An Efficient and High-Precision Nonlinear Co-Rotational Beam Method for Wind Turbine Blades Considering Tapering Effects and Anisotropy

Abstract

The size and flexibility of offshore turbine blades manufactured from composite materials have continuously increased in recent years. In this context, accurate and efficient aeroelastic analyses are important for designing and safely assessing long, flexible blades. Existing linear beam models need to be revised to offer accurate estimates of the geometric nonlinear effects triggered by large displacements. Nonlinear, geometrically exact beam models that have already been extensively used for the above purpose are generally difficult to converge and inefficient. We propose a novel co-rotational beam model for the nonlinear analysis of wind turbine blades. The method adopts vector complement to resolve rotation vector singularity problems. A complete anisotropic cross-sectional stiffness matrix and Timoshenko beam elements are introduced to capture full coupling effects. The method also considers the anisotropy and taper effects caused by the non-uniformity of chord length and material distributions. We established the nonlinear aeroelastic model of the DTU 10 MW turbine, and the results showed that the taper effect dramatically reduced the blade torsion angle by up to 31.44% under rated wind speed. Meanwhile, static beam experiments demonstrate that the accuracy error of the current method is only 1.78%, which is significantly lower than the 17.8% error of the conventional finite element beam method.

1. Introduction

Offshore wind power technology has made enormous progress in the past decade, and wind power is now considered an integral part of the new energy field [1]. Increasing the stand-alone capacity is crucial for reducing kilowatt-hour costs and improving energy utilization efficiency. The blade length has increased from 86 m at 10 MW [2] to 135 m at 22 MW [3]. Meanwhile, wind turbine blades have become increasingly flexible due to limitations in materials technology and the need to cut down costs, although this comes at the risk of causing large deformations.

The blades of offshore wind turbines are subjected to coupled aerodynamic, elastic, and centrifugal loads. This results in large periodic deformations in both the flapwise and edgewise directions, accompanied by significant structural torsion. Accurate prediction of structural torsion is especially important for turbine performance. Accurate prediction of the blade’s structural torsion is especially important. Under extreme winds, the structural torsion will alter the angle of attack, coupling aerodynamics and the structure, causing dramatic aerodynamic changes [4]. Aerodynamics also impact the load check and structural design of blades. Ignoring large blade deformations will inevitably lead to inaccurate estimations of the aeroelastic coupling mechanism and unreliable structural design of the turbines, which finally pushes up the overall manufacturing cost. The blade design of wind turbines involves considering various working conditions, which raises a higher demand for model efficiency. In this context, a nonlinear structural analysis method capable of accurate and efficient estimation is highly important for further developments in long, flexible turbine blades.

Dynamic models of blades can be built using the assumed mode method (AMM) and finite element method (FEM). For AMM, FEM is first employed to calculate the modal shapes and frequencies, and a finite number of modal orders are chosen for model reduction and the improvement of computational efficiency. However, due to the linear superimposition of modes, AMM can hardly consider the high-order vibration and large nonlinear deformations of blades, and its accuracy heavily depends on the number of modes selected. An earlier version of FAST [5] and Bladed [6], the leading wind turbine design software, use AMM to model blades.

When applied to the structural dynamic analysis of blades, FEM consists of 1D beam elements and 3D shell elements. One study has shown that the 1D beam elements capture the dynamic responses of blades correctly and involve far less complex calculations than 3D beam elements [7]. The beam method is deemed necessary for the high-efficiency calculation of long, flexible components of the wind turbine, such as blades and towers. However, most beam methods use Euler–Bernoulli beam elements with a linear assumption, making it impossible to consider complex coupling (e.g., bending–torsion coupling) and large deformations. Therefore, the beam method usually leads to inexact estimates of long, flexible blades.

Kim et al. [8] developed an anisotropic beam finite element for wind turbine blades made of composite material in 2013, coupled to HAWC2 to account for large deformations and the structural coupling of blades. Their method integrated Timoshenko beam elements with multibody dynamics (MBD) to capture the geometric nonlinear effects of blades [9]. In 2014, Wang et al. [10] established a nonlinear aeroelastic analysis method for blades based on the geometrically exact beam method (GEBT) and the blade element momentum theory (BEM). A complete $6\times 6$ cross-sectional stiffness matrix was used to give full consideration to the complex coupled dynamics of blades [11,12]. However, this method involved an independent interpolation of beam section rotation and beam axial displacement, which inevitably resulted in shear lock [13], impairing computational accuracy. Additionally, the GEBT-based model [14] involved many parameters and had poor stability. A shorter time step was usually needed, resulting in low efficiency and its ineligibility for widespread use in the engineering design of blades. Although both methods considered large nonlinear deformation and the bending–torsion coupling of the anisotropic blade, they neglected the taper effect caused by the non-uniformity of blade chord length and structure.

The taper effect can be described as follows. Given the differences in chord length and materials along the blade orientation, the shear and elastic axes on the two cross-sections of the blade’s beam element are not parallel. The difference in the orientation between the two axes results in bending–torsion coupling. The Bladed model utilizes the above-mentioned model and models the blade as a continuum composed of several linear beam elements. Hence, a dynamic model considering geometric nonlinearity is established [6] but it only uses the $4\times 4$ cross-sectional stiffness matrix without fully considering the coupled effect. Moreover, the accuracy of this method depends on the number and meshing method of the elements, thereby introducing a factor of uncertainty.

A nonlinear co-rotational method (CRM) has recently emerged in wind power generation and has attracted academic interest [14,15,16]. The co-rotational formulation was initially introduced by Wempner [17] and Belytschko et al. [18,19]. CRM is unique in that solving the total Lagrangian (TL) and the updated Lagrangian (UL) is no longer necessary with this method, which simplifies the calculation process. Meanwhile, CRM calculates beam elements’ rigid and elastic deformation displacements separately. Under a local elastic deformation coordinate system, Euler–Bernoulli and Timoshenko beam elements are considered mature methods for improving computational efficiency using mature beam elements. Under a local coordinate system of CRM, the nonlinearity problem of the blade material is fully considered, which is of particular importance for manufacturing longer blades from composite materials [20]. Thus, CRM implies that local deformation satisfies the linear displacement–strain relationship, ensuring high accuracy when a linear assumption is adopted under the local coordinate system. According to another study, CRM outperformed GEBT in terms of dynamic calculation accuracy after discretization into the same number of elements [21].

Nevertheless, CRM still has some deficiencies when applied to wind turbine blades. On the one hand, the complex geometric features of blades induce other coupling phenomena, such as bending–torsion coupling. Using an incomplete cross-sectional stiffness matrix and Euler–Bernoulli beam elements makes it impossible to consider anisotropy and some unusual coupling phenomena. Moreover, Moon et al. [15] pointed out that the taper effect of blades would affect the material features over the blade cross-sections, resulting in low computational accuracy. On the other hand, a rotation vector involving fewer variables has been used to represent the singularity problem of finite rotation in space to improve precision [22].

This study proposes a nonlinear analytical method that accounts for blades’ taper effect and anisotropy with high efficiency and accuracy. A new possibility is opened up for the high-accuracy analysis of wind turbine blades.

Our method uniquely combines three innovations: (1) resolving the singularity problem using rotation vector complement, as proposed by Cardona et al. [23]; (2) implementing a complete 6 × 6 cross-sectional stiffness matrix with Timoshenko beam elements; and (3) incorporating taper effects into the co-rotational framework, which previous CRM [24] methods have not adequately addressed.

In this way, we fully consider the taper effect induced by the non-uniform chord length distribution of the blade and couple the taper effect to CRM, which improves the computational accuracy and efficiency for the dynamic blade response. The results from our method were compared against the numerical calculations, confirming the model’s accuracy. The proposed model was further applied to the blades of the DTU10MW turbine [2].

The rest of this study is structured as follows: Section 2 describes the sources of the singularity of finite rotation and proposes a solution to the singularity problem. The cross-sectional stiffness matrix is introduced during the derivation of beam kinematics. Section 3 presents the formula for the coupled cross-sectional stiffness matrices, considering anisotropy and the taper effect. It also introduces the Timoshenko beam element that considers the complete $6\times 6$ cross-sectional stiffness matrix. Section 4 demonstrates the reliability of the proposed method by comparing the experimental results against the numerical calculations and discusses the centrifugal hardening effect and the aeroelastic response of the DTU10MW turbine under the action of the taper effect. Finaly, Section 5 concludes this study.

2. Methodology

As shown in Figure 1, the proposed CRM method consists of three core models to solve the above-mentioned problems: a solution to the singularity problem of finite rotation, a motion description of the nonlinear co-rotational beam and the fully coupled anisotropic Timoshenko beam element model, and a cross-sectional stiffness matrix model considering the taper effect. In this section, the derivation process of beam kinematics reveals the source of singularity and provides a solution. Section 3 refines the method for obtaining the element stiffness matrix during the deviation process.

2.1. Parameterization of Finite Rotation

This study stipulates that italics indicate variables, non-italics show constants and coefficients, and bold indicates vectors or matrices. For example, $\mathbf{R}$ represents a coefficient matrix. For any vector ${\mathit{x}}_{\mathbf{0}}$ in space, its coordinate after rotating around any unit vector$\mathit{n}$ by an angle of $\theta$ is derived as follows:

$${\mathit{x}}_{\mathit{n}}=\mathbf{R}{\mathit{x}}_{\mathbf{0}}$$

(1)

where $\mathbf{R}$ is a 3 $\times 3$ rotation matrix, and $\mathbf{R}$ is an orthogonal matrix; hence, ${\mathbf{R}}^{-1}={\mathbf{R}}^{\mathbf{T}}$. This feature is the cornerstone for deriving the formula below. The rotation vector is defined as follows:

$$\mathsf{\theta }=\theta \mathit{n}=[{\theta }_x{\theta }_y{\theta }_z{]}^T$$

(2)

From Equation (2), it is easy to obtain $\theta =‖\mathit{\theta }‖=\sqrt{{{\theta }_x}^2+{{\theta }_y}^2+{{\theta }_z}^2}$. Furthermore, based on Rodrigues’ formula, we write the relationship between the rotation matrix, rotation vector, and rotation angle thus:

$$\mathbf{R}=\mathit{exp}\left(\tilde{\mathit{\theta }}\right)={\mathbf{I}}_{3\times 3}+{\displaystyle \frac{\mathit{sin}\theta }{\theta }}\tilde{\mathit{\theta }}+{\displaystyle \frac{1-\mathit{cos}\theta }{{\theta }^2}}\tilde{\mathsf{\theta }}\tilde{\mathsf{\theta }}$$

(3)

where $\mathbf{I}$ is an identity matrix; $\tilde{\mathsf{\theta }}$ is the obliquely symmetric matrix of vector $\mathsf{\theta }$ and is denoted by $\mathbf{S}(\mathsf{\theta })$:

$$\mathbf{S}\left(\mathsf{\theta }\right)=\left[\begin{array}{ccc}0& -{\theta }_z& {\theta }_y\\ {\theta }_z& 0& -{\theta }_x\\ -{\theta }_y& {\theta }_x& 0\end{array}\right]$$

(4)

For the dynamic simulation of the beam structure, it is sometimes necessary to derive the rotation vector from the rotation matrix in reverse. Therefore, an inverse transformation exists for Equation (3) that turns the rotation matrix back to the rotation vector. The inverse transformation operator is written as follows:

$$\mathbf{D}(\mathbf{R})={\displaystyle \frac{\mathit{arccos}(\frac{Tr\left(\mathbf{R}\right)-1}{2})}{2\times \mathit{sin}(\mathit{arccos}(\frac{Tr\left(\mathbf{R}\right)-1}{2}))}}[R_{32}-R_{23}{R}_{13}-R_{31}{R}_{21}-R_{12}{]}^T$$

(5)

where $Tr\left(R\right)$ is the trajectory of the matrix $\mathbf{R}$, i.e., the sum of the diagonal elements; $R_{ij}$ is the element in the $i$-th row and the $j$-th column of $\mathbf{R}$.

The angular speed of the rotation vector can be described as the angle variation over an infinitesimal time interval. Thus, the angular speed is given by:

$$\mathsf{\omega }=\mathbf{T}(\mathsf{\theta })\stackrel{.}{\mathsf{\theta }}$$

(6)

where the tangential calculation matrix $\mathbf{T}(\mathsf{\theta })$ is derived as:

$$\mathbf{T}(\mathsf{\theta })={\mathbf{I}}_{3\times 3}+{\displaystyle \frac{1-\mathit{cos}\theta }{{\theta }^2}}\tilde{\mathsf{\theta }}+{\displaystyle \frac{\theta -\mathit{sin}\theta }{{\theta }^3}}\tilde{\mathsf{\theta }}\tilde{\mathsf{\theta }}$$

(7)

Similarly, the acceleration of the rotation angle is derived as a time derivative of the angular speed as follows:

$$\stackrel{.}{\mathsf{\omega }}=\stackrel{.}{\mathbf{T}}(\mathsf{\theta })\stackrel{.}{\mathsf{\theta }}+\mathbf{T}(\mathsf{\theta })\stackrel{..}{\mathsf{\theta }}$$

(8)

where $\dot{\mathbf{T}}(\mathsf{\theta })$ can be obtained via symbolic computation or by referring to the literature [20]. The time derivative of the rotation vector is represented using Equation (6):

$$\dot{\mathsf{\theta }}={\mathbf{T}}^{-1}\left(\mathsf{\theta }\right)\mathsf{\omega }$$

(9)

2.2. Singularity Problem of Finite Rotation

The advantages of describing finite rotation with the rotation vector include reducing the number of parameters and improving the computational efficiency, although the singularity problem of finite rotation will occur. Many studies have been conducted concerning the singularity problem of the rotation vector [21,25,26]. We solve the singularity problem using the rotation vector and its complement to achieve the high-accuracy calculation of flexible blades. In Equation (9), the inverse matrix of the tangential calculation matrix is represented as follows:

$${\mathbf{T}}^{-1}\left(\mathsf{\theta }\right)={\displaystyle \frac{\theta /2}{\mathit{tan}\left(\theta /2\right)}}{\mathbf{I}}_{3\times 3}+\left(1-{\displaystyle \frac{\mathit{\theta }/2}{\mathit{tan}\left(\mathit{\theta }/2\right)}}\right){\displaystyle \frac{\mathsf{\theta }{\mathsf{\theta }}^{\mathbf{T}}}{{\mathit{\theta }}^2}}-{\displaystyle \frac{1}{2}}\tilde{\mathsf{\theta }}$$

(10)

This equation represents the fundamental relationship for converting rotation matrices back to rotation vectors, which is essential for tracking large rotations in flexible blade analysis. The domain of this function is $\mathbb{R}\in Q\&\mathbb{R}\ne 2k\pi (k=0,\pm 1,\pm 2\dots \pm n)$, where $Q$ is the set of rational numbers and $2k\pi (k=0,\pm 1,\pm 2\dots \pm n)$ is denoted by $\psi$. The physical meaning of the rotation vector described by Euler’s rotation theorem implies that any rotation in space can be defined independently with an angle and a rotation axis. However, rotations around the same axis but with a difference in the angle by $\psi$ cannot be differentiated. The rotations need to be mapped for differentiation$,$ and the mapping involves the amplitude of the rotation vector. As shown in Figure 2, the complementary mapping vector is defined as follows [20]:

$${\mathsf{\theta }}_{\mathbf{I}}=\left(\mathbf{1}-{\displaystyle \frac{\mathbf{2}\mathit{\pi }}{\mathit{\theta }}}\right)\mathsf{\theta }$$

(11)

It can be deduced from Figure 2 that ${\theta }_I=2\pi -\theta$, and, based on the rotation direction and the right-handed spiral rule, the rotation axis ${\mathbf{n}}_{\mathbf{I}}=-\mathbf{n}$. Since the rotations in space described by the two angles, namely, a₁ and a_2, are consistent except for the difference in rotation axis, we have:

$$\mathbf{R}=\mathit{exp}\left(\tilde{\mathsf{\theta }}\right)=\mathit{exp}\left({\tilde{\mathsf{\theta }}}_{\mathbf{I}}\right)$$

(12)

When the rotation angle approaches $\pi$, Equation (11) is used to replace the original rotation vector so that the rotation angle is always maintained at $0\le \theta <\mu \pi$ (where $\mu$ is a constant approaching 1, thereby resolving the singularity problem). Here, $\mu$ is defined as eliminating the numerical problem in the computer program. When the rotation vector is defined using Double Precision, we let $\mu$ = 1-1 × 10⁻²⁰.

2.3. Beam Kinematics

For a single beam element, by default, the subscript $i$ represents the node number $(i=1\mathrm{a}\mathrm{n}\mathrm{d}2)$. The superscript $g$ is the variable defined in the global coordinate system, and the superscript $l$ is the variable defined in the local coordinate system. To capture the displacement of the beam element and that of nodes on the element, we first define the motion of a two-node 3D beam element in space under the global and local coordinate systems. As shown in Figure 3, ${\mathbf{R}}_{\mathbf{0}}$ is a rotation matrix. Rotary mapping is performed for the global orthogonal coordinate system ${\mathbf{E}}_{\mathbf{r}}$ onto the coordinate system ${\mathbf{E}}_{\mathbf{0}}$ of the element node 1. In that case, the beam element does not have elastic deformation or rigid displacement. Therefore, ${\mathbf{R}}_{\mathbf{0}}$ is only related to the initial spatial position of the beam and is defined as the initial undeformed configuration. For a wind turbine blade, the discretized beam element has two nods. For each node $i$, the displacement vector under the global coordinate system is defined as ${\mathit{u}}_{\mathbf{i}}^{\mathit{g}}$, and the displacement vector under the local coordinate system is ${\mathit{u}}_{\mathit{i}}^{\mathit{l}}$. The overall displacement vector of element ${\mathit{u}}^{\mathit{l}}=[{\mathit{u}}_{\mathbf{1}}^{\mathit{l}}{\mathit{u}}_{\mathbf{2}}^{\mathit{l}}{]}^T$. The initial position vector of an elemental node is defined as ${\mathit{s}}_{\mathit{i}}^{\mathit{x}\mathit{y}\mathit{z}}$.

It can be inferred from Figure 3 that the nodal motion of the beam element in the global coordinate system can be described using the two methods below.

Method 1: The rotation matrix ${\mathbf{R}}_{\mathbf{0}}$ describes and defines the beam element’s initial coordinate system ${\mathbf{E}}_{\mathbf{0}}$. Due to the elastic and rigid motions of nodes, the rotation vector of the node ${\mathbf{t}}_{\mathbf{i}}^{\mathbf{g}}(i=1\mathrm{a}\mathrm{n}\mathrm{d}2)$ after nodal motion can be further represented by the node’s rotation matrix ${\mathbf{R}}_{\mathbf{g}\mathbf{i}}$:

$${\mathbf{t}}_{\mathbf{i}}^{\mathbf{g}}={\mathbf{R}}_{\mathbf{g}\mathbf{i}}{\mathbf{R}}_{\mathbf{0}}$$

$${\mathbf{R}}_{\mathbf{0}}={\left(e_0^1{e}_0^2{e}_0^3\right)}^T$$

(13)

The three basis vectors of the rotation matrix ${\mathbf{R}}_{\mathbf{0}}$ are defined as follows:

$$\left\{\begin{array}{c}{e}_0^1=\left(s_2^{xyz}-s_1^{xyz}\right)/l_0\\ e_0^2={\displaystyle \frac{\left(e_r^3\times e_0^1\right)}{\left|e_r^3\times e_0^1\right|}}\\ e_0^3=e_0^1\times e_0^2\end{array}\right.$$

(14)

where $l_0=\left|{\mathit{s}}_{\mathbf{2}}^{\mathit{x}\mathit{y}\mathit{z}}-{\mathit{s}}_{\mathbf{1}}^{\mathit{x}\mathit{y}\mathit{z}}\right|$, and the rotation matrix of a node is represented by:

$${\mathbf{R}}_{\mathbf{g}\mathbf{i}}=\mathbf{exp}\left({\mathsf{\theta }}_{\mathbf{i}}^{\mathbf{g}}\right)$$

(15)

where ${\mathsf{\theta }}_{\mathbf{i}}^{\mathbf{g}}$ is the rotation vector of the node.

Method 2: According to the co-rotational formulation for the beam, rigid displacement consists of translational and rotational motions of the element under a local coordinate system. The local coordinate system after the rigid displacement of the element is defined as ${\mathbf{R}}_{\mathbf{r}}=\left[{\mathbf{r}}_{\mathbf{1}}^{\mathbf{l}}{\mathbf{r}}_{\mathbf{2}}^{\mathbf{l}}{\mathbf{r}}_{\mathbf{3}}^{\mathbf{l}}\right]$. When the element undergoes elastic deformation, the rotation vector of a node is given by:

$${\mathbf{t}}_{\mathbf{i}}^{\mathbf{g}}={\mathbf{R}}_{\mathbf{r}}{\mathbf{R}}_{\mathbf{l}\mathbf{i}}$$

(16)

Combining Equations (13) and (16), we obtain the rotation matrix of the node under the local coordinate system:

$${\mathbf{R}}_{\mathbf{l}\mathbf{i}}={{\mathbf{R}}_{\mathbf{r}}}^{-\mathbf{1}}{\mathbf{R}}_{\mathbf{g}\mathbf{i}}{\mathbf{R}}_{\mathbf{0}}={{\mathbf{R}}_{\mathbf{r}}}^{\mathbf{T}}{\mathbf{R}}_{\mathbf{g}\mathbf{i}}{\mathbf{R}}_{\mathbf{0}}$$

(17)

The basis vector of the rotation matrix ${\mathbf{R}}_{\mathbf{r}}$ of rigid displacement is represented by:

$$\left\{\begin{array}{c}{e}_0^1={\displaystyle \frac{d_2^g+s_2^{xyz}-d_1^g-s_1^{xyz}}{\left|d_2^g+s_2^{xyz}-d_1^g-s_1^{xyz}\right|}}={\displaystyle \frac{l_n}{\left|l_n\right|}}\\ e_0^3={\displaystyle \frac{e_0^1\times q_{gavg}}{q_{gavg}}}\\ e_0^2=e_0^3\times e_0^1\end{array}\right.$$

(18)

where ${\mathit{l}}_{\mathit{n}}$ is the displacement vector of the beam element (excluding the angular displacement); ${\mathit{q}}_{\mathit{g}\mathit{a}\mathit{v}\mathit{g}}$ is the basis vector of the midpoint of the line connecting two nodes, as represented by:

$$q_{gavg}={\displaystyle \frac{1}{2}}\left({\mathit{R}}_{\mathit{g}\mathbf{1}}{\mathit{R}}_{\mathbf{0}}\left[010\right]+{\mathit{R}}_{\mathit{g}\mathbf{2}}{\mathit{R}}_{\mathbf{0}}\left[010\right]\right)$$

(19)

This means that the local coordinate system is built on the line connecting the two nodes and that the origin of the coordinate system is located at node 1. Therefore, this coordinate only describes the axial displacement of node 2 and the torsional displacement of the two nodes, and there are 7 degrees of freedom in total. Considering Equations (5) and (17), the displacement vector in the local coordinate system is given by:

$$u_i^l=\left[l_n\mathbf{D}\right({\mathbf{R}}_{\mathbf{l}\mathbf{1}}\left)\mathbf{D}\right({\mathbf{R}}_{\mathbf{l}\mathbf{2}}\left)\right]$$

(20)

This transformation enables the separation of rigid body motion from elastic deformation, which is the core advantage of the co-rotational formulation for handling geometric nonlinearity. According to the energy conservation law, the virtual work done by virtual displacement in the local coordinate system is equal to that in the global coordinate system. Therefore, the generalized nodal internal force ${\mathit{f}}^{\mathit{l}}$ of the element in the local coordinate system is related to the generalized nodal internal force ${\mathit{f}}^{\mathit{g}}$ of the element in the global coordinate system as follows:

$${\mathbf{f}}^{\mathbf{l}}\mathsf{\delta }{\mathbf{u}}^{\mathbf{l}}={\mathbf{f}}^{\mathbf{g}}\mathsf{\delta }{\mathbf{u}}^{\mathbf{g}}$$

(21)

As it is assumed that the blade element satisfies the linear displacement–strain relation in the local coordinate system, ${\mathit{f}}^{\mathit{l}}$ can be linearly represented by ${\mathit{f}}^{\mathit{g}}$ and a transformation matrix $\mathit{B}$ as follows:

$$f^g={\mathit{B}}^T{f}^l={\mathbf{B}}^{\mathbf{T}}{\mathbf{K}}_{\mathbf{e}\mathbf{l}\mathbf{e}}{\mathbf{u}}^{\mathbf{l}}$$

(22)

where ${\mathbf{K}}_{\mathbf{e}\mathbf{l}\mathbf{e}}$ is the stiffness matrix of the beam element. The above equation is differentiated as follows:

$$\delta f^g=\delta {\mathit{B}}^T{f}^l+{\mathbf{B}}^{\mathbf{T}}\delta f^l=K_g\delta u^g$$

(23)

Combining Equations (22) and (23) results in the stiffness matrix of the element (tangential stiffness matrix) under the global coordinate system:

$${\mathbf{K}}_{\mathbf{g}}={\mathbf{B}}^{\mathbf{T}}{\mathbf{K}}_{\mathbf{e}\mathbf{l}\mathbf{e}}\mathbf{B}+{\displaystyle \frac{\mathsf{\delta }{\mathbf{B}}^{\mathbf{T}}}{\mathsf{\delta }{\mathbf{u}}^{\mathbf{g}}}}{\mathbf{f}}^{\mathbf{l}}={\mathbf{B}}^{\mathbf{T}}{\mathbf{K}}_{\mathbf{e}\mathbf{l}\mathbf{e}}\mathbf{B}+{\mathbf{K}}_{\mathbf{G}}\left({\mathbf{f}}^{\mathbf{l}}\right)$$

(24)

where ${\mathit{K}}_{\mathit{G}}$ is the geometric stiffness matrix, a function of the element’s internal force that is related to the deformation of the beam element. The calculation methods for the transformation matrix $\mathit{B}$ and the stiffness matrix and geometric stiffness matrix ${\mathit{K}}_{\mathit{G}}$ of elements have already matured and can be found in the literature [20,27]. Whether the element’s stiffness matrix is correct depends on whether the cross-sectional stiffness matrix and the shape function are correctly calculated, which will be the topic of the following section. From Equation (24), we obtain the tangential stiffness matrix of each element and the internal force vector ${\mathit{f}}^{\mathit{g}}$ in the global coordinate system. We obtain the beam’s overall stiffness matrix and mass matrix through assembly and the implementation of boundary conditions.

3. Taper Effect Model and Anisotropic Beam Model

As mentioned previously, the cross-sectional stiffness matrix is crucial for flexible beam dynamic modeling; however, the cross-section of a wind turbine blade usually displays complex features, and the elastic center of the beam element does not coincide with the shear center. This gives rise to complex bending–torsion coupling and a nonlinear dynamic response. It also means that a non-diagonal matrix better represents the constitutive relation between a local beam element’s strain and internal load. This section focuses on deriving and introducing the cross-sectional stiffness matrix of the anisotropic blade. The blade elements’ taper effect and anisotropy are considered to realize high-precision simulation.

The first part of this section derives the cross-sectional stiffness matrix of the anisotropic blade based on geometric relations. It introduces the coupling relation between shear and torque and that between bending and shear. The second part derives the bending–torsion coupling term for the stiffness matrix based on the taper effect unique to the wind turbine blade and the torsional stiffness variation term around the elastic axis due to deformation. The third part derives the expression of the element stiffness matrix based on the cross-sectional stiffness matrix and the anisotropic Timoshenko beam elements.

3.1. Cross-Sectional Stiffness Matrix of the Anisotropic Blade

For anisotropic beam cross-section, the local constitutive relation of the beam node is represented as follows:

$$\left[\begin{array}{c}{F}_x\\ F_y\\ F_z\\ M_x\\ M_y\\ M_z\end{array}\right]=\left[\begin{array}{llllll}EA\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 0\hfill & G_{y}A\hfill & \hfill & \hfill & Sym\hfill & \hfill \\ 0\hfill & 0\hfill & G_{z}A\hfill & \hfill & \hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & G{I}_x^s\hfill & \hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & E{I}_y\hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & E{I}_z\hfill \end{array}\right]\left[\begin{array}{l}{u}_x\hfill \\ u_y\hfill \\ u_z\hfill \\ {\theta }_x\hfill \\ {\theta }_y\hfill \\ {\theta }_z\hfill \end{array}\right]=\overline{{\mathbf{K}}_{cs}}\left[\begin{array}{l}{u}_x\hfill \\ u_y\hfill \\ {\gamma }_z\hfill \\ {\theta }_x\hfill \\ {\theta }_y\hfill \\ {\theta }_z\hfill \end{array}\right]$$

(25)

where $\overline{{\mathit{K}}_{\mathit{c}\mathit{s}}}$ is the local 6 × 6 cross-sectional stiffness matrix, which represents the material properties of the element. $E$ is the Young modulus; $A$ is the cross-sectional area; $G$ is the shear modulus; $G_y$ and $G_z$ are the shear moduli along the $y$- and z-directions, respectively; $I_x^s$ is the polar second moment of area; $I_y$ and $I_z$ are the components of the second moment of area in the $y$ and $z$ directions, respectively. The above Equation (25) can be also written as follows:

$$\mathbf{F}=\overline{{\mathbf{K}}_{\mathbf{c}\mathbf{s}}}\mathit{\epsilon }$$

(26)

Generally speaking, the cross-sectional structure of the blade can be simplified as in Figure 4. Since the origin of the local coordinate system of the element is situated on the elastic axis, the stiffness matrix needs to be transformed to the elastic axis. The transformational relation is defined as follows:

$${\bar{\mathbf{K}}}_{\mathbf{n}}=\mathbf{T}{\bar{\mathbf{K}}}_{\mathbf{c}\mathbf{s}}{\mathbf{T}}^{\mathbf{T}}$$

(27)

where $\mathbf{T}$ is defined as:

$$\mathbf{T}=\left[\begin{array}{cc}I& 0\\ Y_S& I\end{array}\right],Y_S=\left[\begin{array}{ccc}0& -z_{ns}& y_{ns}\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$

(28)

A new local constitutive relation is introduced based on the above matrix transformation. Employing shift, we introduce the coupling between shear strain and torque and the coupling between bending and shear stress. Hence, the new local cross-sectional stiffness matrix $\overline{K_{cs}}$considering the coupling between torsion and shear stress is represented as follows:

$$\overline{{\mathbf{K}}_{\mathbf{c}\mathbf{s}}}=\left[\begin{array}{llllll}EA\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 0\hfill & G{A}_y\hfill & \hfill & \hfill & Sym\hfill & \hfill \\ 0\hfill & 0\hfill & G{A}_z\hfill & \hfill & \hfill & \hfill \\ 0\hfill & -z_{ns}G{A}_y\hfill & y_{ns}G{A}_z\hfill & G{I}_x^n\hfill & \hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & E{I}_{y1}\hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & E{I}_{z1}\hfill \end{array}\right]$$

(29)

where the torsional stiffness $G{I}_x^n$around the elastic axis is related to the torsional stiffness $G{I}_x^s$ of the initial shear center in the following way:

$$\mathit{G}{\mathit{I}}_{\mathit{x}}^{\mathit{n}}=\mathit{G}{\mathit{I}}_{\mathit{x}}^{\mathit{s}}+\mathit{G}{\mathit{A}}_{\mathit{y}}{\mathit{z}}_{\mathit{n}\mathit{s}}^2+\mathit{G}{\mathit{A}}_{\mathit{z}}{\mathit{y}}_{\mathit{n}\mathit{s}}^2$$

(30)

Furthermore, we consider the structural properties of the blade. The blade cross-section usually has an initial twist angle $\Omega$. The third-party code used for aerodynamic calculations consists of aerodynamic forces on the turbine plane and perpendicular to the turbine plane. Therefore, the influence of the cross-sectional torsion angle should be considered. The structural properties of the beam are projected onto the coordinates of the aerodynamic forces to realize precise coupling between the structure and the aerodynamic forces. Given the above factors, the bending stiffness in Equation (29) is further transformed using the formula below:

$$\left\{\begin{array}{l}E{I}_z=E{I}_{z1}{\cos }^2\Omega -E{I}_{y1}{\sin }^2\Omega \hfill \\ E{I}_y=E{I}_{z1}{\sin }^2\Omega +E{I}_{y1}{\cos }^2\Omega \hfill \end{array}\right.$$

(31)

3.2. Cross-Sectional Stiffness Matrix of the Blade Considering the Taper Effect

For blades manufactured from composite materials, the chord length varies in the spanwise direction. Therefore, the shear and elastic axes of the beam element are not parallel, resulting in the taper effect. We modified the taper effect model proposed by Bossanyi [6] and applied the model to long flexible blades. As shown in Figure 5, the shifts of the shear center on the element’s two endpoints, a and b, in the $y_l$and $z_l$ directions are $y_a,z_a$ and $y_b,z_b$, respectively. The x-direction is the spanwise direction and coincides with the elastic axis.

To account for the taper effect, we introduce the projection relative deviation of the shear axes of element modes in the two directions and the element length $l^e$ to describe the position differences of the two axes on the element. This parameter is described by:

$$\left\{\begin{array}{c}{y}_p={\displaystyle \frac{\Delta y_{ns}}{l^e}}={\displaystyle \frac{y_b-y_a}{l^e}}\\ z_p={\displaystyle \frac{\Delta z_{ns}}{l^e}}={\displaystyle \frac{z_b-z_a}{l^e}}\end{array}\right.$$

(32)

The element length $l^e$ changes due to deformation. Therefore, the cross-sectional stiffness matrix should be updated every moment during the calculation, resulting in low efficiency. As the blade elements only have a local small strain and the deformation generally occurs in the flapwise and edgewise directions, the axial displacement is insignificant. Therefore, the length changes in these directions can be neglected. Based on the above assumption, the stiffness matrix of the element at node b can be further extended from Equation (30) to the following form:

$$\overline{{\mathbf{K}}_{\mathbf{c}\mathbf{s}}}=\left[\begin{array}{llllll}EA\hfill & \hfill & \hfill & \hfill & \hfill & \hfill \\ 0\hfill & G{A}_y\hfill & \hfill & \hfill & Sym\hfill & \hfill \\ 0\hfill & 0\hfill & G{A}_z\hfill & \hfill & \hfill & \hfill \\ 0\hfill & -z_{ns}G{A}_y\hfill & y_{ns}G{A}_z\hfill & G{I}_x^n\hfill & \hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & -y_{p}E{I}_{y1}\hfill & E{I}_{y1}\hfill & \hfill \\ 0\hfill & 0\hfill & 0\hfill & -z_{p}E{I}_{z1}\hfill & 0\hfill & E{I}_{z1}\hfill \end{array}\right]$$

(33)

where:

$$\mathit{G}{\mathit{I}}_{\mathit{x}}^{\mathit{n}}=\mathit{G}{\mathit{I}}_{\mathit{x}}^{\mathit{s}}+\mathit{G}{\mathit{A}}_{\mathit{y}}{\mathit{z}}_{\mathit{n}\mathit{s}}^2+\mathit{G}{\mathit{A}}_{\mathit{z}}{\mathit{y}}_{\mathit{n}\mathit{s}}^2+{\mathit{y}}_{\mathit{p}}^2\mathit{E}{\mathit{I}}_{\mathit{y}}+{\mathit{z}}_{\mathit{p}}^2\mathit{E}{\mathit{I}}_{\mathit{z}}$$

(34)

Equation (33) has an additional bending–torsion coupling term compared with the stiffness matrix so that the beam model can correctly capture the blade’s bending–torsion coupling caused by the taper effect and the changes in torsional stiffness around the elastic axis induced by deformation. This feature is crucial for the aeroelastic calculation of the flexible blade since the changes in the blade’s angle of attack due to bending–torsion coupling and torsional stiffness cause intense variation in aerodynamic forces [28,29].

3.3. Anisotropic Timoshenko Beam Elements

Based on the assumption that the stress–strain relationship in the local coordinate system is linear, linear beam elements are chosen in the local coordinate system. For example, the Euler–Bernoulli beam elements and the Timoshenko beam elements can be used to improve computational efficiency dramatically. To realize high-precision simulation, we introduced a new Timoshenko beam element that fully accommodates the 6 × 6 cross-sectional stiffness matrix. The novel element was introduced by Bazoune et al. [30] and later modified by Stäblein et al. [23]. We assumed that the beam cross-section was flat and not warped by neglecting the higher-order terms in the Timoshenko beam. The strain vector at the beam node is represented by:

$$\mathit{\epsilon }={\left\{{\displaystyle \frac{\partial u}{\partial x}},{\displaystyle \frac{\partial v}{\partial x}}-{\theta }_z,{\displaystyle \frac{\partial w}{\partial x}}+{\theta }_y,{\displaystyle \frac{\partial {\theta }_x}{\partial x}},{\displaystyle \frac{\partial {\theta }_y}{\partial x}},{\displaystyle \frac{\partial {\theta }_z}{\partial x}}\right\}}^T$$

(35)

Using the cross-sectional stiffness matrix of the anisotropic beam, the overall elastic energy of the beam element can be expressed as follows:

$$U={\displaystyle \frac{1}{2}}{\int }_{x=0}^L{\mathit{\epsilon }}^T{\mathit{K}}_{\mathrm{cs}}(x)\mathit{\epsilon }dx$$

(36)

To obtain the relationship between energy and the element’s degree of freedom and to derive the stiffness matrix of the element, we need to estimate the relationship between the strain vector and the node displacement. Hence:

$$\mathit{\epsilon }=\mathit{B}\left(\mathit{x}\right)d$$

(37)

where $\mathit{d}=[{\mathit{u}}_1^{\mathit{l}},{\mathit{u}}_2^{\mathit{l}}{]}^{\mathit{T}}$ is the displacement vector of the element, and $\mathit{B}\left(\mathit{x}\right)$ is the strain–displacement matrix [31]. From Equation (36), we obtain the stiffness matrix of the element:

$${\mathit{K}}_{ele}={\int }_0^L\mathit{B}(x{)}^T{\mathit{K}}_{cs}\left(x\right)\mathit{B}\left(x\right)dx$$

(38)

4. Validation and Discussion

4.1. Static Validation

To facilitate understanding and improvement, we have ensured that all calculation results in this section can be reproduced using the open program.

4.1.1. Case 1: Experimental Validation of the Cantilever Beam Made of Composite Material

A beam made of composite material with a thin-walled rectangular cross-section was considered. The beam length was 0.762 m, and the beam was manufactured from a graphite/epoxy resin composite, a material commonly used for wind turbine blades. The laminated structure resembled the blade’s multi-layer stacked structure, resulting in a strong bending–torsion coupling effect. The present case was intended to validate the proposed method’s ability to capture the composite material’s strong bending–torsion coupling effect. The beam model used was introduced by Chandra et al. [32], with the beam’s cross-sectional mass matrix being a $6\times 6$ identity matrix. The beam’s stiffness matrix has the following form:

$$K=\left[\begin{array}{cccccc}11.387\times {10}^5& & & & & \\ 2.909\times {10}^5& 4.189\times {10}^5& & & sym& \\ 0& 0& 3.122\times {10}^5& & & \\ -30.458& -11.932& 0& 62.692& & \\ 12.674& 8.689& 0& -21.741& 35.146& \\ 0& 0& 12.302& & -0.370& 80.594\end{array}\right]$$

(39)

In this case, the beam was discretized with 20 nodes. A concentrated load p = 4.448 N was imposed at the free end along the z-direction, and the bending and torsion moments of the beam in the spanwise direction were calculated. Figure 6 shows the calculated results of the bending angle at each node along the beam axis. For the sake of convenience, CRB (Co-Rotational Beam) in the text below represents the beam model without considering the taper effect, and CRB-T (Co-Rotational Beam with Taper effect) is the beam model considering the taper effect.

The beam model proposed by Chandra et al. [32] was selected as it represents a well-established experimental benchmark for composite beam bending–torsion coupling, which is widely used in the literature for validation. The FEM method developed by Smith and Chopra [33] was chosen as it provides high-precision numerical results using shell elements, serving as an independent numerical verification of our beam-based approach.

It was found that the beam ends had significant torsion angles. This is because the beam torsion was caused by the coupling of bending in two directions, as demonstrated by Equation (39).

Table 1. Composite beam tip bending displacements in Case 1 via FEM [33], experiments [32], and CRB.

	Experimental [32]	FEM [33]	Present Study with No Taper Effect	Present Study with Taper Effect Account
Data	≈2.30	2.71514	2.67781	2.25932
Relative error (%)	-	17.8%	16.4%	−1.78%

compares the calculated and experimental results on the torsion angle of the beam endpoints.

As shown in Table 1, the result calculated using the proposed CRB approach without considering the taper effect differed from that of the beam FEM [33] by 0.037 (2.678 vs. 2.715), being closer to the experimental data [32], namely, 2.30. However, the relative errors of either CRB or FEM were above 16.4% compared with the experiment [32]. With the taper effect considered, the relative error between the proposed model and the experiment was only 1.78%. This case also proved the non-negligibility of the taper effect in an anisotropic beam with nonuniform material distribution. For the sake of convenience, CRB in the text below represents the beam model without considering the taper effect, and CRB-T is the beam model considering the taper effect.

4.1.2. Case 2: Pre-Bent Cantilever Beam

Large flexible blades are generally present and have airfoil-shaped cross-sections. This case was designed to validate the proposed CRB’s ability to consider the beam’s pre-bent structure and whether the model captures the geometric large nonlinear deformations of the blade. Bathe et al. [34] proposed a pre-bent beam structure for our validation. Figure 7 shows the beam’s section shape parameters and a concentrated load acting on the tip. The inner radius of the beam was 100 m, and the inner angle was 45°. The beam’s cross-sectional stiffness matrix is given below:

$$K={10}^5\times \left[\begin{array}{cccccc}100.0& & & & & \\ 0& 50.0& & & sym& \\ 0& 0& 50.0& & & \\ 0& 0& 0& 7.0& & \\ 0& 0& 0& k_{54}& 8.333& \\ 0& 0& 0& 0& 0& 8.333\end{array}\right]$$

(40)

Compared with the method given in [35], the proposed one implied a refinement of the non-diagonal element$k_{45}=k_{54}={\alpha }_{coupling}\sqrt{k_{44}\times k_{55}}={\alpha }_{coupling}\sqrt{GJ\times E{I}_y}$ in Equation (40), with ${\alpha }_{coupling}=-0.3$. In this case, the beam was discretized by 20 nodes. Figure 8 compares beam displacement with and without considering the bending–torsion coupling. Large displacements and torsions can be observed in Figure 8a,b. A comparison between Figure 8b,d shows that the angle of attack at the beam tip ${\theta }_1=12.08^{\circ }$ under the bending–torsion coupling was far smaller than the value of $25.22^{\circ }$ obtained without considering this coupling. The difference in the angle of attack thus incurred might bring about an enormous difference in aerodynamic load. Therefore, developing a beam model that considers bending–torsion coupling is highly necessary.

Table 2. Comparison of pre-bent cantilever tip displacements.

	u1 (m)	u2 (m)	u3 (m)	θ1 (rad)	θ2 (rad)	θ3 (rad)
Present uncoupled model	−12.18	−7.18	40.49	0.44	−0.71	−0.01
Simo and Vu-Quoc [36]	−11.87	−6.96	40.08	-	-	-
Uncoupled relative error (%)	2.57	3.09	1.02	-	-	-
Present coupled model	−10.650	−6.578	38.779	0.201	−0.794	−0.117
Stäblein and Hansen [31]	−10.660	−6.530	38.740	0.196	−0.810	−0.120
Coupled relative error (%)	−0.10	0.74	0.10	2.51	−1.94	−2.37

compares beam tip displacement using the proposed method, the coupled beam model [35], and the uncoupled beam model [36]. The proposed beam model correctly predicted beam displacements.

4.1.3. Case 3: Frequency and Modal Validation of the Anisotropic Cantilever Beam

For large blades, order reduction of the beam model can be achieved using AMM to improve the computational efficiency. We further validated the natural frequencies of each anisotropic beam element using the box girder model formulated by Hodges et al. [37]. The beam consisted of six layers of one-way slabs made of a T300/5280 graphite/epoxy composite [23]. Other information about the beam properties can be found in the literature [8]. The cross-sectional mass matrix $M$and stiffness matrix $K$ shown below were used for solving the modal properties:

$$K={10}^2\times \left[\begin{array}{cccccc}5.0576\times {10}^4& 0& & & & \\ 0& 7.7444\times {10}^3& & & sym& \\ 0& 0& 2.9558\times {10}^3& & & \\ -1.7196\times {10}^2& 0& 0& 1.5041& & \\ 0& 83.270& 0& 0& 2.4577& \\ 0& 0& 90.670& 0& 0& 7.4529\end{array}\right]$$

(41)

$$M=\left[\begin{array}{cccccc}0.13098& & & & & \\ & 0.13098& & & & \\ & & 0.13098& & & \\ & & & 2.5824\times {10}^{-5}& & \\ & & & & 6.5010\times {10}^{-6}& \\ & & & & & 1.9323\times {10}^{-5}\end{array}\right]$$

(42)

The beam was discretized into 19 co-rotational elements by 20 nodes, and some frequencies were calculated using modal decomposition.

Table 3. Eigenfrequencies of a coupled cantilever obtained with the present model compared to results by Hodges et al. [37], Armanios and Badir [38], Kim et al. [8], and Stäblein [23].

Mode No.	Frequency (Hz)					Relative Error (%)
	This Study	[37]	[38]	[23]	[8]	[37]	[38]	[23]	[8]
1 vertical	2.94	3.00	2.96	2.99	2.98	−2.00	−0.68	−1.67	−1.34
1 horizontal	5.08	5.19	5.10	5.18	5.12	−2.12	−0.39	−1.93	−0.78
2 vertical	18.41	18.54	18.54	18.75	18.65	−0.70	−0.70	−1.81	−1.29
2 horizontal	31.78	32.88	31.98	32.36	32.02	−3.35	−0.63	−1.79	−0.75
1 torsional	180.08	180.32	177.05	180.10	-	−0.13	1.71	−0.01
2 torsional	541.34	544.47	531.15	542.05	-	−0.57	1.92	−0.13

shows a cross-validation comparison of the results from the proposed model, the FEM analysis of beam elements by Hodges et al. [37], the beam model proposed by Armanios and Badir [38], and the Timoshenko finite element beam models developed by Kim et al. [8] and Stäblein [23]. The flapwise and edgewise vibration modes and torsional modes are commonly studied for blade modeling; therefore, the frequencies of these modes were compared in the present study. The maximum error between the calculated result using the proposed method and those from the previous studies was 3.35%, indicating good agreement.

4.2. Validation of Nonlinear Aeroelastic Response of the DTU10MW Wind Turbine

Based on the above analysis, we obtain the stiffness matrix of the blade under an external excitation. The modal calculation case in Section 4.1 demonstrates the proposed method’s accuracy in estimating the beam’s global mass matrix and stiffness matrix. It also validates the high precision of the proposed method in flexible beam structures with large deformations and bending–torsion coupling. Nevertheless, the non-uniformity of spanwise structure and aerodynamic chord length of a real blade usually leads to anisotropy and taper effect. This section is intended to validate the accuracy and practicability of the proposed method in a real blade. The DTU 10 MW Reference Wind Turbine, a successor to the NREL 5-MW baseline wind turbine [39], was chosen as a case study. Blade element momentum theory (BEM) and the above-mentioned anisotropic beam theory were implemented to construct the nonlinear fluid–solid coupling model and estimate the large deflections of the blade under varying wind speeds.

Table 4. Parameters of the DTU 10 MW wind turbine.

Parameters	Values	Parameters	Values
Number of blades	3	Cut-in wind speed	4 m/s
Rotor diameter	178.3 m	Cut-out wind speed	90.0 m/s
Hub diameter	5.6 m	Rotor speed	9.6 rpm
Rated wind speed	11.4 m/s	Hub overhang	7.1 m

shows some parameters of the turbine model. For aerodynamic and structural information on blades, please refer to the official report on DTU [2].

The parametric aeroelastic model was built in the program developed in this study. A comparison was made between the calculated results from this model and HAWC2. Section 4.2.1 validates the aerodynamic calculations in the aeroelastic model. Section 4.2.2 validates the natural frequencies and modes of the blade and discusses the blade frequencies corresponding to different beam displacements induced by the varying rotational speed. Section 4.2.3 validates the dynamic response of the wind turbine subjected to a steady-state wind. The results were consistent with those from HAWC2 13.1, thus validating the accuracy and reliability of the aeroelastic model and the high precision of the proposed beam model.

4.2.1. Nonlinear Aeroelastic Model

Blade element momentum (BEM), which only involves a simple calculation procedure, was used to solve the aerodynamic load on the blade. The theory of BEM and the related derivation process are not the focus of this section, and no more details are provided here. The methodology and theory of aerodynamic modeling based on BEM can be found in the literature [10,40]. The program was written based on BEM, and the calculation results were compared against those from HAWC2, a widely acknowledged commercial code for analyzing wind turbines. Figure 9 shows the axial and tangential inducible factors and aerodynamic angle of attack of the blade under the varying steady-state wind speed (6, 10, and 14 m/s) in the spanwise direction. The results strongly agreed with those from HAWC 13.1, indicating the reliability of the aerodynamic model proposed in this study.

Based on the BEM aerodynamic load calculation and the nonlinear beam method proposed in Section 2 and Section 3, an aeroelastic coupling calculation program named BeamL was developed using C# and Python 3.9.10. The calculation of the nonlinear aeroelastic model primarily includes the following steps:

(1)

Read the input file of the blade to obtain aerodynamic information such as blade twist and chord length, as well as structural data needed in the blade nodes and section stiffness matrix. These are then stored in a .dat file.

(2)

Construct the corotational beam model of the blade, generate the section stiffness matrix considering anisotropic and taper effects under the local coordinate system, and initialize the Blade Element Momentum (BEM) theory.

(3)

Calculate the aerodynamic and centrifugal loads on the blade at each time step and transfer them into the nonlinear beam model.

(4)

Use the nonlinear HHT-α and incremental Newton–Raphson methods to solve for the displacement and velocity of the blade under external loads at the current time step. The nonlinear HHT-α time integration scheme uses α = −0.05 for numerical damping control. The Newton–Raphson solver uses a force residual tolerance of 1 × 10⁻⁶ N and displacement increment tolerance of 1 × 10⁻⁸ m, with a maximum of 20 iterations per time step.

(5)

Reverse-transfer the displacement and velocity to the blade aerodynamic force calculation module and solve for the new aerodynamic forces.

(6)

Repeat steps 3–5 until the end of the simulation time.

The flowchart of the nonlinear aeroelastic model is shown in Figure 10:

4.2.2. Modes and Centrifugal Hardening Effect of the Blade

Accurate estimates of the blade’s stiffness matrix, mass matrix, and inertia matrix in the inertial coordinate system are prerequisites for calculating the blade’s dynamic response. The blade’s modal shapes are closely related to its stiffness and mass matrices. The modal frequencies and shapes reflect whether the nonlinear stiffness matrix is correctly calculated. Due to large deformations of the blade, the stiffness matrix changes dynamically as the blade displacement changes, further affecting the blade’s natural frequencies and modal shapes. The blade is a rotary machine and undergoes centrifugal hardening during rotation, which changes the overall stiffness matrix of the blade with a large deformation. As the blade length increases, the centrifugal hardening effect is expected to become more pronounced [41]. Thus, one should study the inherent properties of the blade in different motion states to validate whether the proposed model correctly captures the centrifugal hardening effect.

Figure 11 provides the first three modal shapes of the blade under static conditions and near-rated rotational speed: first-order flapwise mode, first-order edgewise mode, and second-order flapwise mode. The results show that the blade’s modal shapes were consistent under the varying rotational speed and did not change dramatically with the blade’s nonlinear large deformation and rotational speed.

Table 5. The calculated values of the first three natural frequencies (first-order flapwise, first-order edgewise, second-order flapwise) of the wind turbine blade at different speeds (0, 9.1, and 10 rpm) and comparison with the numerical results from HAWC2 and Pereira [42].

		Calculated Value			Relative Error (%)
Rotational Speeds	Mode.no	Present Study	HAWC2	Pereira	HAWC2	Pereira
0	Flapwise 1	0.6	0.61	0.61	−1.64	−1.64
	Edgewise 1	0.92	0.93	0.93	−1.08	−1.08
	Flapwise 2	1.71	1.74	1.74	−1.72	−1.72
9.1	Flapwise 1	0.63	0.65	-	−3.08	-
	Edgewise 1	0.93	0.93	-	0.00	-
	Flapwise 2	1.73	1.78	-	−2.81	-
10.0	Flapwise 1	0.64	0.65	-	−1.54	-
	Edgewise 1	0.93	0.94	-	−1.06	-
	Flapwise 2	1.75	1.79	-	−2.23	-

further analyzes the changes in the natural frequencies of the blade under varying rotational speeds. A cross-validation comparison was performed between our results and those from HAWC2 and Pereira et al. [42]. There was a good agreement with those from HAWC2; the maximum error was 3.076%, which occurred for the first-order flapwise frequency.

Our study also validated the model’s ability to capture the blade’s centrifugal hardening effect. Figure 12 shows that the natural frequency varied under different rotational speeds. The higher the rotational speed, the higher the natural frequency. Figure 12a shows that, as the turbine’s rotational speed increased from 9 to 10 rpm, the first-order flapwise frequency of the blade increased from 0.63 to 0.65 Hz, the variation rate being 3.17%. The variation rates of the first-order edgewise frequency and the second-order flapwise frequency of the blade were 1.08% and 1.15%, respectively, significantly lower than the first-order flapwise frequency.

4.2.3. Aeroelastic Response and Taper Effect

This section focuses on the flapwise and edgewise displacements of the blade and the structure’s torsion angle, all of which are important parameters in the bending–torsion coupling design of the blade. Bending–torsion coupling may induce edgewise and flapwise instability and flutter [28,43,44], which deserve extra attention. For the validation study, the model considering the taper effect was implemented, and the results were compared against those from the model neglecting the taper effect. The structural beam model of the blade was discretized with 27 nodes, and the blade’s aeroelastic response was estimated under varying wind speeds using the aeroelastic model. Figure 13 compares the calculated displacements of each node in the blade along the spanwise direction using the proposed method vs. Bladed and HAWC2 models. For comparison, nonlinear models were implemented in the Bladed and HAWC2 models. The Bladed model only considered the non-fully-coupled stiffness matrix, apart from the taper effect. In HAWC2, a completely coupled stiffness matrix was implemented, but the taper effect could not be considered. Figure 13 shows that, without considering the taper effect, the flapwise and edgewise displacements of the blade under the varying wind speed, as calculated from the proposed model, agreed well with those generated by the HAWC2 and Bladed models. However, the edgewise displacement of the blade tip calculated by the proposed CRB-T model differed from those by the Bladed and HAWC2 models. Compared with Bladed model, the maximum error was 7.5%, occurring under the rated wind speed. Compared with the HAWC2 model, the maximum error was 4.73%, also occurring under the rated wind speed. This was because the Bladed model could not consider the full coupling effect, while the HAWC2 model failed to consider the taper effect. The calculated torsion angles in Figure 13 indicate significant differences between the results from CRB and CRB-T. Equation (34) of the taper effect model shows that, as an additional torsional stiffness was introduced, CRB-T produced a significantly smaller torsion angle than CRB and HAWC2 models. The same phenomenon also occurred between the Bladed and HAWC2 models, and the torsion angle estimated by the former was much smaller than by the latter. This highlights the non-negligibility of the taper effect and the fully coupled matrix and, hence, the accuracy and importance of the CRB-T model.

Next, we consider the quantitative influence of the taper effect on bending–torsion coupling. Figure 14 shows the comparison of blade tip displacements under varying wind speeds. It is known that the flapwise and edgewise displacements increased due to the taper effect, and the percentage increase rose as the wind speed increased. This was because the relative thickness decreased from the blade root to the tip. The beam elements not considering the taper effect would result in overestimating the flapwise and torsional stiffness and, hence, overestimating flapwise and edgewise displacements. Figure 14 indicates that the taper effect dramatically reduced the torsion angle, and the maximum reduction occurred near the rated wind speed of 10 m/s. The torsion angle was reduced by 0.34°, and the relative reduction was 31.44%, indicating the non-negligibility of the taper effect.

The 31.44% torsion reduction occurs because the taper effect introduces additional torsional stiffness through non-parallel shear and elastic axes. This reduction has significant implications: (1) improved fatigue life due to reduced cyclic torsional stress, (2) modified control system requirements as pitch-to-feather response changes, and (3) potential for blade optimization considering actual torsional behavior. The finding suggests current blade designs may be over-conservative in torsional stiffness, offering opportunities for material optimization and weight reduction.

To further validate the robustness and computational efficiency of the proposed method, we computed the time-domain response of wind turbine blades under steady wind conditions at 12 m/s. The steady wind condition was selected primarily to eliminate the influence of controllers and pitch systems on blade dynamics. The number of blade element discretizations was kept consistent with the GEBT method in FAST [5], totaling 40 elements. Additionally, we compared the actual simulation time consumption between the BeamL program using the CRB-T method and FAST under different time step sizes.

Table 6. Numerical simulation parameter settings for the DTU10MW blade based on the CRB-T method and FAST-GEBT method.

	Present		FAST [5]
Aerodynamic calculation method	BEM		BEM
Structural dynamics method	CRM-T		GEBT
Number of elements	40		40
Time step/s	0.05	0.005	0.005	0.05
Real calculation time	410	2813	3262.8	Error

presents the efficiency comparison results under various parameter settings. The results demonstrate the following: when using the same time step size as GEBT, the proposed method achieves a 13% speedup. When larger time steps are employed, the GEBT method diverges and requires smaller time step sizes to capture convergent solutions, leading to significantly increased computational costs. Combined with the blade tip displacement curves shown in Figure 15, the proposed method can adopt larger time steps (0.05 s) while maintaining high computational accuracy, thus exhibiting superior computational efficiency and numerical robustness. In contrast, the FAST method is prone to divergence when solving nonlinear equations.

5. Conclusions

This study proposed a new high-precision co-rotational formulation to resolve the large nonlinear deformation of long flexible blades of wind turbines. This method offers a solution to the singularity problem of finite rotation in space, as described by the rotation vector, by using the complement of vector and achieving a high-precision beam simulation. This model considers the taper effect induced by the non-uniform chord length distribution of the blade and is further coupled to the Timoshenko beam method that considers the anisotropic beam cross-section. On this basis, we build a high-efficiency, high-precision nonlinear beam model accounting for the anisotropy of the highly flexible blades. Unlike the previous co-rotational beam methods, the proposed model correctly captured both the bending–torsion coupling effect and the centrifugal hardening effect of the blade. A series of benchmark calculation tests and validation were performed, and a nonlinear aeroelastic response model of the blade was built based on the blade element momentum theory. The proposed model was then applied to the aeroelastic analysis of the DTU 10 MW turbine. The following conclusions were drawn from the numerical validation study:

• The validation in the composite material beam showed that the error of the proposed nonlinear beam model was not above 2.66% compared with the geometrically exact beam model. The validation case also suggested that the beam model correctly captured the blade’s anisotropy and bending–torsion coupling effect.
• The validation in the geometrically nonlinear pre-bent beam with large deformation indicated good agreement between the new beam model and some previous ones. When the bending–torsion coupling was considered, the maximum error was 2.51%, confirming the method’s ability to capture the geometric nonlinearity induced by large deflection.
• This study was concerned with the centrifugal hardening effect of the blade of the DTU 10 MW turbine under different rotational speeds. The results showed that, if geometric nonlinearity was considered, the first three natural frequencies of the blade increased as the rotational speed increased. However, the modal shapes were insensitive to the varying rotational speed. The variation rates of the first-order edgewise frequency and the second-order flapwise frequency of the blade were 1.08% and 1.15%, respectively, significantly lower than the variation frequency (3.17%) of the first-order flapwise frequency. Compared with the validation results from HAWC2, the maximum error of frequencies calculated using the present model was 3%. As the megawatt wind turbines become larger and more flexible, the centrifugal hardening effect of the blade will only be intensified.
• Based on the proposed model that considers the taper effect, we estimated the aeroelastic response of the blade under varying wind speeds. Our findings suggested that compared with not considering the taper effect, the blade’s flapwise and edgewise displacements increased by small margins when the taper effect was considered, the maximum increase being 4.5%. Our study also showed that considering the taper effect dramatically reduced the estimated torsion angle of the blade, and the maximum reduction occurred under the rated wind speed, reaching values as high as 31.43%. The above results prove the non-negligibility of the taper effect of blades.

Future work will focus on extending the CRB-T model to shell-like composite structures, integrating with high-fidelity CFD-based aeroelastic simulations, coupling it with active blade control systems, and applying it to floating offshore wind turbine platforms where geometric nonlinearity becomes even more critical.