## 5 Numerical Verification
### Numerical Results  
The tower is divided into 17 segments, as shown in Figure 22, and the relevant parameters are listed in Table 5. Thus, for numerical analysis, a simplified model with 17 corresponding nodes is employed. The location of each node is set at the center of the corresponding segment. Therefore the \(r\)th lumped mass is the mass of the \(r\)th segment. The mode shapes used herein are derived from the finite element analysis of the prototype structure, in which seventeen mode shapes with corresponding natural frequencies are obtained. The modal damping ratios of the first two modes in the Y direction are set as 4.4%, which is identified from the experimental dynamic response results. Thus, the modal mass, modal damping and modal stiffness matrixes can be computed, according to Equations (18)-(20).  
Figure 22: Segments of prototype structure.  
In the case of the along-wind direction, by using the aforementioned theoretical approach in the frequency domain analysis, the dynamic responses of the tower with and without the ECTMD are attained. For different mass ratios, the reduction ratios varying with the damping ratio of ECTMD and frequency ratio are illustrated in Figure 23. There exists an optimal frequency ratio and ECTMD damping ratio, whereby the maximum reduction ratio can be achieved. The maximum reduction ratios for mass ratios of 0.02, 0.04, and 0.063 are 15.8%, 21.2%, and 25.1% respectively. For the mass ratio of 0.063 used in the experiment, it is found that the optimal frequency ratio and ECTMD damping ratio in theory are 0.98 and 0.11 respectively.  
By using the parameters in the experiment, the theoretical results are compared with the experimental results in the along-wind direction, as shown in Figure 24 and Table 6. It can be found from Figure 24 that the theoretical reduction ratios generally match well with the experimental values as the frequency ratio and damping ratio of the ECTMD change, though the experimental reduction ratios are a little lower. The theoretical highest reduction ratio corresponds to the frequency ratio of 0.97 and the ECTMD damping ratio of 0.11, whereas the experimental highest reduction ratio corresponds to the frequency ratio of 0.97 and the ECTMD damping ratio of 0.14. Under the frequency ratio of 0.97 and the ECTMD damping ratio of 0.14, the experimental reduction ratio is 23.1%, whereas the theoretical reduction ratio is 24.8%.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Segment No.** & **Segment Mass (kg)** & **Segment Height (m)** & **Shape Coefficient** & **Windward Area (m\({}^{2}\))** \\ \hline  
\hline \hline \end{tabular}
\end{table}
Table 5: Segment parameters of the prototype tower.  
Figure 23: Reduction ratio over damping ratio and frequency ratio. (**a**) Mass ratio = 0.02; (**b**) Mass ratio = 0.04; (**c**) Mass ratio = 0.063.  
Table 6 shows that at wind speeds of 24.7 and 22.6 m/s, the theoretical results of acceleration and reduction ratio match well with the experimental results. In case of the wind speed of 18.5 m/s, the theoretical results of acceleration are a litter lower, whereas the reduction ratio computed via the frequency domain approach is slightly higher than the experimental results. Generally speaking, the comparison shows that the theoretical approach in frequency domain introduced herein is capable of producing a good prediction of the dynamic responses of the tower with or without ECTMD.