# The Influence of Chamfered and Rounded Corners on Vortex-Induced Vibration of Super-Tall Buildings
## 3 Reduction Effect of Chamfered Corner on VIV  
_Aeroelastic Model with Chamfered Corner_  
Figure 3 shows the cross-sectional diagram of the chamfered-corner model. The chamfer ratio is 0%, 2.5%, 5%, 7.5%, and 12.5% successively, where the chamfer ratio is defined as the ratio of the chamfer size of one corner to the cross-sectional width. Considering the time and economic costs, the model skeleton was repeatedly used and the chamfer ratios were adjusted by the corresponding shapes of strips (See Figure 4). There is a red laser spot near the top of the model in Figure 4, which is the measured height of the wind-induced displacement.  
Figure 3: Cross-sections of MDOF model with chamfered corner (unit: mm): (**a**) chamfer ratio 0%; (**b**) chamfer ratio 2.5%; (**c**) chamfer ratio 5%; (**d**) chamfer ratio 7.5%; (**e**) chamfer ratio 12.5%.  
Figure 2: Wind tunnel and measuring instrument of this study: (**a**) aerodynamic contour of the wind tunnel; (**b**) simulation of the wind field; (**c**) mean wind velocity; (**d**) turbulence intensity; (**e**) laser displacement meter.  
Table 1 shows the structural parameters of different chamfer models. In Table 1, \(n_{0}\) is the primary frequency of each model, \(M\) is equivalent mass, \(\xi\) is structural damping ratio, \(Sc\) is Scruton number, and \(R_{c}\) is the chamfer ratio. As seen from Figure 4 and Table 1, the cross-sectional shapes of each model are smooth from top to bottom and the structural parameters of each model have little difference, which indicates that the above method of adjusting the chamfer ratio is satisfactory. As the structural parameters of all these test cases are the nearly same except the chamfer ratio, it can be reasonably stated that the difference in wind-induced vibration between these cases was caused by the differences in chamfer ratio.  
The generalized mass (\(M\)) and Scruton number (\(Sc\)) can be expressed by Equations (1) and (2), respectively.  
\[M=\frac{\int_{0}^{H}m(z)\phi^{2}(z)d_{z}}{\int_{0}^{H}\phi^{2}(z)d_{z}} \tag{1}\]  
\[Sc=\frac{2M\xi_{S}}{\rho_{a}D^{2}} \tag{2}\]  
where \(m(z)\) and \(\phi(z)\) are the mass per unit length and the mode shape of the model, respectively; \(H\) is the height of the model; \(\rho_{a}\) is the air density, and \(D\) is the width of the model.  
The dynamic parameters in Table 1 are determined by the free vibration of MDOF models. Take Case 1 as an example: Figure 5a shows the time history of free vibration displacement obtained by a repeated hammering test. The attenuation time histories of other cases can be obtained by the same method. The time histories of normalized free vibration displacement are shown in Figure 5b and the PSD of free vibration displacement is shown in Figure 5c, where \(\sigma_{y}\) is the RMS displacement. As seen from Figure 5, the free vibration attenuation results of the aeroelastic model are very stable and satisfactory, and the free vibration results of each case are very close.  
\begin{table}
\begin{tabular}{c c c c c c} \hline
**Case** & \(\mathbf{n_{0}}\) & \(\mathbf{M}\) & \(\xi\) & \(Sc\) & \(\mathbf{R_{c}}\) \\ \hline
1 & 9.9 Hz & 2.1 kg/m & 2.4\% & 8.1 & 0\% \\
2 & 9.6 Hz & 2.2 kg/m & 2.3\% & 8.0 & 2.5\% \\
3 & 9.6 Hz & 2.2 kg/m & 2.2\% & 7.8 & 5\% \\
4 & 9.2 Hz & 2.1 kg/m & 2.4\% & 7.9 & 7.5\% \\
5 & 9.2 Hz & 2.0 kg/m & 2.6\% & 8.3 & 12.5\% \\ \hline \end{tabular}
\end{table}
Table 1: The structural parameters of models with different chamfer ratio.  
Figure 4: Some photos of the MDOF model and coated strips for adjusting the chamfer ratio.