# Influence of the Flexible Tower on Aeroelastic Loads of the Wind Turbine
## 2 Analytical Study
### Tower Finite Element Modeling  
In this paper, the top mass of the tower is simplified to a concentrated mass block and attached to the top of the tower. As for the thin-walled tower with a variable cross-section, a two-node Euler-Bernoulli beam is used to discretize the finite element of the wind turbine tower, and a cubic Hermite polynomial is chosen as the form function of each beam unit. Figure 2 shows a beam unit of the wind turbine tower with length \(l\), mass per unit length \(m(x)\), and bending stiffness \(EI(x)\). The two nodes of this unit are located at the two endpoints of the beam; through these two nodes, the finite unit can be assembled into a single structure. If only plane displacement is considered, each node of the beam unit has only two degrees of freedom, which are lateral displacement and rotation. The relationship between the displacement of the beam unit and the four degrees of freedom can be expressed as:  
\[u(x,t)=\sum_{i=1}^{4}u_{i}(t)\psi_{i}(x) \tag{2}\]  
where \(\psi_{i}(x)\) is defined as the unit displacement due to the occurrence of unit displacement \(u_{i}(x)\) while keeping the other degrees of freedom at zero. \(\psi_{i}(x)\) satisfies the following boundary conditions:  
Figure 1: Simplified model of the tower system. \[\left. \begin{array}{l}i=1:\psi_{1}(0)=1,\psi_{1}^{\prime}(0)=\psi_{1}(l)=\psi_{1} ^{\prime}(l)=0\\ i=2:\psi_{2}^{\prime}(0)=1,\psi_{2}(0)=\psi_{2}(l)=\psi_{2}^{\prime}(l)=0\\ i=3:\psi_{3}(l)=1,\psi_{3}(0)=\psi_{3}^{\prime}(0)=\psi_{3}^{\prime}(l)=0\\ i=4:\psi_{4}^{\prime}(0)=1,\psi_{4}(0)=\psi_{4}^{\prime}(0)=\psi_{4}(l)=0\\ \end{array} \right\} \tag{3}\]  
The equilibrium equation for a beam of equal cross-section considering only the load acting at both ends without taking into account the shear deformation is:  
\[EI\frac{d^{4}u}{dx^{4}}=0 \tag{4}\]  
Its general solution is: \(u(x)=a_{1}+a_{2}\left(\frac{x}{\tau}\right)+a_{3}\left(\frac{x}{\tau}\right)^{ 2}+a_{4}\left(\frac{x}{\tau}\right)^{3}\). The constant \(a_{i}\) can be determined by substituting the boundary conditions, which gives:  
\[\left. \begin{array}{l}\psi_{1}(x)=1-3\left(\frac{x}{\tau}\right)^{2}+2 \left(\frac{x}{\tau}\right)^{3}\\ \psi_{2}(x)=l\left(\frac{x}{\tau}\right)-2l\left(\frac{x}{\tau}\right)^{2}+l \left(\frac{x}{\tau}\right)^{3}\\ \psi_{2}(x)=3\left(\frac{x}{\tau}\right)^{2}-2\left(\frac{x}{\tau}\right)^{3} \\ \psi_{4}(x)=-l\left(\frac{x}{\tau}\right)^{2}+l\left(\frac{x}{\tau}\right)^{3} \end{array} \right\} \tag{5}\]  
where \(\psi_{1}(x)\), \(\psi_{2}(x)\), \(\psi_{3}(x)\), and \(\psi_{4}(x)\) are the cubic Hermite interpolation functions, which ensure the continuity of deflection and cornering between beam units at the boundary. Using the principle of imaginary displacement for a beam unit with bending stiffness \(EI(x)\) and length \(l\), the general expressions for the stiffness influence coefficient \(k_{ij}\) and the mass influence coefficient \(m_{ij}\) of the beam unit can be derived as follows:  
\[k_{ij}=\int_{0}^{l}EI(x)\psi_{i}^{n}(x)\psi_{j}^{n}dx \tag{6}\]  
\[m_{ij}=\int_{0}^{l}m(x)\psi_{i}(x)\psi_{j}dx \tag{7}\]  
Figure 2: Beam cell model, (**a**) beam cell degrees of freedom, (**b**) interpolation function. For cells with \(EI(x)=EI\) and uniformly distributed masses (\(m(x)\equiv\overline{m}\)), when \(i,j=1,2,3\), and \(4\), the analytical solutions of the cell stiffness matrix and the consistent mass matrix can be calculated as:  
\[\overline{k_{e}}=\frac{EI}{l^{3}}\left[ \begin{array}{cccc}12&6l&-12&6l\\ 6l&4l^{2}&-6l&2l^{2}\\ -12&-6l&12&-6l\\ 6l&2l^{2}&-6l&4l^{2}\end{array} \right] \tag{8}\]  
\[\overline{m_{e}}=\frac{ml}{420}\left[ \begin{array}{cccc}156&22l&54&-13l\\ 22l&4l^{2}&13l&-3l^{2}\\ 54&13l&156&-22l\\ -13l&-3l^{2}&-22l&4l^{2}\end{array} \right] \tag{9}\]  
The wind turbine tower is always subjected to the gravitational force of the top mass of the tower, and this axial pressure also has a certain degree of influence on the stiffness of the tower.