## 3 Results and Discussions
### Power Spectral Densities from Aeroelastic Tests  
Power spectral densities of displacements at hub height were obtained to investigate the characteristics of vortex shedding condition of tower-only and wind turbine cases. One example of the power spectral densities of the octagonal tower-only case is shown in Figure 8 for some reduced velocities. The dotted vertical line indicates the system's natural frequency. For reduced velocity \(U^{*}\) = 1.21 (Figure (a)a), two clear peaks were found. The left peak corresponded to the vortex shedding frequency \(f_{s}\) (\(\blacktriangledown\) in the figure) and the right peak to the system's natural frequency \(f_{0}\) (\(\bullet\) in the figure), showing a slightly greater power for the system's natural frequency. When the reduced velocity increased near \(U^{*}\) = 6.25 (Figure (b)b), only one peak was found, indicating that a lock-in mechanism occurred. As the reduced velocity further increased, a peak corresponding to the vortex shedding frequency was again able to be found in the high-frequency range (Figure (c)c). The peak in the high-frequency range was quite small, and the reason for this can be understandable when the power spectral density of the displacement is calculated from the displacement mechanical admittance and the power spectral density of the generalized force [16]. The small peak in the high-frequency range after the lock-in range resulted from the characteristics of the displacement admittance function, i.e., it gave an almost constant value when the frequency was lower than the system's natural frequency, but it dropped quickly right after the system's natural frequency and approached zero. This quick drop was attributed to the small peak in the high-frequency range. Similar discussions can be made for other polygonal towers, but it was found that the peaks in vortex shedding frequency were largely suppressed for polygonal cross-sectional towers. For the reduced velocity where instability vibrations were observed, the peak powers corresponding to the system's natural frequencies greatly increased, as also explained by [PERSON] [17]. The reduced shedding frequencies \(f_{s}/f_{0}\), which were identified from the peaks in power spectral densities, are plotted in Figure 9 with reduced velocities \(U^{*}\) for several polygonal towers. The obvious lock-in range where the vortex shedding frequency was close to the system's natural frequency deviating from the linear trend was clearly found. The lock-in range for the square and octagonal towers began near reduced velocity \(U^{*}\) = 6.25, and for the dodecagonal and tetradecagonal towers started near reduced velocity \(U^{*}\) = 4.32, showing a larger lock-in range. For the dodecagonal and tetradecagonal towers, even though lock-in began at a smaller reduced velocity, peak powers in power spectral densities were smaller than those for the square and octagonal towers. When the reduced shedding frequencies \(f_{s}/f_{0}\) were plotted with reduced velocities \(U^{*}\), the slopes shown in Figure 9 can be interpreted as Strouhal number \(S_{t}\), which is a dimensionless parameter commonly used to examine the characteristics of vortex shedding frequency. The \(S_{t}\) of the square tower (circular symbol) was about 0.14, the \(S_{t}\) of the octagonal tower (square symbol) was about 0.16, the \(S_{t}\) of the dodecagonal tower (star symbol) was about 0.17,  
Figure 7: Maximum fluctuating displacements for wind turbines installed on seven polygonal towers [15]. and the \(S_{t}\) of the tetradecagonal tower (triangular symbol) was about 0.18, showing larger Strouhal number with increasing number of sides. Figure 10 shows the power spectral densities of the wind turbine installed on the square, square helical, and circular towers for wind direction \(\theta_{uzd}=0^{\circ}\) and azimuth angle \(\theta_{azi}=0^{\circ}\).