# Wind-Induced Response Control of High-Rise Buildings Using Inerter-Based Vibration Absorbers
## 3 Description of the 340 m Tall Building and Wind Tunnel Testing  
The primary structure has 69 stories and total height of 340 m. This benchmark building represents the _Qiaokou_ tower, located in Wuhan City, China, built in 2012. The photograph of the building and the main peculiarities of its structural configuration are presented in Figure 2.  
The mass distribution of the primary structure, including dead load and live load, has been extracted from the finite element model of the building and is reported in Figure 3a. Similarly, the distribution of the lateral stiffness along the \(x\)-axis, which is much smaller than that along the \(y\)-axis, has been extracted from the finite element model as well, and is presented in Figure 3b. Mass and stiffness matrices can be established in terms of mass and lateral stiffness distributions. It is worth noting that the developed model of the case study building accounts for the primary structural elements only. The presence of secondary structural components (here not considered for simplicity) could slightly modify the results in terms of interstory displacements and could add some stiffening contributions in the overall building model. Figure 2: Photograph of the 340 m tall building (**left**) and plan view (**right**). The full damping matrix \(\mathbf{C}_{s}\) of the original structure (without vibration absorber) has been calculated from the modal damping matrix \(\mathbf{C}_{\text{mod}}\)[62]:  
\[\mathbf{C}_{s}=\left(\mathbf{\Phi}^{T}\right)^{-1}\mathbf{C}_{\text{mod}}\left( \mathbf{\Phi}\right)^{-1} \tag{11}\]  
where \(\mathbf{\Phi}\) is the modal shape matrix of the original structure. The modal damping matrix \(\mathbf{C}_{\text{mod}}\in\mathbb{R}^{n\times n}\) is a diagonal matrix collecting the modal damping ratios and can be calculated as follows  
\[\mathbf{C}_{\text{mod}}(k,k) =2\xi_{k}\omega_{k}\Big{(}\varphi_{k}^{T}\mathbf{M}_{s}\mathbf{ \varphi}_{k}\Big{)};\ k=1,2,\ldots,69 \tag{12}\]  
where \(\omega_{k}\mathbf{\varphi}_{k}\) are the \(k\)th natural frequency and vibration mode, respectively. The \(k\)th modal damping ratio of the system \(\xi_{k}\) is taken equal to 1% for \(k\) = 1, 2, 3; 4% for \(k\) = 4, 5, 6; 6% for \(k\) = 7, 8, 9, 10; 9% for \(k\) = 11, 12, \(\ldots\), 20; 12% for \(k\) = 21, 22, \(\ldots\), 40; 15% for \(k\) = 41, 42, \(\ldots\), 60; 18% for \(k\) = 61, 62, \(\ldots\), 69. These values were selected based on available field-recorded of high-rise steel framed buildings in the [0-7] Hz frequency range [63]. The main dynamic parameters of the primary structure are listed in Table 1.  
The synchronous multi-point pressure tests of the building with existing surrounding conditions were performed in boundary layer wind tunnel tests (shown in Figure 4) under a simulated C type wind field corresponding to China load code for the design of building structures [64], which reflects the characteristics of the wind field in urban areas. The profiles of mean wind speed and turbulence intensity are shown in Figure 5a. The reference coordinates of the wind tunnel test and wind-induced response analysis are shown in Figure 5b. 24 wind directions are considered in this study, which are identified by a \(\beta_{\text{w}}\) angle (between wind axis and \(x\)-axis) ranging from \(0^{\circ}\) to \(345^{\circ}\) at an interval of \(15^{\circ}\). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Total Mass \(M\)** & **First-Order Natural** & **First-Order** & **First-Order Damping** \\  & **Frequency along \(x\)-axis \(\omega_{1}\)** & **Generalized Mass** & **Ratio (Assumed)** \\ \hline
231,659 t & 0.176 Hz & 61,287 t & \(1\%\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main dynamic parameters of the primary structure. Figure 3: Dynamic properties of the benchmark building extracted from the finite element model: (**a**) mass distribution; (**b**) lateral stiffness distribution.