# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 5 Seismic Responses  
To illustrate the behavior of the different DDSs designed in Section 4, we have carried out a proper set of numerical simulations using the full-scale 180-component of El Centro 1940 seismic record as ground acceleration disturbance (see Figure 8). Specifically, for the nonactuated MBS and the damping configurations DC1, DC2 and DC3, we have computed the maximum absolute interstory drifts  
\[|\dot{r}_{i}^{j}|_{\max}=\max_{0\leq t\leq T_{w}}|\dot{r}_{i}^{j}(t)|,i=1,\ldots,5,\,j=1,\ldots,5, \tag{80}\]  
the maximum absolute story total-accelerations  
\[|\dot{a}_{i}^{j}|_{\max}=\max_{0\leq t\leq T_{w}}|\dot{a}_{i}^{j}(t)|,i=1, \ldots,5,\,j=1,\ldots,5 \tag{81}\]  
and the maximum interbuilding approachings \begin{table}
\begin{tabular}{l c c c c c c c} \hline \hline
**Conf.** & **Act. Build.** & \(H_{\infty}\) **Norm** & **opt. vars.** & **Generations** & **funct. aval.** & **Time (s)** & **Parallel Time (s)** \\ \hline Plain conf. & — & 0.8090 & — & — & — & — & — \\ DC1 & \(1,3,5\) & 0.0897 & 47 & 206 & 41,400 & 273.5 & 67.9 \\ DC2 & \(2,4\) & 0.0970 & 34 & 161 & 32,400 & 242.8 & 57.5 \\ DC3 & \(1\) & 0.1457 & 37 & 288 & 57,800 & 409.0 & 103.8 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Computational design characteristics of the linked DDS configurations DC1, DC2 and DC3. \[(\vec{r}_{i}^{j})_{\text{max}}=\max_{0\leq t\leq T_{w}}\vec{r}_{i}^{j}(t),i=1, \ldots,5,\,j=1,\ldots,4, \tag{82}\]  
where \(\vec{r}_{i}^{j}(t)\), \(a_{i}^{j}(t)\) and \(\vec{r}_{i}^{j}(t)\) are the output variables defined in Equations (33), (37) and (40), respectively, and \(T_{w}\) denotes the total duration of the seismic disturbance. The obtained interstory-drift peak-values are presented in Figure 9. A global view of the plots in that figure indicates that the three designed damping configurations are all able to produce an overall reduction of the interstory-drift peak-values when compared with the response of the nonactuated MBS (black solid lines with rectangles). In a more detailed inspection of the DC1 response (red solid lines with circles), it can be appreciated that the interstory-drift peak-value reduction is particularly effective in the actuated buildings \(\mathcal{B}^{(1)}\), \(\mathcal{B}^{(3)}\) and \(\mathcal{B}^{(5)}\). A slightly poorer performance can be observed in the nonactuated buildings \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(4)}\), whose seismic protection is provided through the linking interbuilding dampers (see Figure 5). A similar behavior can be observed in the response of the damping configuration DC2 (blue dashed lines with asterisks), where \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(4)}\) are actuated buildings and \(\mathcal{B}^{(1)}\), \(\mathcal{B}^{(3)}\) and \(\mathcal{B}^{(5)}\) are nonactuated (see Figure 6). In this case, it is worth noting the loss of performance in the upper level of building \(\mathcal{B}^{(2)}\), which is an effect that has been observed in previous works [22] and can be associated to the action of the interbuilding links. Finally, for the single-actuated-building configuration DC3 (green dotted lines with triangles), the best results are also attained in the actuated building \(\mathcal{B}^{(1)}\), and a moderate but gradual increase of the interstory-drift peak-values can be observed in the nonactuated buildings as we move away from \(\mathcal{B}^{(1)}\). Also in this case, a loss of performance associated to the interbuilding links can be appreciated in the upper level of the actuated building \(\mathcal{B}^{(1)}\). The obtained story total-acceleration peak-values and maximum interbuilding approaches are displayed in Figures 10 and 11, respectively, using the same colors, line styles and symbols. The plots in Figure 10 confirm that, despite having only included the interstory drifts in the optimization index, the considered \(H_{\infty}\) design approach can produce a notable reduction of the acceleration peak-values. That reduction is more relevant in the actuated buildings and smaller but yet significant in the nonactuated ones.