# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 2 Mathematical Model
### Interstory and Interbuilding Damping Models
To compute the dynamical response of the buildings subjected to the action of interbuilding damping systems we define the interbuilding damping matrix \(\widetilde{\mathbf{C}}^{(j)}\) associated to the damping system \(\widetilde{\mathcal{C}}^{(j)}\) as  
\[\widetilde{\mathbf{C}}^{(j)}=\widetilde{\mathbf{P}}^{(j)}\;\mathrm{diag}[ \dot{c}_{1}^{j},\ldots,\dot{c}_{\hat{n}_{j}}^{j}]\;\big{\{}\widetilde{\mathbf{ P}}^{(j)}\big{\}}^{T}, \tag{12}\]  
where \(\mathrm{diag}[\dot{c}_{1}^{j},\ldots,\dot{c}_{\hat{n}_{j}}^{j}]\) is the diagonal matrix defined by the list of damping coefficients \(\dot{c}^{(j)}\) and the location matrix \(\widetilde{\mathbf{P}}^{(j)}\in\mathbb{R}^{n\times\hat{n}_{j}}\) contains the columns of the identity matrix \(\mathbf{I}_{n}\) indicated in the list of damper positions \(\hat{p}^{(j)}\). Thus, for example, the system of additional interbuilding dampers \(\widehat{\mathcal{C}}^{(2)}\) implemented between buildings \(\mathcal{B}^{(2)}\) and \(\mathcal{B}^{(3)}\) in Figure 1 contains \(\hat{n}_{2}=3\) dampers with coefficients \(\dot{c}^{(2)}=\left[\dot{c}_{1}^{2},\dot{c}_{2}^{2},\dot{c}_{3}^{2}\right]\), which are located at the story levels \(\hat{p}^{(2)}=[1,3,5]\). The corresponding placement and diagonal matrices are, respectively,  
\[\widetilde{\mathbf{P}}^{(2)}=\left[ \begin{array}{ccc}1&0&0\\ 0&0&0\\ 0&1&0\\ 0&0&0\\ 0&0&1\end{array} \right],\quad\mathrm{diag}[\dot{c}_{1}^{2},\dot{c}_{2}^{2}, \dot{c}_{3}^{2}]=\left[ \begin{array}{ccc}\dot{c}_{1}^{2}&0&0\\ 0&\dot{c}_{2}^{2}&0\\ 0&0&\dot{c}_{3}^{2}\end{array} \right] \tag{13}\]  
and the matrix of additional interbuilding damping is  
\[\widetilde{\mathbf{C}}^{(2)}=\left[ \begin{array}{ccccc}\dot{c}_{1}^{2}&0&0&0 &0\\ 0&0&0&0&0\\ 0&0&\dot{c}_{2}^{2}&0&0\\ 0&0&0&0&0\\ 0&0&0&0&\dot{c}_{3}^{2}\end{array} \right]. \tag{14}\]  
For an \(m\)-building system equipped with the set of interstory and interbuilding damping systems \(\widehat{\mathcal{C}}^{(1)},\ldots,\widehat{\mathcal{C}}^{(m)},\widehat{ \mathcal{C}}^{(1)},\ldots,\widehat{\mathcal{C}}^{(m-1)}\), the dynamical response of building \(\mathcal{B}^{(j)}\) can be described by the model  
\[\mathbf{M}_{b}\tilde{\mathbf{q}}^{(j)}(t)+\mathbf{f}_{d}^{(j)}(t)+\mathbf{K}_ {b}\mathbf{q}^{(j)}(t)=-\mathbf{M}_{b}[\mathbf{1}]_{n\times 1}\tilde{w}(t), \tag{15}\]  
where the term \(\mathbf{f}_{d}^{(j)}(t)\) denotes the total damping force acting on \(\mathcal{B}^{(j)}\). Using the damping matrix \(\widetilde{\mathbf{C}}^{(j)}\) defined in Equation (12), the total damping force \(\mathbf{f}_{d}^{(1)}(t)\) acting on the initial building \(\mathcal{B}^{(1)}\) can be written in the following form:  
\[\mathbf{f}_{d}^{(1)}(t)=\left[\mathbf{C}_{b}+\widehat{\mathbf{C}}^{(1)}+ \widetilde{\mathbf{C}}^{(1)}\right]\dot{\mathbf{q}}^{(1)}(t)-\widetilde{ \mathbf{C}}^{(1)}\dot{\mathbf{q}}^{(2)}(t); \tag{16}\]for an interior building \(\mathcal{B}^{(j)},j=2,\ldots,m-1\), we have  
\[\mathbf{f}_{d}^{(j)}(t)=-\widetilde{\mathbf{C}}^{(j-1)}\dot{\mathbf{q}}^{(j-1)}( t)+\left[\mathbf{C}_{b}+\widehat{\mathbf{C}}^{(j)}+\widetilde{\mathbf{C}}^{(j-1) }+\widetilde{\mathbf{C}}^{(j)}\right]\dot{\mathbf{q}}^{(j)}(t)-\widetilde{ \mathbf{C}}^{(j)}\dot{\mathbf{q}}^{(j+1)}(t); \tag{17}\]  
and for the final building \(\mathcal{B}^{(m)}\), we obtain  
\[\mathbf{f}_{d}^{(m)}(t)=-\widetilde{\mathbf{C}}^{(m-1)}\dot{\mathbf{q}}^{(m-1 )}(t)+\left[\mathbf{C}_{b}+\widehat{\mathbf{C}}^{(m)}+\widetilde{\mathbf{C}}^ {(m-1)}\right]\dot{\mathbf{q}}^{(m)}(t). \tag{18}\]  
**Remark 3**. : _For notational convenience, we will assume that a nonactuated building \(\mathcal{B}^{(j)}\) is equipped with an empty system of interstory dampers \(\widehat{\mathcal{C}}^{(j)}=[\hat{p}^{(j)},\hat{c}^{(j)}]\), where \(\hat{p}^{(j)}\) and \(\hat{c}^{(j)}\) are empty lists and the number of dampers is \(\hat{n}_{j}=0\). In that case, we agree that \(\widehat{\mathbf{C}}^{(j)}=[\mathbf{0}]_{n\times n}\).