# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 2 Mathematical Model
### Interstory and Interbuilding Damping Models  
The dynamical response of building \(\mathcal{B}^{(j)}\) can be improved by introducing a system of \(0<\hat{n}_{j}\leq n\) additional interstory dampers \(\widehat{\mathcal{C}}^{(j)}=[\hat{p}^{(j)},\hat{c}^{(j)}]\), where \(\hat{c}^{(j)}=[\hat{c}^{j}_{1},\ldots,\hat{c}^{j}_{\hat{n}_{j}}]\) is the list of damping coefficients and \(\hat{p}^{(j)}=[\hat{p}^{j}_{1},\ldots,\hat{p}^{j}_{\hat{n}_{j}}]\) is a list that contains the story levels at which the dampers are implemented. The dynamical response of building \(\mathcal{B}^{(j)}\) equipped with the additional interstory damping system \(\widehat{\mathcal{C}}^{(j)}\) can be modeled in the form  
\[\mathbf{M}_{b}\hat{\mathbf{q}}^{(j)}(t)+\big{[}\mathbf{C}_{b}+\widehat{ \mathbf{C}}^{(j)}\big{]}\hat{\mathbf{q}}^{(j)}(t)+\mathbf{K}_{b}\mathbf{q}^{( j)}(t)=-\mathbf{M}_{b}[\mathbf{1}]_{n\times 1}\hat{w}(t), \tag{8}\]  
where the damping matrix \(\widehat{\mathbf{C}}^{(j)}\) can be computed as  
\[\widehat{\mathbf{C}}^{(j)}=\widehat{\mathbf{P}}^{(j)}\operatorname{diag}[ \hat{c}^{j}_{1},\ldots,\hat{c}^{j}_{\hat{n}_{j}}]\left\{\widehat{\mathbf{P}}^ {(j)}\right\}^{T} \tag{9}\]  
by considering the diagonal matrix \(\operatorname{diag}[\hat{c}^{j}_{1},\ldots,\hat{c}^{j}_{\hat{n}_{j}}]\) and the location matrix \(\widehat{\mathbf{P}}^{(j)}\in\mathbb{R}^{n\times\hat{n}_{j}}\) formed by the columns of the matrix \(\mathbf{P}\) indicated in the placement list \(\hat{p}^{(j)}\). Thus, for example, the system of additional interstory dampers in Figure 2 contains \(\hat{n}_{j}=3\) dampers located at the story positions \(\hat{p}^{(j)}=[1,3,4]\). The corresponding placement and diagonal matrices are, respectively,  
\[\widehat{\mathbf{P}}^{(j)}=\left[ \begin{array}{ccc}1&0&0\\ 0&-1&0\\ 0&1&-1\\ 0&0&1\\ 0&0&0\end{array} \right],\quad\operatorname{diag}[\hat{c}^{j}_{1},\hat{c}^{j}_ {2},\hat{c}^{j}_{3}]=\left[ \begin{array}{ccc}\hat{c}^{j}_{1}&0&0\\ 0&\hat{c}^{j}_{2}&0\\ 0&0&\hat{c}^{j}_{3}\end{array} \right] \tag{10}\]  
and the matrix of additional interstory damping has the following form:  
Figure 2: Schematic mechanical model of a five-story building equipped with a distributed system of \(\hat{n}_{j}=3\) supplemental interstory dampers implemented at the story levels \(\hat{p}^{j}_{1}=1\), \(\hat{p}^{j}_{2}=3\) and \(\hat{p}^{j}_{3}=4\). \[\widehat{\mathbf{C}}^{(j)}=\left[ \begin{array}{ccccc}\dot{c}_{1}^{j}&0&0&0&0\\ 0&\dot{c}_{2}^{j}&-\dot{c}_{2}^{j}&0&0\\ 0&-\dot{c}_{2}^{j}&\dot{c}_{2}^{j}+\dot{c}_{3}^{j}&-\dot{c}_{3}^{j}&0\\ 0&0&-\dot{c}_{3}^{j}&\dot{c}_{3}^{j}&0\\ 0&0&0&0&0\end{array} \right]. \tag{11}\]  
Improving the dynamical response of nonactuated buildings and reducing the bounding risk between adjacent buildings can be achieved by introducing proper systems of additional interbuilding dampers \(\widehat{\mathcal{C}}^{(j)}=[\hat{p}^{(j)},\hat{c}^{(j)}]\), where \(\hat{c}^{(j)}=[\hat{c}_{1}^{j},\ldots,\hat{c}_{\hat{n}_{j}}^{j}]\) is the list of damping coefficients, \(\hat{p}^{(j)}=[\hat{p}_{1}^{j},\ldots,\hat{p}_{\hat{n}_{j}}^{j}]\) is a list that contains the story levels between buildings \(\mathcal{B}^{(j)}\) and \(\mathcal{B}^{(j+1)}\) at which the interbuilding dampers are implemented, and \(\hat{n}_{j}\) is the number of dampers in \(\widehat{\mathcal{C}}^{(j)}\).