# 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  
To compare the vibration mitigation effect of optimal TMD, TMDI and TID, performance-based optimizations for parameters of TMDI and TID were conducted to obtain the best TMDI/TID scheme. At the same time, the influence of variations of parameters in preset intervals on the vibration mitigation effect is also investigated.  
In the mathematical model of TMD/TMDI/TID-equipped structure as described in Equations (1) and (2), vibration absorbers are introduced to control the wind-induced response along the \(x\)-axis, which is the most critical direction due to the lower lateral stiffness of the building (higher oscillations are expected). As the largest displacement and acceleration responses at the top floor of primary structure occurs at wind direction of 90\({}^{\circ}\), the peak displacement and acceleration at the top floor induced by aerodynamic forces at this wind direction are selected to be the two individual objective functions.  
From Equation (2), there are totally six parameters of TMDI, i.e., \(\mu\), \(\beta\), \(\upsilon\), \(\zeta\), \(p\) and \(t\), that need to be fixed to calculate the wind-induced responses of TMDI-equipped structure (5 parameters for TID). The two mass related notation, i.e., mass ratio and inertance ratio, are defined as \(\mu=m_{t}/M\) and \(\beta=b/M\), respectively. The frequency ratio is defined as:  
\[\upsilon=\frac{\omega_{t}}{\omega_{1}}=\sqrt{\frac{k_{t}}{(m_{t}+b)}}/\omega_ {1} \tag{13}\]  
where \(\omega_{t}\) is the circular frequency of TMD, TMDI or TID, and \(\omega_{1}\) is the first order circular frequency of the primary structure. The damping ratio is defined as:  
\[\zeta=\frac{c_{t}}{2\sqrt{(m_{t}+b)k_{t}}} \tag{14}\]  
The other two discrete parameters are the topologies of inerter and the installation floor of vibration absorber as denoted in Equation (2).  
To shed light on the better vibration mitigation effect of inerter-based vibration absorbers, the physical mass ratio \(\mu_{phy}\) of TMD, TMDI and TID are defined below and fixed to the same value equal to 0.5%:  
\[\mu_{phy}=\mu+\beta/200=m_{\text{TMD, TMDI or TID}}/M=0.5\% \tag{15}\]  
Figure 6: Time histories of aerodynamic forces on typical stories corresponding to 90\({}^{\circ}\) wind direction.  
In the above-introduced physical mass ratio, not only the physical mass of the TMD (named as the attached mass), but also the physical mass of the inerter device is taken into consideration in the TMDI and TID schemes. Indeed, when dealing with large values of inertance (apparent mass of the inerter) in the order of tons, which might be the case for high-rise buildings, the physical mass of the inerter turns out to be non-negligible, whereas the majority of the literature studies ignored this term. In Equation (15), the physical mass ratio is constrained to be 0.5% based on the same threshold of TMD mass ratio proposed in [36], and the ratio of the inertance coefficient and physical mass of inertance devices is assumed to be 200 following the previous research about the \"mass-amplification\" effect of inerter [65]. Therefore, the mass-related parameter of the three vibration absorbers can be determined, i.e., TMD (\(\mu=0.5\%\)), TMDI (\(\mu=0.25\%\), \(\beta=50\%\)) and TID (\(\beta=100\%\)). This makes it possible to compare three different configuration schemes sharing a common physical mass ratio for wind-induced response mitigation of the benchmark building. According to the conclusion from previous research that inerter devices spanning more stories lead to a better mitigation effect of the TMDI, the value of the topologies is determined to be \(p=4\) in the optimization procedure, based on practical considerations like in the pendulum-like TMD scheme implemented in the Taipei 101 (spanning from 87 th floor to 91 st floor).