# Theoretical Study on a Cable-Bracing Inerter System for Seismic Mitigation
## 3 Parametric Study
### Parametric Analysis Results  
Based on the analysis indexes mentioned above, a series of numerical cases were considered. In these cases, the inherent damping ratio for the main frame \(\zeta=0.02\) was assumed. Three inertance-mass ratios, namely, \(\mu=0.01\), \(0.1\) and \(1.0\), were used to study the vibration control effects by continuously varying \(\kappa\) and \(\xi\) within specified ranges. To describe the controlling index of the SDOF with a CBIS, contour plots were illustrated with \(\xi\) on the \(x\)-axis, \(\kappa\) on the \(y\)-axis, and \(\max\{\left|H_{U}(i\beta)\right|\omega_{\mathrm{s}}^{2}\}\) as the height, as shown in Figure 5.  
Figure 5: Displacement amplification factor of a CBIS-equipped SDOF structure with changes in \(\kappa\in[0.01,100]\) and \(\xi\in[0.01,1]\).  
The displacement amplification factor of any point in the \(\kappa\)-\(\xi\) space is determined by the frequency response function of Equation (25). The magnitude is represented by the color intensity. The lowest displacement amplification factor is in the corner of the \(\kappa\)-\(\xi\) space, where the two parameters reach their upper bounds. As shown in Figure 5a-c, the desirable solution of the dynamic response ratio is determined by the feasible upper bounds of the CBIS parameters in the given ranges. Mathematically, the optimal configuration for vibration control requires the stiffness ratio and additional damping to be as large as possible. However, both are impossible to realize in actual engineering. It is necessary to introduce appropriate boundary conditions or constraints for practical optimization processes.  
When the additional damping ratios are fixed, for example \(\xi\) = 0.05, 0.1, closed contour lines can be obtained (as shown in Figure 6). A very low point implies that the parameter set for optimal control can always be found in the inner part of every contour. This means that optimal solutions lie within the inner part of the parametric space. When the stiffness ratio closes to 1, and the inertance-mass ratio closes to 0.1, the displacement response reaches its lowest point. In this process, the optimal additional damping ratio \(\xi\) remains unknown. Therefore, the selection of a rational parameter set for the design of CBIS based on only the displacement response is difficult. However, for many situations, the optimization will involve a recursive process in which the optimal configuration keeps updating with a prescribed additional damping ratio \(\xi\) until the mitigation effect satisfies the objective.  
In Figure 6, the displacement responses reach their optimal points when the stiffness ratio closes to 1, and the inertance-mass ratio closes to \([0.1,0.5]\),where the force response of the inerter system is relatively large (as shown in Figure 7). In the optimization design, it is unreasonable to consider only the structural displacement, or the force response provided by the inerter element. Therefore, both the displacement responses and the inerter element's force should be considered in the design of CBIS. The demand-oriented multi-objective optimum design method will be introduced in the next section, and the control force response of the CBIS will be brought into the optimization process as the secondary objective.  
Figure 6: Displacement amplification factor of a CBIS-equipped SDOF structure with changes in \(\kappa\in[0.01,100]\) and \(\mu\in[0.01,1]\).