2.5.2. Thrust Wind Load

The wind load applied to the blades and turbine will produce lateral load, which can be considered as the concentrated load *Fb*, calculated according to Equation (51) [35]:

$$F\_b = 1/2\rho\_a \pi R\_T^2 V\_T^2 C\_T(\lambda\_s) \tag{51}$$

where *VT* is the wind speed at the tower top, *RT* is the blade radius, and *CT* is the thrust coefficient, which is related to the tip speed ratio *λs*, as shown in Figure 5.

**Figure 5.** Thrust coefficient *CT*-tip speed ratio *λ<sup>s</sup>* curve [35].

#### 2.5.3. Tower Dynamic Response

The tower can be divided into n segments, numbered 1~*n* from the bottom to the top. As shown in Figure 6, for the *i* (*i* = 1, 2 ··· *n*)th segment, the number of its bottom point is *i*, and the number of its top point is *i* + 1. Consider each segment as equal cross-section beam, where *Hi* is the length of the *i*th segment. *Hi* is valued differently between segments to account for the varied cross-section geometry. The mass of each segment is concentrated to the bottom point, and the 2 × 2 mass matrix of each segment can be obtained, including the mass and moment of inertia. The mass of blades is considered as the concentrated mass point *mn*+<sup>1</sup> at the tower top. Similarly, the distributed loads applied to each segment are concentrated to the bottom point, and the load matrix of each segment can be obtained, including the force and bending moment. The load matrix at the tower top includes the thrust wind force.

**Figure 6.** Tower discretization.

The swaying-rocking equation of this system with 2(*n*+1) degrees of freedom can be written in matrix form [25]:

$$
\begin{bmatrix}
\overline{\mathbb{K}}\_{11} & \overline{\mathbb{K}}\_{12} & & & \\
\overline{\mathbb{K}}\_{21} & \overline{\mathbb{K}}\_{22} & \mathbb{K}\_{23} & & \\
& \ddots & \ddots & \ddots & \\
& & \overline{\mathbb{K}}\_{i,j-1} & \overline{\mathbb{K}}\_{i,j} & \overline{\mathbb{K}}\_{i,j+1} & \\
& & & \ddots & \ddots & \ddots & \\
& & & & \overline{\mathbb{K}}\_{n,n-1} & \overline{\mathbb{K}}\_{n,n} & \overline{\mathbb{K}}\_{n,n+1} \\
& & & & \overline{\mathbb{K}}\_{n+1,n} & \overline{\mathbb{K}}\_{n+1,n+1}
\end{bmatrix}
\begin{Bmatrix}
u\_{1} \\ \boldsymbol{u}\_{2} \\ \vdots \\ \boldsymbol{u}\_{i} \\ \vdots \\ \boldsymbol{u}\_{n} \\ \boldsymbol{u}\_{n+1}
\end{Bmatrix} = 
\begin{Bmatrix}
F\_{1} \\ F\_{2} \\ \vdots \\ F\_{i} \\ F\_{i} \\ \vdots \\ F\_{n} \\ F\_{n+1}
\end{Bmatrix} \tag{52}
$$

where *ui* is the point displacement matrix, including the horizontal displacement and rotation angle. The stiffness in Equation (52) can be calculated according to equations below. For the first line:

$$\overline{K}\_{11} = K\_{11} + K^\* - \omega^2 m\_1 \tag{53}$$

$$
\overrightarrow{K}\_{12} = K\_{12} \tag{54}
$$

For the second to *n*th line:

$$
\overline{K}\_{i,i-1} = K\_{i,i-1} \tag{55}
$$

$$
\overline{K}\_{\dot{i},\dot{i}} = K\_{\dot{i},\dot{i}} - \omega^2 m\_{\dot{i}} \tag{56}
$$

$$
\overline{K}\_{i,i+1} = K\_{i,i+1} \tag{57}
$$

For *n* + 1th line:

$$\mathbb{X}\_{n+1,n} = \mathbb{K}\_{n+1,n} \tag{58}$$

$$\mathcal{K}\_{n+1,n+1} = \mathcal{K}\_{n+1,n+1} - \omega^2 m\_{n+1} \tag{59}$$

where

$$\mathcal{K}\_{i,i-1} = \begin{bmatrix} -12E\_p I\_{i-1} / H\_{i-1}^3 & -6E\_p I\_{i-1} / H\_{i-1}^2 \\ 6E\_p I\_{i-1} / H\_{i-1}^2 & 2E\_p I\_{i-1} / H\_{i-1} \end{bmatrix} \tag{60}$$

$$K\_{i,i} = \begin{bmatrix} 12E\_p I\_i / H\_i^3 & 6E\_p I\_i / H\_i^2 \\ 6E\_p I\_i / H\_i^2 & 4E\_p I\_i / H\_i \end{bmatrix} + \begin{bmatrix} 12E\_p I\_{i-1} / H\_{i-1}^3 & -6E\_p I\_{i-1} / H\_{i-1}^2 \\ -6E\_p I\_{i-1} / H\_{i-1}^2 & 4E\_p I\_{i-1} / H\_{i-1} \end{bmatrix} \tag{61}$$

$$K\_{i,j+1} = \begin{bmatrix} -12E\_p I\_i / H\_i^3 & 6E\_p I\_i / H\_i^2 \\ -6E\_p I\_i / H\_i^2 & 2E\_p I\_i / H\_i \end{bmatrix} \tag{62}$$

When *i* = 1, Equation (60) is invalid, and the second part of Equation (61) should be removed. When *i* = *n* + 1, Equation (57) is invalid, and the first part of Equation (61) should be removed. Substitute the foundation impedance matrix *K*∗ into Equation (53), the equation can then be solved, and the displacement of the tower can be obtained.

#### **3. Validation**

The calculation result is compared with the FEM result to validate the correctness of the proposed calculation model. ABAQUS simulation software is used in this paper. A pile group supported OWT [25] is used for validation. The total mass of blades and turbine is 177.1 ton, and the tower is divided into three segments, as shown in Table 1. The pile group consists of seven piles, as shown in Figure 2. The pile diameter is 1.7 m and the pile wall thickness is 30 mm. The elastic modulus of the steel is 210 GPa and the density of the steel is 7800 kg/m3. The pile length embedded in the soil is 30 m nad the pile length submerged in the seawater is 20 m. The elastic modulus of the soil is 40 MPa, the Poisson's ratio of the soil is 0.3, and the density of the soil is 1800 kg/m3. As shown in Figure 7, the offshore wind turbine model is established.

**Table 1.** Tower parameters.


**Figure 7.** FEM numerical simulation model.

Since the overall displacement of the offshore wind turbine is relatively small, linear modal analysis is used to calculate its structure natural frequency. The deformation of soil foundation is also small during the dynamic analysis; therefore, the change of soil foundation stiffness is not considered, and the small strain linear elastic model is used for the soil.

For the tower, the eight-node S8R shell element is used. The tower is divided into three parts, as shown in Table 1. For each part, the tower diameter changes linearly with increasing height. To prevent the separation between tower parts, a bonding constraint is added between the interfaces, including the interface between the bottom tower part and the cap. The blades are simplified as the concentrated force applied to the tower top. The pile foundation is modeled as solid element to better simulate the pile–soil interaction. For the side face of the soil, the lateral displacement is constrained; for the bottom side of the soil, displacements of all directions are constrained. The pile–soil interaction is set as small sliding, penalty contact, while the coefficient is set as 0.4.

The dead weight is applied to the model according to parameters presented in Table 1, and the wave load is also applied to the structure. Here, we use FORTRAN to write a subroutine to accurately input wave load according to Equation (4). The offshore wind turbine is fine meshed. As for the soil, the soil around the pile is refined, as shown in Figure 8.

**Figure 8.** Meshing of the model.

After establishing the FEM model, the modal analysis is performed, and the first 10 structure natural frequencies are obtained. Here, we focus on the first lateral structure natural frequency. Then, we applied a 10 kN horizontal harmonic thrust wind load to the tower top, and the tower top displacement and load frequency curve can be obtained.

As shown in Figure 9, the FEM result and the calculating result are in good agreement, with only some differences being observed for the maximum tower displacement, which validates the correctness of the calculating result. The structure natural frequency can also be obtained from Figure 9, which falls within the "soft-stiff" design frequency range.

**Figure 9.** Comparison between the calculation result and FEM result.
