# Wind-Induced Response Control of High-Rise Buildings Using Inerter-Based Vibration Absorbers
## 4 Wind-Induced Response Mitigation Using Inerter-Based Vibration Absorbers
### Effects of the Inerter-Based Vibration Absorbers on Wind-Induced Displacements  
After setting the parameters of the TMDI and TID systems, the \(\mathbf{M},\mathbf{C},\mathbf{K}\) matrices can be calculated by using Equation (2), respectively, hence the time histories of wind-induced displacements of the TMDI- and TID-equipped benchmark buildings can be analyzed corresponding to wind speed of 40.07 m/s (50-years return period stipulated by survivability limit state design) for each of the 24 wind directions from 0\({}^{\circ}\) to 345\({}^{\circ}\) according to Equation (8). Figure 13 represents a segment of the displacement time-history response (for an overall duration of 20 min) at the 69 th story corresponding to 90\({}^{\circ}\) wind direction. Figure 12: Acceleration transfer function at the top floor for different structural control schemes. Extreme wind-induced displacement response can be evaluated in terms of mean and Root Mean Square (RMS) value for each of the 24 wind directions considered in this study. Figure 14 presents variation of extreme displacement responses (as per Equation (10)) along the building height corresponding to three typical wind directions (\(0^{\circ}\), \(45^{\circ}\) and \(90^{\circ}\)). By inspection of Figure 14, it appears clear that TMD, TMDI and TID significantly decrease the wind-induced extreme displacement responses. The mitigation effects of TID are slightly better than that of TMD and TMDI. This demonstrates that the inerter plays a significant role in the vibration mitigation of the structure. Figure 14: _Cont._  
Figure 13: Time histories of displacement at the 69 th story along the \(x\)-axis corresponding to \(90^{\circ}\) wind direction. Figure 15 shows the variation of mean and extreme top-floor displacements corresponding to a variety of wind directions ranging from 0\({}^{\circ}\) to 345\({}^{\circ}\). The maximum absolute displacement is smaller than 1/1500 of the height of the benchmark building, which justifies the linear elastic behavior assumption made in this paper for the building dynamic model. Figure 15 indicates that three vibration absorbers have no effects on mean displacement responses, which in fact coincide with those of the OS. The mean displacement responses approach zero for wind direction corresponding to 90\({}^{\circ}\) and 270\({}^{\circ}\) wind-direction, because this corresponds to the across-wind response induced by the vortices shedding at both edges of the windward side. For any other wind direction, TMD, TMDI and TID significantly suppress wind-induced top-floor displacement responses along the \(x\)-axis. Based on Figures 13-15, it can be concluded that TID with same physical mass ratio achieved a slightly better vibration mitigation effect to the TMD and TMDI in terms of displacement response. Figure 14: Profile of extreme wind-induced displacement response at the \(x\)-axis along the building height corresponding to three different wind directions: (**a**) 0\({}^{\circ}\) wind direction. (**b**) 45\({}^{\circ}\) wind direction. (**c**) 90\({}^{\circ}\) wind direction. Figure 15: Mean and extreme top-floor displacement response at the \(x\)-axis corresponding to wind directions ranging from 0\({}^{\circ}\) to 345\({}^{\circ}\). To quantify the mitigation effect of TMD, TMDI and TID, a factor of vibration-absorbing \(F_{uu}\) is defined as follows  
\[F_{uu}(\%)=\left|\frac{R_{\text{OS}}-R_{\text{T}}}{R_{\text{OS}}}\right|\times 100 \tag{20}\]  
where \(R_{\text{OS}}\) represents the response of the original structure, while \(R_{\text{T}}\) denotes the corresponding response indicator for the structure with TMD\(\backslash\)TMDI\(\backslash\)TID. Some relevant results for a few emblematic wind directions and loading conditions are listed in Table 4. In particular, Table 4 presents the vibration-absorbing factor for some typical conditions.