# Theoretical Study on a Cable-Bracing Inerter System for Seismic Mitigation
## 3 Parametric Study
### Multi-Objective \(H_{2}\) Norm Optimum Design  
Three unknown parameters can be designed optimally according to performance demands. An optimization method is proposed to achieve the desired performance levels (structure's displacement \(u\)) with low control output force (CBIS's force \(f_{\text{d}}\)). In other words, the goal is to suppress the displacement of the structure as thoroughly as possible while minimizing the output force of the inerter system. Therefore, both the displacement response and the CBIS force should be considered in the design of this inerter system. Two dimensionless response variation ratios [32] are defined as: namely, the displacement reduction ratio, \(\gamma_{U}\) and the force ratio, \(\gamma_{F_{\text{d}}}\), which can be expressed as:  
\[\gamma_{U}(\zeta,\mu,\xi,\kappa)=\frac{\sigma_{U}}{\sigma_{U_{0}}}=\frac{ \sqrt{\int_{-\infty}^{\infty}\left|H_{U}(i\omega)\right|^{2}\!d\omega}}{\sqrt {\int_{-\infty}^{\infty}\left|H_{U_{0}}(i\omega)\right|^{2}\!d\omega}}, \tag{26}\]  
\[\gamma_{F_{\text{d}}}(\zeta,\mu,\xi,\kappa)=\frac{\sigma_{F_{\text{d}}}}{ \sigma_{F_{\text{d}0}}}=\frac{\sqrt{\int_{-\infty}^{\infty}\left|H_{F_{\text{ d}}}(i\omega)\right|^{2}\!d\omega}}{\sqrt{\int_{-\infty}^{\infty}\left|H_{F_{ \text{d}0}}(i\omega)\right|^{2}\!d\omega}}. \tag{27}\]  
In these expressions, \(\gamma_{U}\) is the ratio of the CBIS-equipped structure's displacement, compared with the primary structure, and \(\gamma_{F_{\text{d}}}\) is the dimensionless force of the CBIS. \(\sigma_{U}\) is the root mean square (RMS) of the output displacement response of the structure equipped with CBIS, and \(\sigma_{U_{0}}\) is the displacement response of the primary structure. \(\sigma_{F_{\text{d}}}\) is the force of the RMS response of the structure equipped with CBIS, and \(\sigma_{F_{\text{d}0}}\) is the force RMS response of the original structure. \(\left|H_{U_{0}}(i\omega)\right|\) and \(\left|H_{F_{\text{d}0}}(i\omega)\right|\) are the displacement and damping force (caused by the inherent damping) transfer function moduli of the original structure, respectively. The optimization of CBIS can be expressed mathematically as:  
\[\underset{\mu,\xi,\kappa}{\text{minimize}}\Big{[}\gamma_{U}(\mu, \xi,\kappa),\gamma_{F_{\text{d}}}(\mu,\xi,\kappa)\Big{]},\] \[\text{subject to}\left\{ \begin{array}{l}\mu_{\text{min}}\leq \mu\leq\mu_{\text{max}}\\ \xi_{\text{min}}\leq\xi\leq\xi_{\text{max}}\\ \kappa_{\text{min}}\leq\kappa\leq\kappa_{\text{max}}\end{array} \right., \tag{28}\]  
where \(\mu\), \(\xi\) and \(\kappa\) are decision variables, and \(\mu_{\text{min}}\), \(\xi_{\text{min}}\), and \(\kappa_{\text{min}}\) are the lower bounds of \(\mu\), \(\xi\) and \(\kappa\), while \(\mu_{\text{max}}\), \(\xi_{\text{max}}\), and \(\kappa_{\text{max}}\) are the upper bounds, respectively. Multi-objective optimization (MOO) is used to find the boundary of the feasible criterion space where all optimal points lie: namely, Pareto Front (shown in Figure 8). On this boundary, there are many points. Based on the demanding  
Figure 7: Force amplification factor of the inerter system. performance, a reasonable optimization parameter for \(\gamma u\leq 40\%\) and the corresponding \(\gamma_{F_{4}}\) can be found. Furthermore, a set of parameters was selected for time domain analysis, as shown in Table 1.  
One of the advantages of a CBIS over a viscous element is that it can enhance the energy dissipation efficiency.