## 2 Theoretical Analysis of CBIS
### Basic Concept of the CBIS
#### 2.1.2 The Cable-Bracing System  
The cable-bracing system proposed in this study consists of a pair of pretensioned diagonal cables and some auxiliary elements (such as the lever, the seesaw system). Figure 2 illustrates three typical forms of cable-bracing systems: (a) simple cable-bracing system [34], (b) cable-bracing lever system [37], and (c) cable-bracing seesaw system [38]. Figure 1: The mechanical schematic of the inerter and the tuned viscous mass damper (TVMD): (**a**) the inerter; (**b**) the TVMD. Figure 2: _Cont._  
In the preliminary design, it is assumed that the stiffness of the auxiliary element is infinite, and the stiffness of the cable-bracing system only depends on the cable stiffness \(k_{\text{c}}\). The cable stiffness \(k_{\text{c}}\) can be determined by the following equation:  
\[k_{\text{c}}=\frac{E_{\text{c}}A_{\text{c}}}{l_{\text{c}}} \tag{3}\]  
where \(E_{\text{c}}\) and \(A_{\text{c}}\) are the elastic modulus and cross-section area of the cable, respectively; \(l_{\text{c}}\) is the cable equivalent length considering the cable-bracing system's displacement transition mechanism. Take the cable-bracing systems in Figure 2 as examples, the equivalent length \(l_{\text{c}}\) of the simple cable-bracing system equals the total length of its one-side cable. Influenced by the displacement transition mechanism, when calculating the equivalent length \(l_{\text{c}}\) of the cable-bracing lever system and cable-bracing seesaw system, their diagonal cable lengths should be weighted according to the principle of stiffness equivalence:  
\[l_{\text{c}}=\left\{ \begin{array}{ll}l_{\text{c1}}+l_{\text{c2}}&\text{ simple cable - bracing system}\\ \eta^{2}l_{\text{c1}}+l_{\text{c2}}&\text{ cable - bracing lever system}\\ 2\Big{(}\frac{\cos(\gamma)}{\sin(\alpha+\gamma)}\Big{)}^{2}l_{\text{c1}}& \text{ cable - bracing seesaw system}\end{array} \right. \tag{4}\]  
Figure 2: Schematics of different cable-bracing system: (**a**) simple cable-bracing system; (**b**) cable-bracing lever system; (**c**) cable-bracing seesaw system. where \(l_{\rm cl}\) and \(l_{\rm c2}\) are one-sided diagonal and horizontal cable lengths, respectively; \(\alpha\), \(\eta\), \(\gamma\) are the cable installation angle, the lever arm ratio, and the seesaw angle, respectively. Applying a small amount of pretension on the cable helps the cable maintain a straight line in the static state. Caused by the cable's tension-only characteristic, the cable's compression will lead it to slack. Thus, the pair of pretensioned cables are stressed alternately during the seismic motion. Figure 2 shows that the cable of the cable-bracing system is in series with the TVMD's tuning spring, which can be replaced by the stiffness \(k_{\rm s}\) as a whole:  
\[k_{\rm s}=\frac{k_{\rm in}k_{\rm c}}{k_{\rm in}+k_{\rm c}} \tag{5}\]  
For simplification, we choose the stiffness \(k_{\rm s}\) as a key parameter in the subsequent discussion. The cable stiffness \(k_{\rm c}\) is determined by the selected cable and Equation (4). Submitting \(k_{\rm s}\) and \(k_{\rm c}\) into Equation (5), the tuning spring stiffness \(k_{\rm in}\) can be obtained. The proportion of the structure displacement transferred to the inerter system displacement can be adjusted through auxiliary elements and the cable-bracing system's installation angle. It reflects the adjustability of the cable-bracing system. In order to compare cable-bracing systems with each other and unify their motion equations, we define the displacement transition ratio as the ratio \(\beta\) of the inerter system displacement and the horizontal displacement of the structure when ignoring the cable flexibility. Take cable-bracing systems in Figure 2 as examples--for the simple cable-bracing system, the inerter system displacement equals the diagonal cable displacement. The lever or seesaw further increase the diagonal cable displacement transferred to the inerter system for the cable-bracing lever system and cable-bracing seesaw system. Thus, the displacement transition ratios are:  
\[\beta=\left\{