## 4 An Illustrative Design Example
### Design of the CBISs  
According to the above analysis, we can conclude that installing the CBIS crossing all layers is the most efficient installation method. However, the more layers crossed, the higher the requirements on the construction and maintenance of the CBIS, and sometimes the number of layers that the CBIS can cross is limited by the building facade requirements. Since these engineering factors are difficult to quantify, we recommend and analyze the cross-all-layers and cross-three-layers installed CBISs as two possible selections (CASE A and CASE B) showed in Figure 9. CASE A and CASE B represent the most efficient installation method and an easier-to-implement installation method, respectively. The vertical distribution of the CICBISs in CASE B is uniform to ensure a reasonable control force distribution. Thus, the total number of CICBISs is selected to be three, and the inerter installation locations are selected as the bottom, third, and sixth layer. Meantime, another design scheme, installing the IICBISs layer by layer on the benchmark structure, is illustrated as well to compare with the CICBISs, as CASE C showed in Figure 9, the IICBISs are divided into three groups, thereby limiting the types of cable-bracing systems to ensure engineering practice. Figure 8: Illustration of the modal inerter-mass ratios: (**a**) the influence of installation location; (**b**) the influence of the crossed layer number. The optimization problems for CASE A, CASE B, and CASE C in Figure 9 can be expressed as:  
\[ \begin{array}{ll}\text{find}&y=\left\{m_{\text{d}},c_{\text{d}},\omega_{\text{ r}},\beta_{1},\beta_{2},\dots,\beta_{\text{n}_{\text{d}}}\right\}\\ \text{to minimize}&J_{\text{F}}(y)\\ \text{subject to}&\left\{\begin{array}{l}J_{\text{P}}(y)\leq J_{\text{target}} \\ \omega_{\text{r}}=\frac{1}{\sqrt{1-\mu}}\omega_{0}\\ m_{\text{d}}=m_{\text{design}}\end{array}\right.\end{array}  \tag{43}\]  
where \(J_{\text{target}}\)= 0.5 determined by the initial analysis and the performance requirements of the primary structure. An inerter with the apparent mass \(m_{\text{design}}\)= \(1\times 10^{6}\)kg is utilized herein, which can be achieved with an actual mass lower than 150 kg, using the same mass-amplifying mechanism of the inerter mentioned in [39]. The parameters of the Kanai-Tajimi power spectrum used to simulate the seismic excitation are selected as \(\omega_{\text{g}}\)= \(4\pi\) rad/s, \(\zeta_{\text{g}}=0.6\), which is suggested by [PERSON] [41] for the firm ground. Table 2 shows the optimal results of CASE A, CASE B, and CASE C obtained using a numerical optimization. Since the unified designed apparent mass is predetermined as \(1\times 10^{6}\)kg, the utilization efficiency of the CICBISs and IICBISs is reflected by the displacement transition ratios. It was found that the average displacement transition ratios of the cross-all-layers and cross-three-layers installed CBISs are about 30.0% and 57.2% of the IICBISs', reflecting the high efficiency of the CICBIS. It is said that when using the cable-bracing system with similar transfer ratios, the designed apparent mass and designed damping coefficient of the CICBIS are much smaller than those of the IICBIS. Furthermore, as the number of the CICBISs used in CASE A and CASE B is smaller than that of the IICBISs in CASE C, the total apparent mass and damping coefficient requirements of the CICBISs are reduced compared with the IICBISs. According to the optimal results shown in Table 2, we can know that the total apparent mass and damping coefficient requirements of cross-all-layers installed CBIS are about 10.0% and 7.5% of the IICBIS's, respectively. The total apparent mass and damping coefficient requirements of the cross-three-layers installed CBISs are reduced to about 33.3% and 27.8% of the IICBIS's, respectively. In fact, the IICBISs in CASE C can be replaced by the traditional TVMDs as well, which have been studied in [27; 28].