# 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
#### 4.1.2 Optimization of Parameters of TID  
Similar to the configurations of the optimization in Section 4.1.1, the optimization of parameters of TID for mitigating peak displacement and acceleration at top floor can be expressed as Equations (18) and (19), respectively. \[\left\{ \begin{array}{ll}\text{minimize }&f_{1}(\
u,\zeta,t)=\hat{D}_{dis}\\ \text{s.t. }&\beta=100\%,-p=-4,\\ &v\in[0.7,1.2],\zeta\in[0,20\%],\\ &t\in[30,58]\end{array} \right. \tag{18}\]  
\[\left\{ \begin{array}{ll}\text{minimize }&f_{2}(\
u,\zeta,t)=\hat{D}_{acc}\\ \text{s.t. }&\beta=100\%,-p=-4,\\ &v\in[0.7,1.2],\zeta\in[0,20\%],\\ &t\in[30,58]\end{array} \right. \tag{19}\]  
The results of optimization of TID in \(v-\zeta-t\) space are displayed in Figure 9:  
Figure 8: Contour of (**a**) peak displacement (**b**) peak acceleration with variable frequency ratio, damping ratio and floor of TMDI installation. Figure 9: _Cont._The similar trends of the distribution of the parameters of TID as that of TMDI can be observed by comparing the Figures 7 and 9. For TID having inertance ratio of 100%, which is twice that of TMDI, the highest reduction of displacement and acceleration responses is achieved when the damping ratios are 12% and 19%, respectively, which are almost twice those of TMDI. The inerter devices in TID scheme produce larger inertia, which corresponds to a better ability to store energy, thus the corresponding requirement for dissipating rate of energy stored in both inerter and attached mass increases at the same time. Based on the same considerations of determining the optimal parameters of TMDI as shown in Figure 10, the configuration of the optimally-designed TID is determined and listed in Table 3. Slightly different from the result of that of TMDI, the point inside the orange square, which represents the selected configuration of TID, lies close to the dashed line due to the limitation of discrete parameter, i.e., floor of TMDI installation. Figure 10: Contour of (**a**) peak displacement (**b**) peak acceleration with variable frequency ratio, damping ratio and floor of TID installation. Figure 9: Peak acceleration at top floor of Tuned Inerter Damper (TID)-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). Apart from the two inerter-based vibration absorbers, the optimal parameters of TMD with fixed mass ratio equal to 0.5% (and, thus, equal physical mass ratio to that of the TMDI and TID) are determined by performing the same optimization procedure. As stated above, the optimal parameters of TMD, TMDI and TID in a comparison group are listed in Table 3. It can be seen that the TMDI and TID scheme benefit a lot from the inerter system on the effective mass. Once the optimal parameters of the TMD/TMDI/TMD are selected, the corresponding mass, damping and stiffness matrices can be determined according to Equation (2). Therefore, the transfer function of displacement and acceleration response as per Equation (5) can be computed to assess the effects of TMD/TMDI/TID in mitigating the wind-induced response. Figures 11 and 12 present the modulus of the transfer function of displacement and acceleration responses, respectively, at the 69 th story (top floor) of the benchmark 340 m tall building subject to the aerodynamic forces consistent with the wind tunnel test measurements. In Figure 11, the first peaks of displacement transfer function of the original structure, structure with TMD, TMDI and TID appear around the first natural frequency (1.10 rad/s, consistent with 0.176 Hz reported in Table 1).