## 2 Theoretical Analysis of CBIS
### MDOF Structure with the CBISs  
The governing motion equations of an uncontrolled MDOF structure's underground motion excitations are:  
\[\mathbf{M}_{\rm p}\mathbf{\ddot{X}}_{\rm p}+\mathbf{C}_{\rm p}\mathbf{\dot{X} }_{\rm p}+\mathbf{K}_{\rm p}\mathbf{X}_{\rm p}=-\mathbf{M}_{\rm p}\mathbf{r}_{ \rm p}\ddot{u}_{0} \tag{10}\]  
where \(\mathbf{X}_{\rm p}\) is the displacement vector of the MDOF structure relative to the ground; \(\mathbf{M}_{\rm p}\), \(\mathbf{K}_{\rm p}\), \(\mathbf{C}_{\rm p}\) denote the mass, stiffness, and damping matrices of the primary structure, respectively; \(\mathbf{r}_{\rm p}\) is the influence coefficient vector. Let \(x_{\rm p,i}\), \(m_{\rm p,i}\), \(k_{\rm p,i}\), and \(c_{\rm p,i}\) denote the relative displacement, mass, stiffness, and damping coefficient of the \(i^{th}\) layer of the primary structure, respectively; the matrices in Equation (10) are defined as:  
\[\mathbf{X}_{\rm p}=\left\{x_{\rm p,1},x_{\rm p,2},\ldots x_{\rm p,n}\right\}^{T} \tag{11}\]  
\[\mathbf{M}_{\rm p}=\mathrm{diag}\big{[}m_{\rm p,1},m_{\rm p,2},\ldots m_{\rm p,n}\big{]} \tag{12}\]  
\[\mathbf{K}_{\rm p}=\mathbf{T}^{T}\mathrm{diag}(k_{\rm p,1},k_{\rm p,2},\ldots k _{\rm p,n})\mathbf{T} \tag{13}\]  
\[\mathbf{C}_{\rm p}=\mathbf{T}^{T}\mathrm{diag}(c_{\rm p,1},c_{\rm p,2},\ldots c _{\rm p,n})\mathbf{T} \tag{14}\]  
\[\mathbf{r}_{\rm p}=\{1,1,\ldots,1\}^{T} \tag{15}\]  
where \(n\) is the number of MDOF structure layers; \(\mathbf{T}\) is an n-dimension square matrix with 1 in the diagonal and \(-1\) in the first off-diagonal, denoting a transformation matrix defining the relative deformation between consecutive floors. Similar to the SDOF structure, the motion equations of the MDOF structure with the CBISs can be simplified as:  
\[\mathbf{M}\mathbf{\ddot{X}}+\mathbf{C}\mathbf{\dot{X}}+\mathbf{K}\mathbf{X}=- \mathbf{M}\mathbf{r}\ddot{u}_{0} \tag{16}\]  
where \(\mathbf{X}\) denotes the displacement vector of the primary structure and the CBISs; \(\mathbf{M}\), \(\mathbf{C}\), and \(\mathbf{K}\) are the mass, damping, and stiffness matrix of the controlled structure, respectively; \(\mathbf{r}\) is the influence coefficient vector. Let \(m_{\rm e,i}\), \(k_{\rm e,i}\), and \(c_{\rm e,i}\) denote the apparent mass, stiffness, and damping coefficient of the \(i\)th CBIS, respectively. The matrices in Equation (16) are defined as:  
\[\mathbf{X}=\big{\{}\mathbf{X}_{\rm p}{}^{T},\mathbf{X}_{\rm e}{}^{T}\big{\}} ^{T},\text{where}\ \mathbf{X}_{\rm e}=\big{\{}x_{\rm e,1},x_{\rm e,2},\ldots x_{\rm e,n_{\rm d}} \big{\}}^{T} \tag{17}\]  
\[\mathbf{M}=\left[ \begin{array}{cc}\mathbf{M}_{\rm p}&\\ &\mathbf{M}_{\rm e}\end{array} \right],\ \text{where}\ \mathbf{M}_{\rm e} =\mathrm{diag}\big{\{}m_{\rm e,1},m_{\rm e,2},\ldots m_{\rm e,n_{ \rm d}}\big{\}} \tag{18}\]  
\[\mathbf{C}=\left[ \begin{array}{cc}\mathbf{C}_{\rm p}&\\ &\mathbf{C}_{\rm e}\end{array} \right],\ \text{where}\ \mathbf{C}_{\rm e} =\mathrm{diag}\big{\{}c_{\rm e,1},c_{\rm e,2},\ldots c_{\rm e,n_{\rm d}} \big{\}} \tag{19}\]  
\[\mathbf{K}=\left[ \begin{array}{cc}\mathbf{K}_{\rm p}+\mathbf{R}_{\rm c} \mathbf{K}_{\rm e}\mathbf{R}_{\rm c}{}^{T}&-\mathbf{R}_{\rm c}\mathbf{K}_{\rm e} \\ -\mathbf{K}_{\rm e}{}^{T}\mathbf{R}_{\rm c}{}^{T}&\mathbf{K}_{\rm e}\end{array}  \right],\ \text{where}\ \mathbf{K}_{\rm e} =\mathrm{diag}\big{\{}k_{\rm e,1},k_{\rm e,2},\ldots k_{\rm e,n_{\rm d}} \big{\}} \tag{20}\]  
\[\mathbf{r}=\left\{\mathbf{r}_{\rm p}{}^{T},0_{1\times\rm n_{\rm d}}\right\}^{T} \tag{21}\]where \(x_{\rm e,i}\) denotes the equivalent inerter displacement of the \(i\)th CBIS; \(n_{\rm d}\) is the total number of the CBIS; \([\mathbf{R}_{\rm c}]_{\rm mx_{\rm d}}\) is the installation matrix, denoting the installation location of the CBIS. Take the \(j\)th CBIS installed between the \(r\)th and \(s\)th layers as an example (as is shown in Figure 5). The installation location of the \(j\)th CBIS is represented with the entries in the \(j\)th column of \(\mathbf{R}_{\rm c}\).