# The Influence of Chamfered and Rounded Corners on Vortex-Induced Vibration of Super-Tall Buildings
## 4 Wind-Induced Response of Aeroelastic Model with Chamfered Corner  
Figure 6a shows the crosswind displacement of aeroelastic models with different chamfer ratios, where \(D\) is the cross-sectional width of the aeroelastic model, and \(\sigma_{y}/D\) is a percent sense. \(V_{r}\) is defined in Equation (3) as the reduced wind speed calculated by the wind speed at the top of the model. Since the structural parameters of each chamfer model are almost the same, the difference in test results of each model can be undoubtedly attributed to the effect of the chamfered corner.  
\[V_{r}=\frac{V}{n_{1}D} \tag{3}\]  
where \(V\), \(n_{1}\), and \(D\) denote mean wind velocity, system vibration frequency in the first mode, and width of the model's windward side, respectively.  
Figure 5: Free vibration result of the aeroelastic models with chamfered corners: (**a**) free vibration displacement of Case 2; (**b**) normalized free vibration displacement; (**c**) normalized PSD of free vibration displacement.  
Figure 6: Crosswind responses and the normalized PSD of models with different chamfer ratios: (**a**) RMS responses; (**b**) normalized PSD of crosswind response.  
As seen from Figure 6a, the RMS displacement of the non-chamfer model is the largest when \(V_{r}\) is close to the critical wind speed of VIV (\(V_{c}\), in short, which can be calculated by the \(St\) number of existing studies shown in Figure 7--\(V_{r}\) corresponding to the maximum VIV displacement will be slightly larger than that calculated by \(St\), but they are all called \(V_{c}\) in this paper for the sake of description). The displacement of the aeroelastic model with a 2.5% chamfer ratio is close to that of the non-chamfer model, which shows that reducing the chamfer ratio to 2.5% has an insignificant effect on VIV. However, when the chamfer ratio is increased to 5%, the crosswind displacement decreases significantly (especially when \(V_{r}\) is close to \(V_{c}\)), and the reduction of VIV response reaches 60%, where the reduction of VIV is computed by comparing experimental RMS displacement when \(V_{r}\) is close to \(V_{c}\). The above results show that the chamfer ratio above 5% has a remarkable effect on suppressing the VIV of super high-rise buildings with square sections. It should be pointed out that the RMS response of the model with the chamfer ratio 2.5% is much larger than that of the non-chamfer case with an unknown reason when the wind speed is larger than VIV wind speed. This phenomenon was not explained in this paper; it may be caused by the dispersion of test data.  
Take Case 1 as an example. Figure 6b shows the normalized PSD of crosswind displacement. As can be seen, there are three spectral peaks in Figure 6b, corresponding to vortex shedding frequency, first-order frequency, and second-order frequency of the aeroelastic model, respectively, where the spectral peak corresponding to second-order frequency is very insignificant. With the increase of wind speed, the spectral peak frequency related to vortex shedding increases correspondingly, and its contribution decreases rapidly. According to the spectral peak position, the first-order frequency of the model is about 10 Hz and the second-order frequency is about 32 Hz. By calculating the band area of the power spectrum corresponding to the first and second-order frequency, respectively, it was found that the contribution of the second-order vibration is less than 1% when the test wind speed is close to VIV wind speed. When the reduced wind speed increases to about 10, the wind-induced vibration is dominated by the first-order vibration. Compared with the contribution of the first-mode vibration, the contribution of the second-order vibration is insignificant.