3}=2.070\), \(m_{4}=2.048\), \(m_{5}=2.661\) and story stiffness coefficients (\(\times 10^{8}\) N/m) \(k_{1}=1.47\), \(k_{2}=1.13\), \(k_{3}=0.99\), \(k_{4}=0.89\) and \(k_{5}=0.84\)[35]. The building damping matrix (in Ns/m)  
\[\mathbf{C}_{b}=10^{5}\times \begin{bmatrix}2.602&-0.924&0&0&0\\ -0.924&2.196&-0.810&0&0\\ 0&-0.810&1.995&-0.728&0\\ 0&0&-0.728&1.867&-0.687\\ 0&0&0&-0.687&1.274\end{bmatrix}  \tag{70}\]  
has been computed as a Rayleigh damping matrix with 2% of relative damping in the first and fifth modes [32]. The DDSs damping devices are assumed to be linear FVDs with maximum damping capacity \(c_{\text{max}}=3.0\times 10^{7}\) Ns/m. The maximum DDS damping capacity has been set to \(c_{\text{tot}}=1.5\times 10^{8}\) Ns/m and the total number of damping devices has been restricted to \(n_{d}=12\). Also, a full-linked configuration has been enforced on all the considered DDSs by forbidding empty interbuilding damping systems. For the first damping configuration (DC1), we select the schemes of allowed interstory damper positions  
\[\vartheta^{(1)}=[1,2,3,4,5],\ \vartheta^{(2)}=[\,],\ \vartheta^{(3)}=[1,2,3,4,5],\ \vartheta^{(4)}=[\,],\ \vartheta^{(5)}=[1,2,3,4,5] \tag{71}\]  
and allowed interbuilding damper positions  
\[\vartheta^{(1)}=[1,2,3,4,5],\ \vartheta^{(2)}=[1,2,3,4,5],\ \vartheta^{(3)}=[1,2,3,4,5],\ \vartheta^{(4)}=[1,2,3,4,5], \tag{72}\]  
which are schematically represented in Figure 5 by blue dashed and red dotted rectangles, respectively. That configuration keeps \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(4)}\) as nonactuated buildings and allows implementing interstory dampers at all the interstory levels of buildings \(\mathcal{B}^{(1)}\), \(\mathcal{B}^{(3)}\) and \(\mathcal{B}^{(5)}\), and interbuilding dampers at all interbuilding positions. In the optimization variable \(v=[\overline{\alpha},d]\), the list of dampers allocations \(\overline{\alpha}\) contains \(\overline{\eta}=35\) binary variables, and the list of damping coefficients \(d\) includes \(n_{d}=12\) continuous variables. For DC1, the solution provided by the GA solver to the optimization problem in Equation (69) includes the optimal allocation list  
\[\overline{\alpha}=[\,\overbrace{1,1,0,0}^{\hat{\alpha}^{(1)}},\,\overbrace{1, 1,0,0}^{\hat{\alpha}^{(3)}},\,\overbrace{0,1,1,0,0}^{\hat{\alpha}^{(5)}},\, \overbrace{0,0,0,1,0}^{\hat{\alpha}^{(1)}},\,\overbrace{0,0,1,0,0}^{\hat{ \alpha}^{(2)}},\,\overbrace{0,0,1,0,0}^{\hat{\alpha}^{(3)}},\,\overbrace{0,0, 1,0,0}^{\hat{\alpha}^{(4)}},\,\overbrace{0,0,1,0,0}^{\hat{\alpha}^{(4)}},\, \overbrace{0,0,1,0,0}^{\hat{\alpha}^{(4)}},\,}\,, \tag{73}\]  
which corresponds to the system of interstory and interbuilding dampers displayed as blue and red small dashpots in Figure 5, respectively, and the damping coefficients collected in the first row of Table 1.  
The second damping configuration (DC2), schematically displayed in Figure 6, allows placing dampers in the interstory positions  
\[\hat{\sigma}^{(1)}=[\;],\;\hat{\sigma}^{(2)}=[1,2,3,4,5],\;\hat{\sigma}^{(3)}=[\; ],\;\hat{\sigma}^{(4)}=[1,2,3,4,5],\;\hat{\sigma}^{(5)}=[\;] \tag{74}\]  
and the interbuilding positions  
\[\hat{\sigma}^{(1)}=[3,4,5],\;\hat{\sigma}^{(2)}=[3,4,5],\;\hat{\sigma}^{(3)}=[3,4,5],\;\hat{\sigma}^{(4)}=[3,4,5], \tag{75}\]  
which constrains the placement of interstory dampers to buildings \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(4)}\) and restricts the placements of interbuilding dampers to the upper three interbuilding levels. In this second case, the total number of optimization variables is reduced to 34, with \(\overline{\eta}=22\) binary allocation variables and \(n_{d}=12\) continuous damping-coefficient variables.