# Wind-Induced Response Control of High-Rise Buildings Using Inerter-Based Vibration Absorbers
## 4 Wind-Induced Response Mitigation Using Inerter-Based Vibration Absorbers
### Optimization of the Parameters of the TMD/TID Scheme Applied to the Benchmark Building
#### 4.1.1 Optimization of Parameters of TMDI  
As stated above, the displacement- and acceleration-based optimization of the three parameters, i.e., frequency ratio, damping ratio and floor of installation, can be expressed as Equations (16) and (17), respectively. The preset intervals of three parameters are determined based on practical considerations and results of previous researches [35; 50]  
\[\left\{\begin{array}{ll}\text{minimize }f_{1}(\upsilon,\zeta,t)=\hat{D}_{dis} \\ s  
From the sliced contour shown in Figure 7, the cool-color area represents the most efficient combination of parameters, leading to the minimum value of the two specific response indicators. The following conclusions can be drawn:  
* For TMDIs which efficiently mitigate the wind-induced displacement responses, the optimal frequency ratio lies around 1.1, which indicates that the frequency of optimal TMDI is close to the first order frequency of the primary structure;
* The minimal peak displacement and acceleration are achieved when the damping ratios are 7% and 10%, respectively;
* As for the optimal floor of installation of TMDI, it can be seen that the best vibration mitigation effect is achieved when the TMDI is installed at the middle-upper portion of the benchmark building (around 44 th floor), and not in the conventional configuration of TMD, i.e., at the topmost floor;
* For acceleration mitigation purpose, the optimal frequency ratio and installation floor of TMDI is slightly larger than that of displacement-oriented optimization. Such differences may be justified in view of the fact that the transfer function of acceleration is \(-\omega^{2}\) times that of displacement, which means that a better mitigation effect of acceleration can be realized by decreasing the value of transfer function at higher frequency around the first peak under the same fluctuating wind excitations.  
Through the optimization results of TMDI at wind direction of 90\({}^{\circ}\) (the most adverse conditions), a set of optimal tuning parameters of TMDI with good trade-off between displacement mitigation and acceleration mitigation was selected by approximately averaging the two corresponding parameters due to the smooth gradient between two optimal schemes of TMDI (as shown in Figure 8). The notations \(\text{Dis}_{\text{min}}\) and \(\text{Acc}_{\text{min}}\) represent two configurations of TMDI that achieve the best displacement and acceleration mitigation effect, respectively.  
Figure 7: Peak displacement (**left**) and acceleration (**right**) at top floor of TMDI-equipped structure at wind direction of 90\({}^{\circ}\) in the \(v-\zeta-t\) space. **(a)** Distributions of peak displacement (**left**) and acceleration (**right**) at constant frequency ratios (\(v\) = 0.7, 0.8, 0.9, 1.0, 1.1, 1.2). **(b)** Distributions of peak displacement (**left**) and acceleration (**right**) at constant damping ratios (\(\zeta\) = 6%, 12%, 18%). **(c)** Distributions of peak displacement (**left**) and acceleration (**right**) at constant floor of TMDI installation (\(t\) = 30, 37, 44, 51, 58).