# Theoretical Study on a Cable-Bracing Inerter System for Seismic Mitigation
## 3 Parametric Study
### Ignoring the Flexibility of the Cable  
To simplify the CBIS analytical model, the flexibility of the cable is neglected as shown in Figure 12.  
Consider this simplified CBIS-equipped SDOF structure as a model with no connection element flexibility. The relationship between the axial deformation of the cable and the rotational angle of the shaft is expressed as:  
\[\varphi(t)=\frac{u(t)\cos\theta}{r_{0}}. \tag{32}\]  
The equation of motion for an SDOF model with a CBIS is given by:  
\[\left(m+\frac{J\cos^{2}\theta}{r_{0}{}^{2}}\right)\ddot{u}(t)+\left(c_{0}+c_{ \mathrm{d}}\cos^{2}\theta\right)\ddot{u}(t)+ku(t)=-ma_{\mathrm{g}}(t). \tag{33}\]  
In Equation (33), \(\frac{J\cos^{2}\theta}{r_{0}^{2}}\) is \(m_{\mathrm{d}}\), which is namely the inertance of the inerter. From Equation (33), it can be understood that the utilization of CBIS induces the elongation of the natural period and increases the damping effect. The mass of the primary structure is affected, while the stiffness of the overall structure remains unaffected. By using the Laplace transformations, the transfer function can be easily obtained as:  
\[H_{U}(s)=\frac{U(s)}{A_{\mathrm{g}}(s)}=\frac{-1}{(1+\mu)s^{2}+(2\zeta\omega_{ \mathrm{s}}\cos^{2}\theta)s+\omega_{\mathrm{s}}^{2}}. \tag{34}\]  
We assume a single-floor frame structure, which can usually be treated as an SDOF structure. The key parameters of the structure and the inerter system are as follows: inherent damping ratio is \(\zeta\) = 0.02, the tilt angle of the cable is \(\theta\) = \(\pi\)/4, the inertance--mass ratio is \(\mu\) = 0.1, 0.2, and the damping  
\begin{table}
\begin{tabular}{c|c|c|c|c|c|c|c|c} \hline \multirow{2}{*}{**Seismic Input**} & \multicolumn{2}{c|}{**El Centro Record**} & \multicolumn{2}{c|}{
\begin{tabular}{c} **The Tohoku** \\ **University Record** \\ \end{tabular} } & \multicolumn{2}{c|}{**Kobe Record**} & \multicolumn{2}{c}{**Chi-Chi Record**} \\ \cline{2-10}  & **Peak** & **RMS** & **Peak** & **RMS** & **Peak** & **RMS** & **Peak** & **RMS** \\ \hline Uncontrolled & 8.77 & 1.74 & 16.59 & 1.85 & 25.47 & 4.75 & 2.44 & 0.35 \\ \hline Controlled & 4.75 & 1.04 & 7.95 & 1.16 & 14.14 & 2.91 & 1.55 & 0.19 \\ \hline Reduction effects (\%) & 45.84 & 40.23 & **52.08** & 37.30 & 44.48 & 38.74 & 36.48 & **45.71** \\ \hline \end{tabular}
\end{table}
Table 2: Acceleration responses at the roof of the test frame (m/s\({}^{2}\)).  
Figure 12: Simplified analytical model of the CBIS-equipped SDOF structure.  
\begin{table}
\begin{tabular}{c|c|c|c|c|c|c|c|c} \hline \multirow{2}{*}{**Seismic Input**} & \multicolumn{2}{c|}{**El Centro Record**} & \multicolumn{2}{c|}{
\begin{tabular}{c} **The Tohoku** \\ **University Record** \\ \end{tabular} } & \multicolumn{2}{c|}{**Kobe Record**} & \multicolumn{2}{c}{**Chi-Chi Record**} \\ \cline{2-10}  & **Peak** & **RMS** & **Peak** & **RMS** & **Peak** & **RMS** & **Peak** & **RMS** \\ \hline Uncontrolled & 0.168 & 0.040 & 0.396 & 0.046 & 0.646 & 0.129 & 0.053 & 0.009 \\ \hline Controlled & 0.077 & 0.019 & 0.176 & 0.026 & 0.327 & 0.081 & 0.032 & 0.005 \\ \hline Reduction effects (\%) & 54.17 & **52.50** & **55.56** & 43.48 & 49.38 & 37.21 & 39.62 & 44.44 \\ \hline \end{tabular}
\end{table}
Table 3: Displacement responses at the roof of the test frame (m).  
ratio is \(\xi\) = 0.01, 0.05, 0.1. By using these parameters, the displacement amplification factor can be plotted as shown in Figure 13.  
Figure 13 shows that inerter systems can significantly suppress the resonant response in a narrow band near the natural frequency of the primary structure but does not impact the other range of frequencies.