## 5 Numerical Verification  
### Methodology  
The frequency domain approach is introduced herein to attain the numerical dynamic response of the tower. As shown in Figure 21, for a structure with \(n\) degree of freedom (DOF), the governing equation of motion is  
\[\left[ \begin{array}{cc}\mathbf{m}+\mathbf{L}\mathbf{L}^{T}m_{d}&\mathbf{L}m_{d }\\ \mathbf{L}^{T}m_{d}&m_{d}\end{array} \right]\left\{ \begin{array}{c}\ddot{ \mathbf{x}}\\ \ddot{v}\end{array} \right\}+\left[ \begin{array}{cc}\mathbf{c}&0\\ 0&c_{d}\end{array} \right]\left\{ \begin{array}{c}\dot{\mathbf{x}}\\ \dot{v}\end{array} \right\}+\left[ \begin{array}{cc}\mathbf{k}&0\\ 0&k_{d}\end{array} \right]\left\{ \begin{array}{c}\mathbf{x}\\ v\end{array} \right\}=\left\{ \begin{array}{c}\mathbf{f}(t)\\ 0\end{array} \right\} \tag{14}\]  
where \(\mathbf{m}\), \(\mathbf{c}\), and \(\mathbf{k}\) are the mass, damping and stiffness matrices of the primary structure respectively, \(\mathbf{x}\) is the displacement vector, \(v=x_{\mathrm{d}}-x_{p}\), \(x_{p}\) is the \(p\)th nodal displacement at the attaching point of the ECTMD, \(x_{\mathrm{d}}\) is the displacement of the ECTMD, \(m_{\mathrm{d}}\), \(c_{\mathrm{d}}\) and \(k_{\mathrm{d}}\) are the mass, damping coefficient and stiffness of the ECTMD respectively, \(\mathbf{L}\) is the location vector of the ECTMD, and \(\mathbf{f}(t)\) is the vector of wind loading. Note that for the simplified model shown in Figure 21, \(\mathbf{m}\) is a diagonal matrix, i.e.,  
\[\mathbf{m}=\mathrm{diag}(m_{1},m_{2},\cdots,m_{n}) \tag{15}\]  
where \(m_{r}\) (\(r=1,\ldots,n\)) is the \(r\)th lumped mass. Figure 19: Variation of RMS acceleration with wind direction and wind speed. (**a**) 18.5 m/s; (**b**) 22.6 m/s; (**c**) 24.7 m/s. Figure 20: Variation of reduction ratio with wind direction and wind speed. (**a**) 18.5 m/s; (**b**) 22.6 m/s; (**c**) 24.7 m/s. The vector of the frequency response function is then obtained by [43]  
\[\left\{ \begin{array}{c}\mathbf{H}_{q}\\ H_{\mathbb{P}}\end{array} \right\}=\mathbf{A}^{-1}\{1,\cdots,1,0\}^{T} \tag{16}\]  
where  
\[\mathbf{A}=\left[ \begin{array}{cc}-\omega^{2}\Big{(}\mathbf{M}+\varphi^{T} \mathbf{L}\mathbf{L}^{T}\varphi m_{d}\Big{)}+i\omega\mathbf{C}+\mathbf{K}&- \omega^{2}\varphi^{T}\mathbf{L}m_{d}\\ -\omega^{2}\mathbf{L}^{T}\varphi m_{d}&-\omega^{2}m_{d}+i\omega c_{d}+k_{d} \end{array} \right] \tag{17}\]  
\(\mathbf{H}_{q}\) is the vector of frequency response function for the primary structure, \(H_{\mathbb{P}}\) is the frequency response function of the ECTMD, \(\mathbf{M}=\varphi^{T}\mathbf{m}\varphi\), \(\mathbf{C}=\varphi^{T}\mathbf{c}\varphi\), and \(\mathbf{K}=\varphi^{T}\mathbf{k}\varphi\) are the modal mass, modal damping and modal stiffness matrices of the primary structure respectively, \(\varphi\) is the mode shape matrix, and \(\omega\) is the circular frequency of the excitation. If \(\mathbf{m}\) and \(\varphi\) are available, then the modal mass matrix, \(\mathbf{M}\), can be obtained in the form of  
\[\mathbf{M}=\text{diag}(M_{1},M_{2},\cdots,M_{n}) \tag{18}\]  
Then the modal stiffness matrix, \(\mathbf{K}\), can be obtained by  
\[\mathbf{K}=\text{diag}(K_{1},K_{2},\cdots,K_{n}),\hskip 14.226378 ptK_{r}= \omega_{r}^{2}M_{r},\hskip 14.226378 ptr=1,2,\cdots n \tag{19}\]  
where \(\omega_{r}\) is the \(r\)th natural circular frequency. Furthermore, the modal damping matrix, \(\mathbf{C}\), is in a form of  
\[\mathbf{C}=\text{diag}(C_{1},C_{2},\cdots,C_{n}),\hskip 14.226378 ptC_{r}=2 \xi_{r}\omega_{r}M_{r},\hskip 14.226378 ptr=1,2,\cdots n \tag{20}\]  
where \(\xi_{r}\) is the modal damping ratio of the \(r\)th mode. If the two modal damping ratios of the prescribed modes are specified, the \(\xi_{r}\) can be thus computed by using Rayleigh damping [44].