Coupling between bending and twist has a significant influence on the aeroelastic response of wind turbine blades. The coupling can arise from the blade geometry (e.g. sweep, prebending, or deflection under load) or from the anisotropic properties of the blade material. Bend–twist coupling can be utilized to reduce the fatigue loads of wind turbine blades. In this study the effects of material-based coupling on the aeroelastic modal properties and stability limits of the DTU 10 MW Reference Wind Turbine are investigated. The modal properties are determined by means of eigenvalue analysis around a steady-state equilibrium using the aero-servo-elastic tool HAWCStab2 which has been extended by a beam element that allows for fully coupled cross-sectional properties. Bend–twist coupling is introduced in the cross-sectional stiffness matrix by means of coupling coefficients that introduce twist for flapwise (flap–twist coupling) or edgewise (edge–twist coupling) bending. Edge–twist coupling can increase or decrease the damping of the edgewise mode relative to the reference blade, depending on the operational condition of the turbine. Edge–twist to feather coupling for edgewise deflection towards the leading edge reduces the inflow speed at which the blade becomes unstable. Flap–twist to feather coupling for flapwise deflections towards the suction side increase the frequency and reduce damping of the flapwise mode. Flap–twist to stall reduces frequency and increases damping. The reduction of blade root flapwise and tower bottom fore–aft moments due to variations in mean wind speed of a flap–twist to feather blade are confirmed by frequency response functions.

Structural coupling of the flap- or edgewise bending and twist of wind turbine blades has a considerable influence on the aeroelastic response. The coupling creates a feedback loop between the aerodynamic forces, which induce bending in the blade, and the angle of attack, which determines the aerodynamic forces.

Bend–twist coupling can arise from the blade geometry (geometric coupling) or
from the anisotropic blade material (material coupling). Geometric coupling
is the result of a curved blade geometry (e.g. from prebend, load deflection,
or sweep) which induces additional torsion when the blade is loaded. Elastic
coupling results from the fibre direction in the spar cap and/or skin of the
blade. If fibre-reinforced plastic laminates are loaded transverse to their
principle axes, normal and shear strains become coupled. The coupling
transcends to the cross-section level, where it can result in the coupling of
beam bending and twist. Bend–twist coupling can be utilized to tailor the
aeroelastic response of wind turbine blades. Early studies on bend–twist
coupled blades investigate twisting towards a larger angle of attack for
flapwise deflection towards the suction side of the blade to reduce lift by
stalling the aerofoil (flap–twist to stall coupling). With the development
towards pitch-regulated
turbines, twisting towards a smaller angle of attack
has also been investigated (flap–twist to feather). The motivation behind
bend–twist coupling in wind turbine blade applications has mainly been load
alleviation. Fatigue load reductions in the range of 10–20 % have been
reported for flap–twist to feather coupled blades

Apart from the intended load alleviation, bend–twist coupling also affects
the aeroelastic modal properties (i.e. frequency, damping, mode shapes) and
stability of the blade.

In this paper the aeroelastic modal properties and stability limits of the
DTU 10 MW Reference Wind Turbine (RWT)

The modal properties of the DTU 10 MW RWT are investigated using the
aero-servo-elastic code HAWCStab2

To allow for the analysis of anisotropic cross-sectional properties the beam
element proposed by

The element coordinate system has its origin at the first node of the
element. The beam axis

Assuming plane sections to remain plane a beam strain vector

The generalized degrees of freedom are obtained by substituting a part of

A consistent mass matrix of the element

The implementation of the anisotropic beam element into the aero-servo-elastic analysis tool HAWCStab2 has been validated against various test cases of previous publications and by comparison of eigenfrequencies and steady-state results of the DTU 10 MW RWT with flap–twist coupled blades.

Bend–twist coupling was introduced by setting entries

Eigenfrequencies of a coupled cantilever obtained with the
present model compared to results by

Tip displacements and rotations (in Wiener–Milenkovic parameter)
of a coupled cantilever obtained with the present model compared to results
by

45

A square unit cross section with a modulus of elasticity of

Comparison of 45

The anisotropic beam element by

Natural frequency comparison of a flap–twist to feather coupled
DTU 10 MW RWT blade (

Power and thrust of the DTU 10 MW RWT with flap–twist to feather
coupled blades (

In an earlier study of a blade section model it has been shown that the
damping of the first edgewise and flapwise modes is mainly influenced by the
work of the lift

The quasi-steady aerodynamic lift of a blade section is

The effects of bend–twist coupling on the modal properties and stability of
wind turbine blades were investigated with the DTU 10 MW RWT developed by

To reduce the coupling-related power loss the blades were pretwisted at a
reference wind speed of 8 m s

Aerodynamic twist along the blade for the reference and flap–twist
to feather and stall coupled blades with coupling coefficients

Pitch angles over wind speed for the reference and flap–twist to
feather and stall coupled blades with coupling coefficients

In this section, the structural and aeroelastic modal properties of bend–twist coupled and pretwisted blades are investigated. First, the results of blade-only analysis are presented, followed by some additional investigations where the turbine dynamics have also been considered. The results focus on the first edgewise and first flapwise blade modes as the effects of bend–twist coupling on the frequency and damping are most distinct for those mode shapes. Also, the first edgewise mode is the lowest damped blade mode and the first to become unstable.

First, the effects of coupling on the structural mode shapes of the unloaded
blade are investigated. Figures

Structural mode shape of the first edgewise mode for the
reference blade, and edge–twist coupled blades with constant coupling
coefficients of

Structural mode shape of the first flapwise mode for the
reference blade, and flap–twist coupled blades with constant coupling
coefficients of

The bend–twist coupling also affects the structural and aeroelastic modal
frequency and damping. The aeroelastic modal properties are compared at
8 m s

Contour plots of structural (left) and aeroelastic (middle)
frequencies and their difference (right) of the first edgewise (top) and
first flapwise (bottom) mode for varying edge–twist (ordinate) and flap–twist
(abscissa) coupling coefficients at 8 m s

Figure

Contour plots of structural (left) and aeroelastic (middle) damping
ratios and their difference (right) of the first edgewise (top) and first
flapwise (bottom) mode for varying edge–twist (ordinate) and flap–twist
(abscissa) coupling coefficients at 8 m s

The effect of bend–twist coupling on frequencies and damping over the
operational range of the turbine has also been investigated.
Figure

Aeroelastic frequency

Figure

Aeroelastic frequency

The effect of flap–twist coupling on the edgewise mode over the operational
range of the turbine has also been examined. Figure

Aeroelastic frequency

Amplitudes

Amplitudes

Amplitudes

Aeroelastic frequency

Next, the modes shapes are investigated to identify the cause of the changes
in aeroelastic damping. Figure

Amplitudes

Aeroelastic frequency

Figure

Frequency response of the flapwise blade root bending moment to
mean wind speed variation between 0.0 and 2.0 Hz for steady-state
operation at mean wind speeds of 5, 10, 15, and 20 m s

Frequency response of the tower bottom fore–aft moment to
mean wind speed variation between 0.0 and 0.5 Hz for steady-state
operation at mean wind speeds of 5, 10, 15, and 20 m s

Figure

The stability of bend–twist coupled blades has been investigated in a runaway
scenario where the wind speed is slowly increased, while the pitch angle is
set to

The mode shapes of the reference and edge–twist coupled blades at a tip speed
of about 140 m s

The blade-only analysis has shown that the damping of the edgewise mode is
sensitive to the pitch angle (see Fig.

Flap–twist to feather coupled blades have been reported to reduce fatigue
loads of the flapwise blade root bending moment of the blade

Figure

Edge–twist coupling has only a small influence on the frequency of the
edgewise mode. Damping increases for edge–twist to feather coupling when the
pitch angle is close to zero (see Figs.

The DTU 10 MW RWT blade becomes unstable due to flutter of the edgewise
mode. Edge–twist to feather coupling reduces the critical inflow speed due to
an increase in the torsional component of the edgewise mode and a torsional
phase angle that is close to the flapwise velocity. The critical inflow speed
increases for edge–twist to stall coupled blades. The formation of an
edge–twist flutter mode, where the torsional component of the edgewise mode
becomes large enough and in phase with the flapwise velocity has previously
been reported by

Flap–twist to feather coupling increases the frequency and reduces the
damping of the first flapwise blade mode (see Fig.

The inflow speed at which the DTU 10 MW RWT becomes unstable due to flutter of the edgewise mode changes little for flap–twist to feather coupling and increases for flap–twist to stall coupling. The effect of flap–twist coupling on the classical flutter (where flapwise and torsional mode coalesce into an unstable mode) and divergence speeds could not be investigated as the first edgewise blade mode of the DTU 10 MW RWT becomes unstable before those speeds are reached.

In this paper the aeroelastic modal properties and stability limits of the DTU 10 MW RWT with bend–twist coupled blades have been investigated. Coupling has been introduced in the cross-section stiffness matrix by means of coupling coefficients. The aeroelastic modal properties and stability limits of both edge- and flap–twist coupled blades have been investigated by means of eigenvalue analysis around a steady-state equilibrium using the aero-servo-elastic tool HAWCStab2. For the analysis with fully coupled cross-section stiffness matrices, an anisotropic beam element has been implemented in HAWCStab2 and validated against previously published test cases.

The damping of the first edgewise mode increases for edge–twist to feather coupling, and it reduces for edge–twist to stall coupling if the pitch angle is close to zero. Outside that region, where the blade is pitched, damping reduces for edge–twist to feather and increases for edge–twist to stall coupled blades. The effect of edge–twist coupling on the edgewise turbine modes (forward and backward whirling and symmetric) is similar to the blade-only mode. Analysis of the edgewise mode shows that geometric coupling due to prebending and load deflection has a significant effect on the torsional component of the edgewise mode shape. Edge–twist to feather coupling reduces the critical inflow speed of the turbine due to an increase in the torsional component and a torsional phase angle that approaches the flapwise velocity.

The results on flap–twist coupled blades confirm the findings of previous studies: flap–twist to feather coupling increases the frequency and reduces the damping, and flap–twist to stall coupling reduces the frequency and increases the damping of the flapwise mode. Flap–twist coupling has little influence on frequency and damping of the edgewise mode. Flap–twist to feather coupling reduces the blade root flapwise moment frequency response to mean wind speed variation, which concurs with fatigue load reduction that have been observed for flap–twist to feather coupled blades. The frequency response of the tower bottom fore–aft moment is also reduced for flap–twist to feather coupled blades. The effect of flap–twist coupling on the classical flutter and divergence speeds could not be investigated as the first edgewise blade mode of the DTU 10 MW RWT becomes unstable before those speeds are reached.

Please contact the corresponding author to obtain the turbine models and data presented.

The authors declare that they have no conflict of interest.

The present work is funded by the European Commission under the programme “FP7-PEOPLE-2012-ITN Marie Curie Initial Training Networks” through the project “MARE-WINT – new MAterials and REliability in offshore WINd Turbines technology”, grant agreement no. 309395.Edited by: Joachim Peinke Reviewed by: two anonymous referees