# Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine
## 3 Calculation Method  
### Calculation of Aerodynamic Loads under Consideration of NAF  
For any nacelle attitude angle, the horizontal axis wind turbine coordinate system shown in Figure 4 is established to facilitate the calculation of the spatial position and wind speed components of each wing section of the blade [14]. 1. The inertial coordinate system \(S\) coincides with the tower bottom coordinate system \(S_{BOT}\), and the nacelle coordinate system \(S_{N}\) coincides with the tower top coordinate system \(S_{top}\). 2. When the azimuth angle \(\theta\) = 0, the wind turbine rotation plane coordinate system \(S_{R}\) coincides with the hub coordinate system \(L_{H}\), and \(S_{R}\) rotates with the azimuth angle \(\theta\). 3. The tower height is \(L\), and the horizontal distance from the top of the tower to the hub is \(L_{H}\). 4. The spindle inclination angle and blade taper angle are \(\beta_{T}\) and \(\beta_{C}\). The transformation relationship between the blade coordinate system and the inertial coordinate system exists both in terms of rotation and translation. The transformation between the coordinate systems is described using the chi-square coordinates [15] as \(S= \begin{bmatrix}X&Y&Z&1\end{bmatrix} ^{T}\). The transformation relationship between the coordinate systems is as follows:  
\[S_{N}=K_{L}S \tag{23}\]  
\[S_{H}=K_{LH}K_{\beta T}K_{DN}R_{N}S_{N} \tag{24}\]  
\[S_{R}=K_{\theta}S_{H} \tag{25}\]  
\[S_{B}=K_{\beta C}S_{R} \tag{26}\]  
where: \(K_{L}\) is the transformation matrix from the inertial coordinate system to the nacelle coordinate system; \(K_{Lh}\) and sub \(K_{\beta T}\) are the translation and rotation transformation matrices from the nacelle to the hub coordinate system, respectively; \(R_{N}\) and \(K_{DN}\) are the attitude matrix and deflection matrix of the nacelle, respectively; \(K_{\theta}\) is the transformation  
Figure 4: Horizontal axis wind turbine coordinate system. matrix from the hub to the rotating coordinate system of the wind wheel, and \(K_{\beta C}\) is the transformation matrix from the wind wheel to the rotating blade. \[K_{L}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&-L\\ 0&0&0&1\end{array} \right] \tag{27}\]  
\[K_{LH}=\left[ \begin{array}{cccc}1&0&0&L_{H}/\cos\beta_{T}\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array} \right] \tag{28}\]  
\[K_{\beta T}=\left[ \begin{array}{cccc}\cos\beta_{T}&0&\sin\beta_{T}&0\\ 0&1&0&0\\ -\sin\beta_{T}&0&\cos\beta_{T}&0\\ 0&0&0&1\end{array} \right] \tag{29}\]  
\[K_{DN}=\left[ \begin{array}{cccc}1&0&0&x_{N}\\ 0&1&0&y_{N}\\ 0&0&1&z_{N}\\ 0&0&0&1\end{array} \right] \tag{30}\]  
\[K_{\theta}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos\theta&-\sin\theta&0\\ 0&-\sin\theta&\cos\theta&0\\ 0&0&0&1\end{array} \right] \tag{31}\]  
\[K_{\beta C}=\left[ \begin{array}{cccc}\cos\beta_{c}&0&-\sin\beta_{c}&0\\ 0&1&0&0\\ \sin\beta_{c}&0&\cos\beta_{c}&0\\ 0&0&0&1\end{array} \right] \tag{32}\]  
From Equations (23) to (32), the transformation relation between the blade coordinate system and the inertial coordinate system can be obtained as:  
\[S=K_{L}^{-1}R_{N}^{-1}K_{DN}^{-1}K_{\beta T}^{-1}K_{LH}^{-1}K_{\theta}^{-1}K_ {\beta C}^{-1}S_{B} \tag{33}\]  
Therefore, the coordinate component \(\left[ \begin{array}{cccc}X&Y&Z\end{array} \right]^{T}\) in the inertial coordinate system for the leaf element coordinates \(\left[ \begin{array}{cccc}X_{B}&Y_{B}&Z_{B}\end{array} \right]^{T}=\left[ \begin{array}{cccc}0&0&r\end{array} \right]^{T}\) is calculated by Equation (34). \[\left[ \begin{array}{cccc}X\\ Y\\ Z\\ 1\end{array} \right]=K_{L}^{-1}R_{N}^{-1}K_{DN}^{-1}K_{\beta T}^{-1}K_{LH}^{-1}K_ {\theta}^{-1}K_{\beta C}^{-1}\left[ \begin{array}{cccc}0\\ 0\\ r\\ 1\end{array} \right] \tag{34}\]  
The coordinate component of the wind speed \(U_{S}=\left[ \begin{array}{cccc}U&0&0\end{array} \right]^{T}\) in the inertial coordinate system in the blade coordinate system can be calculated by the following equation. \[\left[ \begin{array}{cccc}U_{BX}\\ U_{BY}\\ U_{BZ}\\ 1\end{array} \right]=R_{N}K_{\beta T}K_{\theta}K_{\beta C}\left[