# Distributed Passive Actuation Schemes for Seismic Protection of Multibuilding Systems
## 2 Mathematical Model
### Overall Multibuilding Model  
By considering the overall vector of displacements  
\[\mathbf{q}(t)=\left[ \begin{array}{c}\mathbf{q}^{(1)}(t)\\ \vdots\\ \mathbf{q}^{(m)}(t)\end{array} \right], \tag{19}\]  
the dynamical response of the overall multibuilding system can be described by the model  
\[\mathbf{M}\,\ddot{\mathbf{q}}(t)+\left(\,\mathbf{C}+\widehat{\mathbf{C}}+ \widetilde{\mathbf{C}}\,\right)\dot{\mathbf{q}}(t)+\mathbf{K}\,\mathbf{q}(t)= -\mathbf{M}\,[\mathbf{1}]_{\overline{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{\mathbf{ \mathbf{ \mathbf{ \mathbf{     }}}}}}}}}}}}}}} \left(t)\right. \tag{20}\]  
where \(\overline{n}=m\cdot n\) is the total number of degrees of freedom; \(\mathbf{M}\in\mathbb{R}^{\overline{n}\times\overline{n}}\) and \(\mathbf{K}\in\mathbb{R}^{\overline{n}\times\overline{n}}\) are the overall mass and stiffness matrices, respectively, which have the following block-diagonal structure:  
\[\mathbf{M}=\text{diag}\left[\mathbf{M}_{b},\overset{(m)}{\ldots},\mathbf{M}_ {b}\right]=\left[ \begin{array}{ccc}\mathbf{M}_{b}&&&\\ &\ddots&\\ &&\mathbf{M}_{b}\end{array} \right],\ \mathbf{K}=\text{diag}\left[\mathbf{K}_{b}, \overset{(m)}{\ldots},\mathbf{K}_{b}\right]=\left[ \begin{array}{ccc} \mathbf{K}_{b}&&&\\ &\ddots&\\ &&&\mathbf{K}_{b}\end{array} \right]; \tag{21}\]  
\(\mathbf{C}\in\mathbb{R}^{\overline{n}\times\overline{n}}\) and \(\widehat{\mathbf{C}}\in\mathbb{R}^{\overline{n}\times\overline{n}}\) are the overall matrices of structural and interstory damping, respectively, which also have a block-diagonal structure  
\[\mathbf{C}=\text{diag}\left[\mathbf{C}_{b},\overset{(m)}{\ldots},\mathbf{C}_ {b}\right]=\left[ \begin{array}{ccc}\mathbf{C}_{b}&&&\\ &\ddots&\\ &&\mathbf{C}_{b}\end{array} \right],\ \widehat{\mathbf{C}}=\text{diag}\left[ \widehat{\mathbf{C}}^{(1)},\ldots,\widehat{\mathbf{C}}^{(m)}\right]=\left[ \begin{array}{ccc}\widehat{\mathbf{C}}^{(1)}&&&\\ &\ddots&\\ &&&\widehat{\mathbf{C}}^{(m)}\end{array} \right]; \tag{22}\]  
and \(\widetilde{\mathbf{C}}\in\mathbb{R}^{\overline{n}\times\overline{n}}\) is a matrix that models the overall interbuilding damping and has the following block-tridiagonal structure:\[\widetilde{\mathbf{C}}=\left[ \begin{array}{ccccc}\widetilde{\mathbf{C}}^{(1)}&- \widetilde{\mathbf{C}}^{(1)}&&&\\ -\widetilde{\mathbf{C}}^{(1)}&\widetilde{\mathbf{C}}^{(1)}+\widetilde{ \mathbf{C}}^{(2)}&-\widetilde{\mathbf{C}}^{(2)}&&\\ &\cdots&\cdots&\cdots&\\ &&&-\widetilde{\mathbf{C}}^{(m-2)}&\widetilde{\mathbf{C}}^{(m-2)}+\widetilde {\mathbf{C}}^{(m-1)}&-\widetilde{\mathbf{C}}^{(m-1)}\\ &&&-\widetilde{\mathbf{C}}^{(m-1)}&\widetilde{\mathbf{C}}^{(m-1)}\end{array}  \right]. \tag{23}\]  
The matrix \(\widetilde{\mathbf{C}}\) can be computed as  
\[\widetilde{\mathbf{C}}=\mathbf{Q}\;\text{diag}\left[\widetilde{\mathbf{C}}^{( 1)},\ldots,\widetilde{\mathbf{C}}^{(m-1)}\right]\mathbf{Q}^{T}, \tag{24}\]  
where \(\mathbf{Q}\in\mathbb{R}^{\overline{n}\times(m-1)n}\) is the block-matrix  
\[\mathbf{Q}=\left[ \begin{array}{ccccc}\mathbf{I}_{n}&&&\\ -\mathbf{I}_{n}&\mathbf{I}_{n}&&\\ &\cdots&\cdots&\\ &&-\mathbf{I}_{n}&\mathbf{I}_{n}\\ &&&-\mathbf{I}_{n}\end{array} \right], \tag{25}\]  
defined by blocks \([\mathbf{Q}]_{i,j}\), \(i=1,\ldots,m\), \(j=1,\ldots,m-1\) with the following form:  
\[ \begin{cases}[\mathbf{Q}]_{i,i}=\mathbf{I}_{n},&i=1,\ldots,m-1,\\ [\mathbf{Q}]_{i+1,i}=-\mathbf{I}_{n},&i=1,\ldots,m-1,\\ [\mathbf{Q}]_{i,j}=[\mathbf{0}]_{n\times n},&\text{otherwise},\end{cases}  \tag{26}\]  
where \([\mathbf{0}]_{n\times n}\) denotes the null matrix of dimension \(n\times n\). For example, the five-building system in Figure 1 includes three non-empty interbuilding damping systems \(\widetilde{\mathcal{C}}^{(1)}\), \(\widetilde{\mathcal{C}}^{(2)}\) and \(\widetilde{\mathcal{C}}^{(3)}\) with \(\tilde{n}_{1}=2\), \(\tilde{n}_{2}=3\), and \(\tilde{n}_{3}=5\) dampers, respectively.