When simulating a wind turbine, the lowest eigenmodes of the rotor blades are
usually used to describe their elastic deformation in the frame of a
multi-body system. In this paper, a finite element beam model for the rotor
blades is proposed which is based on the transfer matrix method. Both static
and kinetic field matrices for the 3-D Timoshenko beam element are derived by
the numerical integration of the differential equations of motion using a
Runge–Kutta fourth-order procedure. In the general case, the beam reference
axis is at an arbitrary location in the cross section. The inertia term in
the motion differential equation is expressed using appropriate shape
functions for the Timoshenko beam. The kinetic field matrix is built by
numerical integration applied on the approximated inertia term. The beam
element stiffness and mass matrices are calculated by simple matrix
operations from both field matrices. The system stiffness and mass matrices
of the rotor blade model are assembled in the usual finite element manner in
a global coordinate system accounting for the structural twist angle and
possible pre-bending. The program developed for the above-mentioned
calculations and the final solution of the eigenvalue problem is accomplished
using MuPAD, a symbolic math toolbox in
MATLAB^{®}. The natural frequencies calculated
using generic rotor blade data are compared with the results proposed from
the FAST and ADAMS software.

Vibration of an elastic system refers to a limited reciprocating motion of a particle or an object of the system. Wind turbines operate in an unsteady environment which results in a vibrating response (see Manwell et al., 2009). They consist of long slender structures (rotor blades and tower), which result in resonant frequencies that should be taken into account during the initial design and construction. When the excitation frequency of the vibrating system is near any natural frequency, an undesirable resonant state occurs with large amplitudes, which may lead to damage or even the collapse of the wind turbine or its components. The vibration response, especially of the rotor blades, depends on the stiffness which is a function of the materials used, the design and the size (see Jureczko et al., 2005). Therefore, the estimation of natural frequencies in the early design phase plays an important role in avoiding resonance.

The eigenmodes associated with the lowest natural frequencies are employed as shape functions to describe the elastic deformation of the rotor blade beam model in the frame of the usual simulation of the wind turbine as a multi-body system. Generally, the first two bending eigenmodes in each direction (flapwise and edgewise) and optionally the two additional torsional eigenmodes are used. The determination of the lowest eigenmodes with sufficient numerical accuracy is the first step with respect to the modal superposition applied to the deformational motion of the rotor blades.

Due to the geometrical complexity of the blade cross section profiles and the extended use of composite materials, the exact calculation of natural frequencies in the design process takes a considerable amount of time and financial expense with regards to the 3-D modelling of the rotor blade using CAD software; hence a simplified finite element beam model is necessary. A twisted rotor blade is simplified into a cantilever beam with a non-uniform cross section. The structural twist angle is implemented by changing the orientation of the principal axis of the blade cross section along the length of the blade.

In the finite element formulation of beams two linear beam theories are
established, the Euler–Bernoulli beam model and the Timoshenko beam model.
Although the Euler–Bernoulli beam theory is widely used, the Timoshenko beam
theory is considered to be better as it incorporates the effects of
transverse shear and the rotational inertia on the dynamic response of the
beam (see Kaya, 2006). In the classical approach of the finite element
formulation for the free vibration analysis of beams, the stiffness and mass
matrices are derived using interpolation functions derived from second- and
fourth-order Hermite polynomials. The stiffness matrix is derived from the
following equation (see Wu, 2013):

Using the above-mentioned standard relations and appropriate shape functions for the Euler–Bernoulli beam and the Timoshenko beam, the stiffness matrix and consistent mass matrix for the finite beam element can be derived. However, an alternative to this procedure, based on the transfer matrix method for the beam theory (see Graf and Vassilev, 2006:69–88 and Stanoev 2007) is developed in the present article. The element stiffness matrix can be derived on the basis of numerical integration of the first-order ordinary differential equation system for the differential beam element. The associated mass matrix can be developed by numerical integration of the inertia term in the differential equation of motion. The numerical integration results in static and kinetic field matrices, from which the element stiffness and mass matrices can be easily assembled.

In the present article, the above-mentioned procedure is used to determine
the element stiffness and element mass matrix for the Timoshenko beam. The
interpolation functions used for deriving the mass matrix are based on
Hermite polynomials according to Bazoune and Khulief (2003). The system
stiffness and mass matrices for the rotor blade are assembled in a global
coordinate basis in the usual finite element manner. The numerical solution
of the associated eigenvalue problem for the system without damping is
computed using computer algebra software (in the frame of
MATLAB^{®}).

In Sects. 2 and 3, the proposed method is described in detail; in Sect. 4 the method is applied on a rotor blade structure with 288 degrees of freedom (DOFs). The results for the natural frequencies and the corresponding eigenmodes are compared with the results calculated using the FAST and ADAMS software.

The general assumptions in the linear beam theory are as follows:

The beam reference (longitudinal) axis is at any arbitrary location of the cross section.

The only stresses that occur are normal stresses

Cross section planes, which are initially normal to the longitudinal axis, will remain plane after deformation.

Deformation at the point (

As seen in Fig. 1, the displacement vector

The virtual work components for internal forces corresponding to normal
stresses and shear stresses are given by the following:

With the introduction of the constitutive relations of Hooke's material law
for the stresses corresponding to the internal forces in Eq. (6) and
expressing the strains using Eqs. (4a)–(4c) and (5a)–(5b), the virtual work
relationship is reformulated as

After the coefficient comparison in Eqs. (6) and (7) the internal forces
corresponding to the normal stresses can be expressed by introducing the
cross sectional stiffness matrix

For the special case where the cross section coordinate system coincides with
the principal axes, the deformation relationship in Eq. (8) reduces to

The finite beam element: internal forces and nodal DOFs.

The virtual work relation in Eq. (7) is rewritten for the case where the beam
coordinate system coincides with the principal axis of the cross section –
see Eq. (11).

In the classical finite element formulation, the beam stiffness matrix and the consistent beam mass matrix are derived by developing an approach for the displacement functions through shape (interpolation) functions, which consist of first- and third-order Hermite polynomials. In this section, an alternative finite element procedure is presented, based on the numerical Runge–Kutta fourth-order integration of the differential motion equations. The integration of the static part (the coefficient matrix in Eqs. 13 and 14 respectively) leads to the static field matrix, while the integration of the inertia terms in the equation of motion Eq. (16) results in a kinetic field matrix. In the last step of the integration, using both field matrices, the respective element stiffness and the element mass matrices can be calculated using simple matrix operations.

The differential equations of motion for the differential beam
element (see Fig. 2) can be written in a matrix
form according to Stanoev (2007) and Müller (2012) as follows:

The inertia matrix in Eq. (18) implies that the distributed
mass

The acceleration term

Shape functions for axial and torsional deformations

Definition of the dimensionless coordinate (

The relationships in Eq. (10a), Eq. (11) and the
corresponding part of Eq. (14) are a starting point to derive approximation
functions for bending deformation in the

A similar method is used for determining the approximation functions

Starting with Eqs. (10b), (11) and (14), the following is obtained:

The special form of the numerical Runge–Kutta fourth-order integration method
applied here is described in detail in Müller and Wolf (1975) and
Schenk (2012). The integration operator is applied to the equations of motion
in Eq. (16), i.e.

Eccentrically applied mass

First flapwise eigenmode.

First edgewise eigenmode.

The numerical integration of the inertia term

Second flapwise eigenmode.

By solving the matrix equation in Eq. (41a) for

Mixed flap/edgewise eigenmode.

The numerical integration according to Runge–Kutta, described in Sec. 3.4,
offers the possibility of including single load or mass quantities within a
beam element. Single eccentric masses can be taken into account at the
integration interval boundaries. In the local coordinate system of the beam
element, at a general position vector

First torsional eigenmode.

The additional inertia matrix

First three calculated (bolded values) flapwise and edgewise eigenfrequencies.

Comparison between Timoshenko and Bernoulli beams with three variants for shear stiffness values.

Third flapwise eigenmode.

Second torsional eigenmode.

The system matrices for a rotor blade beam model are assembled in the usual
finite element manner employing the developed element matrices

The programming code for the procedure and the graphic plots described/shown
below was written in MuPAD, a symbolic math toolbox in
MATLAB^{®} (see Kusuma Chandrashekhara, 2018).
The code was verified using realistic data for a wind turbine rotor blade.
The blade structural data belong to a 5 MW reference wind turbine designed
for offshore development (Jonkman et al., 2009). The blade is 63 m in length
and is divided into 48 beam elements. The blade structural data consist of
distributed mass (

The mode shapes and the corresponding eigenfrequencies for the first flapwise and edgewise eigenmodes as well for two torsional eigenmodes are as shown in Figs. 5–11.

In Table 2 the calculated natural frequencies for three different variants of
the shear correction coefficients are shown, approximated as

The proposed Timoshenko beam element in the 3-D description has been developed on the basis of the transfer matrix method. Both static and kinetic field matrices for the beam element are derived by applying a Runge–Kutta fourth-order numerical integration procedure on the differential equations of motion in a special way. Appropriate shape functions for the Timoshenko beam have been used to approximate the inertia forces in the motion differential equation. The beam element stiffness and mass matrices are assembled by matrix operations from the derived element field matrices. The usual finite element equations of motion for the rotor blade model are cast in the general case accounting for the structural twist angle and possible pre-bending. Therefore, in the general case the rotor blade beam model represents a polygonal approximated space curve.

For the sake of verification, the natural frequencies and associated eigenmodes are calculated using real-life rotor blade data with the incorporation of realistic twist angle data. The first two edgewise and flapwise eigenfrequencies obtained are compared with the proposed results from FAST and ADAMS software given in Jonkman et al. (2009). It can be observed that the deviation of the results of the Timoshenko beam model from FAST is comparatively lower and is in good agreement with FAST; thus, it can be stated that the approach presented regarding alternative finite element formulation works well.

One key input parameter for the Timoshenko beam model is the shear stiffness.
As it was not the main goal of the present work to determine an
appropriate shear correction coefficient for realistic rotor blade data, the
numerical example was performed with a very rough approximation for

Data used in this work as
rotor blade structural data for the numerical example are publicly available
at ^{®} program code written to calculate
the rotor blade is available upon request.

ES led the underlying master thesis (Kusuma Chandrashekhara, 2018) and
developed the overall method theoretically by the use of the sources
referenced. SKC developed and wrote the code for the Timoshenko shape
functions used in the inertia term. He composed the
MATLAB^{®} code using different existent code
parts and optimized many of the procedures. SKC also performed the
calculation and the graphic plots of the numerical example.

The authors declare that they have no conflict of interest. Edited by: Lars Pilgaard Mikkelsen Reviewed by: two anonymous referees