## 4 An Illustrative Design Example
### Effectiveness of the CICBISs  
First, the performance of the CICBISs in terms of attenuating the structure response under the harmonic excitation is analyzed from the perspective of the displacement frequency response functions (FRFs). Taking the fourth layer's inter-layer drift FRFs, where the maximum inter-layer drift occurs, and the top layer's absolute displacement FRFs as examples, the FRFs, as is shown in Figure 10, are normalized by the uncontrolled peak response and the black dash line is the FRFs of the uncontrolled structure. The fourth layer FRF curves show that the damping effect of CASE A, CASE B, and CASE C on the first-order response is significant, and their curves almost coincide. It means the CICBISs and IICBISs have a similar control effectiveness on the maximum inter-layer drift layer. The top layer absolute displacement FRF curves show that the structure response is mainly controlled by the first-order modal, i.e., whether the CICBISs or IICBISs can effectively control the overall displacement of the structure. The cross-all-layers installed CBIS can accurately control the first-order modal response without affecting other modes. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & & \(m_{\text{d}}\) (kg) & \(c_{\text{d}}\) (kN/m/s) & \(\omega_{\text{r}}\) & \(\beta\) & \(k_{\text{d}}\) (kN/m) \\ \hline CASE A & CICBIS-All & 1,000,000 & 1605.899 & 3.259 & 0.508 & 10,620.172 \\ \hline \multirow{3}{*}{CASE B} & CICBIS-1 & & & & 0.909 & \\  & CICBIS-2 & 1,000,000 & 1970.397 & 3.270 & 0.964 & 10,695.612 \\  & CICBIS-3 & & & & 1.040 & \\ \hline \multirow{3}{*}{CASE C} & IICBIS-1 & & & & 1.535 & \\  & IICBIS-2 & 1,000,000 & 2129.588 & 3.281 & 1.799 & 10,766.132 \\ \cline{1-1}  & IICBIS-3 & & & & 1.754 & \\ \cline{1-1} \cline{2-6}  & TVMD-1 & 2,354,898 & 5014.963 & 3.281 & 1.000 & 25,353.142 \\ \cline{1-1}  & TVMD-2 & 3,234,898 & 6889.000 & 3.281 & 1.000 & 34,827.336 \\ \cline{1-1}  & TVMD-3 & 3,078,151 & 6555.193 & 3.281 & 1.000 & 33,139.778 \\ \hline \end{tabular}
\end{table} 
Table 2: Optimal results of CASE A, CASE B, CASE C. \begin{table}
\begin{tabular}{c c c c} \hline  & \(\alpha\) (\({}^{\circ}\)) & \(\eta\) & \(\beta\) \\ \hline CICBIS-All & 46.009 & 1.116 & 0.508 \\ CICBIS-1 & 20.054 & 0.968 & 0.909 \\ CICBIS-2 & 17.361 & 1.010 & 0.964 \\ CICBIS-3 & 17.361 & 1.090 & 1.040 \\ IICBIS-1 & 5.959 (8.881 \({}^{1}\)) & 1.543 (1.554 \({}^{1}\)) & 1.535 \\ IICBIS-2 & 5.959 & 1.809 & 1.799 \\ IICBIS-3 & 5.959 & 1.764 & 1.754 \\ \hline \end{tabular} \({}^{1}\) The bottom layer parameters.  
\end{table} 
Table 3: Design parameters of the cable-bracing system. Furthermore, Figure 11 shows the RMS responses of the controlled and uncontrolled structures under the stochastic excitation proposed for optimization, where the black dash line is the uncontrolled structure response and the red dash line is the target drift limitation. The RMS inter-layer drift angles in Figure 11a show that both the designed CICBISs and IICBISs have reached the target drift reduction ratio \(J_{\text{target}}\)\(=0.5\). Consistent with the conclusions reflected in the FRF curves, CASE A, CASE B, and CASE C have almost the same control effect on the structure displacement, except for a slightly difference in the top layer inter-layer drift angle. It is caused by the lack of control device in CASE B. From the RMS control forces of the CICBISs and IICBISs, shown in Figure 11b, it can be concluded that the cross-layer installation will not increase the control force. Table 4 shows the cost index \(J_{\text{F}}\) of each design case. It can be seen that the cost indexes of CASE A and CASE B are about 8.2% and 26.3% of CASE C's, respectively, which is mainly caused by the reduced number of the inerter systems used in CASE A and CASE B. \begin{table}
\begin{tabular}{c c c c} \hline  & **CASE A** & **CASE B** & **CASE C** \\ \hline cost index \(J_{\text{F}}\) (kN) & 718.234 & 2299.150 & 8727.719 \\ \hline \end{tabular}
\end{table}