## 2 Theoretical Analysis of CBIS
### MDOF Structure with the CBISs
Only the \(r\)th and \(s\)th entries of the column are \(-1\) and \(1\), representing the \(j\)th inerter installation location and the \(j\)th cable connection position, respectively, and the rest are all \(0\). \(\mathbf{R}_{\rm c}\) can express all types of CBIS studied herein (\(s-r=1\) denotes the IICBIS; \(s-r>1\) denotes the CICBIS). For quantifying the CICBIS advantages, we derived the generalized two-degree-of-freedom equation of the CBIS-equipped MDOF structure based on the Hamiltonian principle and defined its effective inerter-mass ratio. Assuming the vibration shape of an MDOF structure equipped with a CBIS represented by the vector \(\phi\), the displacement vector \(\mathbf{X}_{\rm p}\) and \(\mathbf{X}_{\rm e}\) can be expressed as:  
\[\left\{ \begin{array}{l}\mathbf{X}_{\rm p}=\alpha_{\rm p}\phi\\ \mathbf{X}_{\rm e}=\alpha_{\rm d}\mathbf{R}_{\rm c}T\phi\end{array} \right. \tag{22}\]  
where \(\alpha_{\rm p}\) and \(\alpha_{\rm d}\) are generalized coordinates of the primary structure and the inerter, respectively. Therefore, the kinetic energy \(E_{\rm T}\), strain energy \(E_{\rm U}\), and dissipated energy \(E_{\rm D}\) of the CBIS-equipped MDOF structure are:  
\[\left\{ \begin{array}{l}E_{\rm T}=\frac{1}{2}\alpha_{\rm p}^{2}\phi^{T} \mathbf{M}_{\rm p}\phi+\frac{1}{2}\dot{\alpha}_{\rm d}^{2}\phi^{T}\mathbf{R}_ {\rm c}\mathbf{M}_{\rm e}\mathbf{R}_{\rm c}T\phi\\ E_{\rm U}=\frac{1}{2}\alpha_{\rm p}^{2}\phi^{T}\mathbf{K}_{\rm p}\phi+\frac{1}{ 2}(\alpha_{\rm p}-\alpha_{\rm d})^{2}\phi^{T}\mathbf{R}_{\rm c}\mathbf{K}_{\rm e }\mathbf{R}_{\rm c}T\phi\\ E_{\rm D}=\frac{1}{2}\dot{\alpha}_{\rm p}^{2}\phi^{T}\mathbf{C}_{\rm p}\phi+ \frac{1}{2}\dot{\alpha}_{\rm d}^{2}\phi^{T}\mathbf{R}_{\rm c}\mathbf{C}_{\rm e }\mathbf{R}_{\rm c}T\phi\end{array} \right. \tag{23}\]  
Assuming \(L=E_{\rm T}-E_{\rm U}\), the Euler-Lagrange equations are:  
\[\left\{ \begin{array}{l}\frac{d}{dt}\Big{(}\frac{\partial}{\partial x_{\rm p }}L\Big{)}-\frac{\partial}{\partial x_{\rm p}}L+\frac{\partial}{\partial x_{ \rm p}}E_{\rm D}=0\\ \frac{d}{dt}\Big{(}\frac{\partial}{\partial x_{\rm d}}L\Big{)}-\frac{\partial} {\partial x_{\rm d}}L+\frac{\partial}{\partial x_{\rm d}}E_{\rm D}=0\end{array} \right. \tag{24}\]  
Substituting Equation (23) into Equation (24) gives:  
\[\left\{ \begin{array}{l}\phi^{T}\mathbf{M}_{\rm p}\phi\dddot{\alpha}_{\rm p }+\phi^{T}\mathbf{C}_{\rm p}\phi\dddot{\alpha}_{\rm p}+\phi^{T}\mathbf{K}_{ \rm p}\phi\dddot{\alpha}_{\rm p}+\phi^{T}\mathbf{R}_{\rm c}\mathbf{K}_{\rm e} \mathbf{R}_{\rm c}T\phi\dddot{\alpha}_{\rm p}+\phi^{T}\mathbf{R}_{\rm c} \mathbf{K}_{\rm e}\mathbf{R}_{\rm c}T\phi\dddot{\alpha}_{\rm d}+\phi^{T} \mathbf{R}_{\rm c}\mathbf{K}_{\rm e}\mathbf{R}_{\rm c}T\phi\dddot{\alpha}_{\rm d }+\phi^{T}\mathbf{R}_{\rm c}\mathbf{K}_{\rm e}\mathbf{R}_{\rm c}T\phi\dddot{ \alpha}_{\rm e}+\phi^{T}\mathbf{R}_{\rm c}\mathbf{K}_{\rm e}\mathbf{R}_{\rm c} T\phi\dddot{\alpha}_{\rm e}+\phi^{T}\mathbf{R}_{\rm c}\mathbf{K}_{\rm e} \mathbf{R}_{\rm c}T\phi\dddot{\alpha}_{\rm e}-\alpha_{\rm p})=0\end{array} \right. \tag{25}\]  
Figure 5: Schematic of an multi-degree-of-freedom (MDOF) structure with a CBIS. Comparing Equation (25) with Equation (9), the generalized mass \(M_{\rm p}\) of the primary structure and the generalized apparent mass \(M_{\rm d}\) of the CBIS can be expressed as:  
\[\left\{ \begin{array}{c}M_{\rm p}=\phi^{\rm T}{\bf M_{\rm p}}\phi\\ M_{\rm d}=\phi^{\rm T}{\bf R_{\rm c}}{\bf M_{\rm e}}{\bf R_{\rm c}}^{\rm T}\phi \end{array} \right. \tag{26}\]  
The performance of the tuned inerter systems is positively related to their mass ratio [5]. In the generalized two-degree-of-freedom equation, the effective inerter-mass ratio \(\mu\) is defined as:  
\[\mu=\frac{M_{\rm d}}{M_{\rm p}}=\frac{\phi^{\rm T}{\bf R_{\rm c}}{\bf M_{\rm e} }{\bf R_{\rm c}}^{\rm T}\phi}{\phi^{\rm T}{\bf M_{\rm p}}\phi} \tag{27}\]  
Note that the installation location vector \({\bf R_{\rm c}}\) only affects the generalized apparent mass of the CBIS, but has no effect on the generalized mass of the primary structure.