# Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine
## 2 Analytical Study
### Tower Finite Element Modeling
Using Hermite cubic polynomials as an interpolation function yields the general form of the unit geometric stiffness influence coefficient:  
\[k_{Gij}=\int_{0}^{l}N(x)\psi_{i}^{\prime}(x)\psi_{j}^{\prime}dx \tag{10}\]  
where \(N(x)\) denotes the axial force acting on the unit. When \(N(x)=N_{G}\) is constant, the consistent geometric stiffness matrix of the unit is obtained as:  
\[\overline{K_{Ge}}=\frac{N_{G}}{30l}\left[ \begin{array}{cccc}36&3l&-36&3l\\ 3l&4l^{2}&-3l&-l^{2}\\ -36&-3l&36&-3l\\ 3l&-l^{2}&-3l&4l^{2}\end{array} \right] \tag{11}\]  
The axial force along the full length of the unit is assumed to be constant. By superimposing the above two effects, elastic and geometric, the total stiffness matrix of the structural unit is obtained, expressed as:  
\[\overline{k}=\overline{k_{e}}-\overline{K_{Ge}} \tag{12}\]  
where the negative sign indicates that the presence of axial load \(N(x)\) increases the deflection of the tower. If units \(m\), \(n\), and \(p\) are connected to node \(i\) of the structure, the stiffness coefficient of this node can be calculated using the direct stiffness method as:  
\[\hat{\overline{k}}_{ii}=\hat{\overline{k}}_{ii}^{(m)}+\hat{\overline{k}}_{ii}^ {(n)}+\hat{\overline{k}}_{ii}^{(p)} \tag{13}\]  
where the notation \({}^{\sim}\) indicates the coefficient in the overall coordinate system. The stiffness matrix of the whole structure can be obtained by Equation (13). Similarly, the consistent mass matrix of the whole structure can be calculated. The damping characteristics of structures generally need to be determined by experimental methods, but it would be quite difficult and impractical to obtain the damping of each structure by experimental tests. Therefore, the dynamic properties of a structure are often analyzed by using a classical damping model. In this paper, the Rayleigh damping model is used, which integrates the effects of mass-proportional damping and stiffness-proportional damping, and its calculation equation is as follows:  
\[C=\eta_{0}M+\eta_{1}K \tag{14}\]  
where \(\eta_{0}\) and \(\eta_{1}\) are scaling factors. \[\left( \begin{array}{c}\eta_{0}\\ \eta_{1}\end{array} \right)=\frac{2\omega_{m}\omega_{n}}{\omega_{n}^{2}-\omega_{m }^{2}}\left( \begin{array}{cc}\omega_{n}&-\omega_{m}\\ -1/\omega_{n}&1/\omega_{m}\end{array} \right)\left( \begin{array}{c}\xi_{m}\\ \xi_{n}\end{array} \right) \tag{15}\]  
where \(\omega_{m}\) and \(\omega_{n}\) are the m-order and n-order frequencies of the vibrating system, and \(\xi_{m}\) and \(\xi_{n}\) are the damping ratios of the corresponding modal frequencies [8]. When \(\xi_{m}=\xi_{n}=\xi\), it can be simplified as follows:  
\[\left( \begin{array}{c}\eta_{0}\\ \eta_{1}\end{array} \right)=\frac{2\xi_{n}^{\prime}}{\omega_{m}+\omega_{n}} \left( \begin{array}{c}\omega_{m}\omega_{n}\\ 1\end{array} \right) \tag{16}\]  
In practical engineering applications, \(\omega_{m}\) is usually taken as the fundamental frequency of the vibrating system, and \(\omega_{n}\) is taken as the higher order frequency that has a significant effect on the dynamic behavior of the system. When the modal damping and modal frequency of the tower system are known, its damping matrix can be determined by Equations (15) and (16). After calculating the mass matrix, stiffness matrix and damping matrix of the whole tower, the multi-degree-of-freedom finite element numerical model of the tower system can be established by treating the top mass of the tower as a boundary condition [10].