## 3 Optimum Seismic Control of an MDOF System with CBISs
### Design Formulation and Procedure  
The design of the CBISs used to attenuate the MDOF structure seismic response includes the total number determination, the placement design, and the parameter optimizations. The total number and placement of the CBISs can be predetermined through an initial analysis of the structure and by considering the building facade limitations. Typically, considering the high efficiency of the CICBISs, there is no need to install them at each layer. The demand-based design method is adopted herein to optimize the parameters of the CBISs to ensure the vibration performance of the structure while reducing the cost of manufacturing and installing the CBISs. For intuitively expressing the optimization process, the design variables representing the CBISs are adopted and defined as the following:  
\[y=\left\{m_{\rm d},c_{\rm d},\omega_{\rm r},\beta_{1},\beta_{2},\ldots,\beta_{ n_{\rm d}}\right\} \tag{36}\]  
where \(m_{\rm d}\), \(c_{\rm d}\) are the designed apparent mass, and the designed damping coefficient of the inerter system, respectively; \(\beta_{\rm i}\) denotes the displacement transition ratio of the \(i\)th cable-bracing systems and \(n_{\rm d}\) denotes the total number of CBISs; \(\omega_{\rm r}\) is the frequency of the CBIS defined as:  
\[\omega_{\rm r}=\sqrt{\frac{k_{\rm d}}{m_{\rm d}}} \tag{37}\]  
where \(k_{\rm d}\) denotes the designed stiffness of the CBISs. The equivalent parameters of each CBIS can be calculated by the following equations:  
\[\left\{ \begin{array}{l}m_{\rm e,i}=m_{\rm d}\cdot\beta_{1}^{2}\\ k_{\rm e,i}=m_{\rm d}\cdot\omega_{\rm r}^{2}\cdot\beta_{1}^{2}\\ c_{\rm e,i}=c_{\rm d}\cdot\beta_{1}^{2}\end{array} \right.\;\;i=1,\ldots,n_{\rm d} \tag{38}\]It is worth noting that the displacement transition ratios reflect the proportional relationship between each CBIS, and their specific value can be determined by presetting the designed apparent mass \(m_{\text{d}}\). The performance of the CBIS in terms of attenuating the MDOF structure seismic response is represented by the drift reduction ratio \(J_{\text{P}}\):  
\[J_{\text{P}}=\max_{i=1}^{n}\left[\frac{\sigma_{\text{u}}(\text{i})/h_{\text{i}} }{\theta_{\text{0 max}}}\right] \tag{39}\]  
where \(\sigma_{\text{u}}(\text{i})\) denotes the \(i\)th RMS story drift of the controlled structure; \(h_{\text{i}}\) denotes the \(i\)th floor height of the structure; \(\theta_{\text{0 max}}\) denotes the maximum RMS story drift ratio of the uncontrolled structure. Since the materials' strength required for the manufacturing of the inerter system and the strength of the anchor nodes are positively related to the control force [7; 15; 16], the RMS control force of a CBIS is herein selected as its cost index. Because the inerter system of each CBIS is the same, the cost index of installing \(n_{\text{d}}\) CBISs \(J_{\text{F}}\) can be estimated as:  
\[J_{\text{F}}=n_{\text{d}}\max_{i=1}^{n_{\text{d}}}\{\sigma_{\text{F}}(i)\} \tag{40}\]  
where \(\sigma_{\text{F}}(i)\) is the RMS control force of the \(i\)th CBIS. The increase in the connection stiffness may achieve a better control effect, while it comes at the cost of tuning effects and tremendously increases the apparent mass and damping coefficient requirements of the inerter system. We limit the frequency of the CBIS as follows to enhance the tuning effect and get a reasonable solution, as [PERSON] et al. did in [27]:  
\[\omega_{\text{r}}=\frac{1}{\sqrt{1-{}_{1}\mu}}{}_{0}\omega_{0} \tag{41}\]  
where \({}_{1}\omega_{0}\) denotes the natural frequency of the uncontrolled structure; \({}_{1}\mu\) is the effective inerter-mass ratio, which can be obtained by submitting the equivalent apparent mass matrix \(\mathbf{M}_{\text{e}}\), \(\mathbf{R}_{\text{c}}\) represents the location matrix of CBISs, and the first-order modal vector \({}_{1}\phi\) into Equation (27).