WESWind Energy ScienceWESWind Energ. Sci.2366-7451Copernicus PublicationsGöttingen, Germany10.5194/wes-2-343-2017Modal properties and stability of bend–twist coupled wind turbine bladesStäbleinAlexander R.alexander@staeblein.comhttps://orcid.org/0000-0003-0804-2005HansenMorten H.https://orcid.org/0000-0002-8868-6152VerelstDavid R.https://orcid.org/0000-0002-3687-0636Technical University of Denmark, Department of Wind Energy,
Frederiksborgvej 399, 4000 Roskilde, DenmarkAlexander R. Stäblein (alexander@staeblein.com)30June20172134336028October20166December20168May201712May2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://wes.copernicus.org/articles/2/343/2017/wes-2-343-2017.htmlThe full text article is available as a PDF file from https://wes.copernicus.org/articles/2/343/2017/wes-2-343-2017.pdf
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.
Introduction
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. investigate the
aeroelastic stability of coupled helicopter composite blades using an
eigenvalue approach. The structure is modelled by a finite-element beam
formulation that integrates the strain energies over the cross section, thus
explicitly considering the fibre layup. The aerodynamic forces are assumed
quasi-steady. A linearization of the rotor blade around a steady-state
equilibrium point is used to obtain the modal properties by means of an
eigenvalue analysis. report reduced
frequencies for edge–twist coupled blades. Twist to feather for edgewise
deflection towards the leading edge increases the damping of the edgewise
mode. Damping reduces for edge–twist to stall. The authors conclude that
edge–twist coupling has an appreciable influence on stability. Twist to
feather for flapwise deflection towards the suction side of the aerofoil
increases the frequency and reduces the damping of the flapwise mode. The
frequency reduces and damping increases for twist to stall.
investigate the aeroelastic stability of
flap–twist to feather and stall coupled blades by casting Theodorsen's
equations of the aerodynamic lift and moment into pseudo-time domain and
applying the principle of virtual work to obtain aerodynamic mass, damping,
and stiffness matrices. The aerodynamic matrices are subsequently combined
with the structural matrices to formulate an eigenvalue problem.
report a moderately reduced flutter
speed for twist to feather coupled blades while divergence becomes critical
for twist to stall. investigate the damping of
a blade section in attached and separated flow. The edge- and flapwise
directions of vibration are prescribed and coupled with in-phase and
counterphase pitch motion. Aerodynamic damping is obtained by integrating the
aerodynamic work over one cycle of oscillation. For attached flow,
edge–twist to feather coupling reduces the damping for edgewise vibration
directions between the inflow and the rotor plane. For edge–twist to stall
coupling the damping increases. Flap–twist to feather coupling reduces
damping while damping increases for flap–twist to stall coupling.
investigates the flutter speed of an
uncoupled and a bend–twist to feather coupled megawatt-sized wind turbine
blade with quasi-steady and unsteady aerodynamic models, applying the
Theodorsen's approach. shows that
quasi-steady flutter speeds are significantly lower than the flutter speeds
obtained with unsteady aerodynamics. The flutter speed of the coupled blade
is moderately lower than of the uncoupled blade.
investigate the effect of finite steady-state
blade deflections on the aeroelastic stability of the NREL 5 MW Reference
Wind Turbine
using an eigenvalue approach. A geometric
nonlinear finite-element beam model and aerodynamic forces obtained from
blade element momentum theory are used to find a steady-state equilibrium of
the turbine. After linearization of the turbine around the steady-state
operational point and adoption of a Beddoes–Leishman-type dynamic stall
model, an eigenvalue analysis is carried out. Rotor dynamics are considered
by means of a Coleman transformation. The eigenvalue approach has been
implemented in the software tool HAWCStab2 .
observe a slight reduction in the flutter
limit for deflected blades due to coupling of the edgewise and torsional
component. investigates the aeroelastic
response of backward swept blades using the eigenvalue approach.
concludes that the backward sweep, which
induces flap–twist to feather coupling, mainly influences the flapwise mode
and has little influence on edgewise vibrations. Aeroelastic frequencies of
the flapwise mode increase while the flapwise damping and the flutter speed
reduce with sweep. investigate the flutter speed
of the NREL 5 MW turbine with flap–twist to feather coupled blades using
time domain analysis. report a slightly
reduced flutter speed if coupling is introduced by changing the fibre
direction of the glass fibres. If coupling is achieved by using carbon fibres
the flutter speed increases due to the higher stiffness of the blade.
investigate the aeroelastic modal
properties and stability limits of an edge- and flap–twist coupled blade
section using eigenvalue analysis. The authors conclude that damping
increases for edge–twist to feather coupling and reduces for twist to stall.
Flap–twist to feather increases the frequency and reduces damping while
twist to stall has the opposite effect.
show that edge–twist coupling can
result in aeroelastic flutter if the torsional component of the coupled
edge–twist mode becomes large enough to enable the formation of an
edge–twist flutter mode. Flap–twist to feather leads to a moderate
reduction of the classical flutter speed, while flap–twist to stall coupling
results in divergence.
In this paper the aeroelastic modal properties and stability limits of the
DTU 10 MW Reference Wind Turbine (RWT)
with bend–twist coupled blades are
investigated. Coupling is introduced in the cross-section stiffness matrix by
means of a coupling coefficient as proposed by
. The aeroelastic modal properties and
stability limits of both edge- and flap–twist coupled blades are
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 the beam element of
has been implemented in HAWCStab2.
MethodsIntroduction
The modal properties of the DTU 10 MW RWT are investigated using the
aero-servo-elastic code HAWCStab2 . HAWCStab2
calculates the steady-state response (including large blade deflections) at
an operational point (a combination of wind speed, rotational speed, and pitch
angle) assuming an isotropic rotor (i.e. no wind shear, yaw, tilt,
turbulence, tower shadow, or gravity). The aerodynamic forces are based on
blade element momentum theory and include tip loss. An analytical
linearization around the steady state is used to determine the modal
frequency and damping of the turbine by means of eigenvalue analysis. The
linearization includes the effects of shed vorticity, dynamic stall, and
dynamic inflow. The periodicity of the system is handled using the Coleman
transformation.
To allow for the analysis of anisotropic cross-sectional properties the beam
element proposed by has been implemented into
HAWCStab2. The two-noded element assumes polynomial shape functions of
arbitrary order where the shape function coefficients are eliminated by
minimizing the elastic energy of the beam while satisfying the boundary
conditions. The beam element formulation is recapitulated in this section.
Kinematic assumptions
The element coordinate system has its origin at the first node of the
element. The beam axis z is along the length L of the element, pointing
towards the second node. Axes x and y define the cross-sectional plane of
the beam. The lateral displacements ux,uy,uz, and the rotations
θx,θy,θz along the beam axis z are expressed as N-1-order polynomials ∑i=0N-1aizi. In matrix notation the
displacements and rotations along the beam can be expressed as
u(z)=N(z)α
where u(z)={ux,uy,uz,θx,θy,θz}T is the vector of the beam displacements and rotations,
N(z)=I6×6,zI6×6,z2I6×6,…,zN-1I6×6
is the polynomial matrix, where I6×6 are
6×6 identity matrices, and α is the vector of the
6N polynomial coefficients, which will be called generalized degrees of
freedom.
Elastic energy and strain displacement relation
Assuming plane sections to remain plane a beam strain vector ε=∂ux∂z-θy,∂uy∂z+θx,∂uz∂z,∂θx∂z,∂θy∂z,∂θz∂zT
can be introduced. Together with the 6×6 cross-section stiffness
matrix Kcs the elastic energy U of the beam can be
written as
U=12∫z=0LεTKcsεdz.
The beam strain vector can be expressed in terms of the generalized
degrees of freedom,
ε=B0N+∂∂zN︸Bα,
where
B0=0000-10000100000000000000000000000000
is a transformation and B the strain-displacement matrix.
Combining Eqs. () and () the elastic energy
becomes
U=12αT∫z=0LBTKcsBdz︸Dα,
where D is the beam element stiffness with respect to the
generalized degrees of freedom α.
Compatibility and order reduction
The generalized degrees of freedom are obtained by substituting a part of
α denoted α1 by the nodal degrees of
freedom d and determine the remaining part of α denoted α2 by
minimizing the elastic energy. Compatibility with the nodal degrees of
freedom d yields
d=Ndα=[N1|N2]α1α2,
where
Nd=[N1|N2]=I6×606×606×6…06×6I6×6LI6×6L2I6×6…LNI6×6
and
α=α1α2=I12×120(6N-12)×12α1+012×(6N-12)I(6N-12)×(6N-12)α2=A1α1+A2α2.
From Eq. () α1 can be rewritten
as
α1=N1-1(d-N2α2).
Substituting Eq. () into Eq. () yields
α=A1N1-1︸Y1d+A2-A1N1-1N2︸Y2α2.
The remaining generalized degrees of freedom α2 are obtained by substituting
Eq. () into the elastic energy Eq. (), and
minimizing with respect to α2, which yields
dUdα2=Y2TDY1︸Pd+Y2TDY2︸-Qα2=0⇒α2=Q-1Pd.
Substituting Eqs. () into () provides
α=(Y1+Y2Q-1P)︸Nαd,
which allows expression of the elastic energy Eq. () with
respect to the nodal degrees of freedom,
U=12dTNαTDNα︸Keld,
where Kel is the element stiffness matrix with respect
to the nodal degrees of freedom.
A consistent mass matrix of the element Mel is obtained
from the kinetic energy,
T=12d˙TNαT∫z=0LNTMcsNdzNα︸Meld˙,
where Mcs is the cross-sectional mass matrix.
Validation
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 K46, which
couples flapwise bending with torsion, and K56, coupling edgewise
bending with torsion, to
K46=γyK44K66,K56=-γxK55K66,
where γx and γy are edge- and flap–twist coupling
coefficients as proposed by , and K44,
K55, and K66 are flapwise bending, edgewise bending, and torsional
stiffness of the cross-section. For a positive definite stiffness matrix the
coupling coefficients have to be |γx/y|<1. For wind turbine blades
values up to 0.2–0.4 are deemed achievable
. Negative coupling
coefficients result in pitch to feather (reducing the angle of attack) for
edgewise/flapwise deflection towards the leading edge/suction side of the
blade. Positive coupling coefficients result in pitch to stall (increasing
the angle of attack).
Eigenfrequencies of a coupled cantilever
present the natural frequencies of a
coupled cantilever box beam. The beam is 2.54 m long and has a height of
16.76 mm (0.66 in) and width of 33.53 mm (1.32 in). The wall
thickness is 0.84 mm (0.033 in) with six layers of unidirectional lamina
stacked (20/-70/20/-70/-70/20) from outside to inside. The material is T300/5208 graphite/epoxy with properties provided by
. The material density is given by
as 1604 kg m-3 (1.501×10-4 lbsec2 in-4). The cross-section stiffness matrix
was taken from and converted to SI
units:
Kcs=5.0576×10600-1.7196×104007.7444×105008.3270×10302.9558×105009.0670×1031.5041×10200sym.2.4577×10207.4529×102.
The cantilever was discretized with 16 elements. Table shows
a comparison of the frequencies obtained with the present beam model, the
beam models by and
, and a finite-element shell model by
. The finite beam element by
is based on a mixed variational
formulation with cross-sectional properties obtained with a virtual work
method by . The formulation by
is based on a variational asymptotic method
and Hamilton's principle.
Eigenfrequencies of a coupled cantilever obtained with the
present model compared to results by ,
(both beam models), and
(shell model).
ModeFreq. (Hz) Rel. diff. (%) PresentHodgesArmaniosKimHodgesArmaniosKim1 vert.2.993.002.962.980.3-1.0-0.31 horiz.5.185.195.105.120.2-1.5-1.22 vert.18.7519.0418.5418.651.5-1.1-0.52 horiz.32.3632.8831.9832.021.6-1.2-1.13 vert.52.4454.6551.9252.174.2-1.0-0.53 horiz.89.4093.3989.5593.394.50.24.51 tors.180.10180.32177.05–0.1-1.7–2 tors.542.05544.47531.15–0.4-2.0–Tip displacements and rotations of a coupled cantilever
present a coupled cantilever beam with a tip
load. The stiffness matrix in the original study is
Kcs=1368.170000088.56000038.7800016.9617.61-0.351sym.59.12-0.370141.47×103.
The beam has a length of 10 m and was discretized by 10 elements. A tip load
of 150 N was applied to the cantilever. The tip displacements and rotations
(in Wiener–Milenkovic parameter) are shown in Table .
Tip displacements and rotations (in Wiener–Milenkovic parameter)
of a coupled cantilever obtained with the present model compared to results
by .
present the geometric nonlinear response of a
45∘ bend cantilever with a radius of 100 m as shown in
Fig. . The test case has been extended with bend–twist
coupled cross-sectional properties by .
45∘ bend cantilever.
A square unit cross section with a modulus of elasticity of 1.0×107 N m-2 was used for the analysis. The beam was coupled with a
constant coefficient of γy=-0.3 along the length. A tip load of
300 N has been applied. Table shows the tip displacement of
the uncoupled beam compared to results by
. And the coupled beam compared to results
of a Timoshenko beam element with anisotropic cross-sectional properties by
.
Comparison of 45∘ bend cantilever tip displacements.
Original test case (uncoupled) and modified test case with bend–twist
coupling.
Displacement (m) Rel. diff. (%) xyzxyzSimo and Vu-Quoc-11.87-6.9640.08–––Present uncoupled-12.15-7.1740.482.33.11.0Stäblein and Hansen-10.66-6.5338.68–––Present coupled-10.65-6.5638.69-0.10.40.0Eigenfrequencies and steady-state results for DTU 10 MW RWT blade
The anisotropic beam element by has previously
been implemented in HAWC2, an aeroelastic time-domain analysis tool for wind
turbines capable of computing structural modal properties at standstill. The
HAWCStab2 implementation was therefore compared to HAWC2 by analysing the
natural frequencies at standstill, as well as steady-state power and thrust
(ignoring wind shear, yaw, tilt, turbulence, tower shadow, and gravity) of a
bend–twist to feather coupled blade for the DTU 10 MW RWT. The blade was
coupled with a constant coefficient of γy=-0.2 along the blade. For
the comparison, only the cross-sectional properties of the blade were
modified. The twist distribution and pitch angle were adopted from the
reference turbine which explains the unusual shape of the power curve. The
first 10 natural frequencies are compared in Table and
the results show only minimal differences. The power and thrust over the
operational wind speed range are compared in Fig. and, again,
the results show only minimal differences.
Natural frequency comparison of a flap–twist to feather coupled
DTU 10 MW RWT blade (γy=-0.2) obtained with HAWCStab2 and HAWC2.
Power and thrust of the DTU 10 MW RWT with flap–twist to feather
coupled blades (γy=0.2) obtained with HAWC2 and HAWCStab2. Note that
the blades have not been pretwisted and the pitch angel has not been adjusted
for this graph.
Aerodynamic damping analysis of blade modes
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 . Changes in the damping
ratio can therefore be explained by looking at the phase angle between the
aerodynamic lift and the flapwise component. Damping reduces if the lift
force is ahead of the flapwise component and it increases if lift is behind.
To facilitate the interpretation of the mode shapes in the results section,
the quasi-steady aerodynamic lift is recapitulated here. For the eigenvalue
analysis in this study, the Beddoes–Leishman-type dynamic stall model in
HAWCStab2 has been used to determine the modal properties of the blades.
The quasi-steady aerodynamic lift of a blade section is
Lqs=πρcWWθ-y˙+c2-eacθ˙+πρc24Wθ˙-y¨+c4-eacθ¨,
where ρ is the density of the air, c the chord length, W the inflow
velocity, θ the angle between inflow and chord, y the flapwise
displacement, and eac the distance between aerodynamic centre
and centre of twist of the section. Time derivatives are denoted ()˙
and ()¨. The first term in Eq. () is the lift due to
the circulatory airflow around the section, and the second term contains the
inertia and centrifugal forces of the apparent mass. By assuming a low
frequency and a small torsional component θ of the first edgewise and
flapwise modes, the time derivative of the torsion θ˙ can be
considered small. By further ignoring inertia and centrifugal forces of the
apparent mass the aerodynamic lift reduces to
Lqs≈πρcWWθ-y˙.
The lift due to the flapwise velocity y˙ is always 90∘ behind
the flapwise component, which explains the high damping of flapwise modes. The
phase angle of the lift in Eq. () therefore depends on the
amplitude and phase angle of the torsional component θ relative to the
flapwise component of the mode shape. If the torsion is ahead of the flapwise
component, damping reduces. Damping increases if torsion is behind the
flapwise component.
Bend–twist coupled DTU 10 MW blade
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
. It is a horizontal-axis, variable-pitch,
variable-speed wind turbine with a rotor diameter of 178 m and a hub height
of 119 m. The structural properties of the blades in terms of 6×6
cross-section stiffness matrices were obtained with BECAS
and the input data provided on the DTU 10 MW RWT
project homepage
http://dtu-10mw-rwt.vindenergi.dtu.dk
.
Bend–twist coupling was introduced by means of edge- and flap-twist coupling
coefficients γx and γy as described earlier.
To reduce the coupling-related power loss the blades were pretwisted at a
reference wind speed of 8 m s-1 using the procedure presented by
. Figure shows the aeroelastic
twist along the blade for the reference and flap–twist to feather and stall
coupled blades with coupling coefficients γy=±0.1 constant along
the blades. The aerodynamic twist of the flap–twist to feather coupled blade
increases towards the blade tip, compared to the reference blade, to
compensate for the coupling-induced twist. The flap–twist to stall coupled
blade has a lower aerodynamic twist towards the blade tip. As a result of the
pretwisting procedure, the steady-state angle of attack and hence the
aerodynamic states are identical for all models at the reference wind speed
of 8 m s-1. After pretwisting the blade, the pitch angles that
optimize power below rated and limit power above rated were recalculated. The
pitch angles have a lower bound of 0∘ and are constrained by a
maximum angle of attack of 8∘ in the outer part of the blade. The
pitch angles over wind speed for the reference and flap–twist coupled blades
are shown in Fig. .
Aerodynamic twist along the blade for the reference and flap–twist
to feather and stall coupled blades with coupling coefficients
γy=±0.1 constant along the blades.
Pitch angles over wind speed for the reference and flap–twist to
feather and stall coupled blades with coupling coefficients γy=±0.1
constant along the blade.
Results
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.
Blade modal properties
First, the effects of coupling on the structural mode shapes of the unloaded
blade are investigated. Figures and
show the structural mode shapes of the first edgewise
mode for edge–twist coupled blades and the first flapwise mode for
flap–twist coupled blades. The edgewise and flapwise amplitudes are similar
for all blades. The structural properties and upwind prebend of the reference
blade result in a tip twist of 0.5∘ towards stall for the edgewise
mode. Edge–twist coupling of the cross-section stiffness matrix results in
an additional tip twist of about 0.5∘ towards feather for
γx=-0.1, or towards stall for γx=+0.1 relative to the
reference blade. The flapwise mode of the reference blade has a tip twist of
about 0.3∘ towards feather. Flap–twist coupling results in an
additional tip twist of about 0.4∘ towards feather for
γy=-0.1, or towards stall for γy=+0.1. The sudden change in
the torsional component at the last element is caused by a forward sweep of
the blade geometry at the tip. When the torsion of the tip node is computed,
some of the flapwise bending is projected onto the spanwise blade coordinate
due to the slightly forward swept element axis.
Structural mode shape of the first edgewise mode for the
reference blade, and edge–twist coupled blades with constant coupling
coefficients of γx=±0.1.
Positive coupling coefficients denote twist to stall for edgewise
deflection towards the leading edge.
The amplitudes are normalized to 1 m tip deflection in the edgewise
direction.
Structural mode shape of the first flapwise mode for the
reference blade, and flap–twist coupled blades with constant coupling
coefficients of γy=±0.1.
Positive coupling coefficients denote twist to stall for flapwise
deflection towards the suction side.
The amplitudes are normalized to 1 m tip deflection in the flapwise
direction.
Aeroelastic frequency and damping over coupling coefficients
The bend–twist coupling also affects the structural and aeroelastic modal
frequency and damping. The aeroelastic modal properties are compared at
8 m s-1, where the aerodynamic steady states are the same for all
blades because this wind speed is used in the pretwisting of the coupled
blades. Figure shows contour plots of the structural
(left column) and aeroelastic (middle column) modal frequencies for the first
edgewise (top row) and first flapwise (bottom row) mode. The difference
between structural and aeroelastic frequency is also plotted (right column)
to show the effect of the aerodynamic forces. The contour plots have been
obtained with a coupling coefficient step size of 0.01, resulting in a total
of 2601 blade models. Each individual blade has been pretwisted to ensure the
same angle of attack along the blade as the reference blade. The structural
and aeroelastic frequencies of the edgewise mode (top row) are mainly
influenced by edge–twist coupling. Flap–twist coupling has only a small
influence. The structural frequency has a maximum around γx=-0.1 and
reduces from there for both twist to feather and stall. The frequency
difference on the top right shows that the aerodynamic forces increase the
frequency for edge–twist to stall coupling and reduce it for edge–twist to
feather. The frequencies of the flapwise mode (bottom row) are mainly
influenced by flap–twist coupling. The structural frequency has a maximum
around γy=0.05 and reduces from there for both twist to feather and
stall. The aerodynamic forces result in a frequency increase for flap–twist
to feather and a reduction for flap–twist to stall. The frequency change due
to flap–twist coupling is somewhat larger than for edge–twist coupling.
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-1 wind speed.
Figure shows contour plots of the structural (left
column) and aeroelastic (middle column) modal damping for the first edgewise
(top row) and first flapwise (bottom row) mode. The difference between
structural and aeroelastic damping is also plotted (right column) to show the
effect of the aerodynamic forces. For the edgewise mode, the structural
damping contributes about 25 % to the aeroelastic damping. The aeroelastic
damping tends to increase for edge–twist to feather and reduce for edge–twist
to stall coupling. For the flapwise mode, the structural contribution to the
aeroelastic damping is negligible. Damping increases for flap–twist to stall
coupling and reduces for flap–twist to feather.
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-1 wind speed.
Aeroelastic frequency and damping over operational range
The effect of bend–twist coupling on frequencies and damping over the
operational range of the turbine has also been investigated.
Figure shows aeroelastic frequency (top) and damping
ratio (bottom) over the operational wind speed range for the first edgewise
blade-only mode for the reference, and edge–twist coupled blades with
coupling coefficients of γx=±0.1. The frequency of the edgewise
mode changes little with the coupling. Damping increases in the wind speed
range between 6 and 11 m s-1 (where the pitch angle is close to zero)
for the edge–twist to feather coupled blade and damping reduces for
edge–twist to stall coupling. Outside that region where the blade is pitched
(see Fig. ) damping reduces for edge–twist to feather coupling
and increases for edge–twist to stall.
Aeroelastic frequency (a) and damping ratio (b)
over
the operational wind speed range of the first edgewise blade-only
mode for the reference and edge–twist coupled blades with coupling
coefficients of γx=±0.1.
The grey solid lines indicate other blade modes of the reference blade.
The grey dashed line indicates the pretwisting reference speed of 8 m s-1.
Figure shows aeroelastic frequency (top) and damping
ratio (bottom) over the operational wind speed range of the first flapwise
blade mode for the reference, and flap–twist coupled blades with coupling
coefficients of γy=±0.1. The frequency of the flapwise mode
increases over the whole operational range for flap–twist to feather coupling
and reduces for flap–twist to stall. Damping reduces over the whole
operational range for flap–twist to feather and increases for flap–twist to
stall.
Aeroelastic frequency (a) and damping ratio (b)
over
the operational wind speed range of the first flapwise blade-only
mode for the reference and flap–twist coupled blades with coupling
coefficients γy=±0.1.
The grey solid lines indicate other blade modes of the reference blade.
The grey dashed line indicates the pretwisting reference speed of 8 m s-1.
The effect of flap–twist coupling on the edgewise mode over the operational
range of the turbine has also been examined. Figure
shows aeroelastic frequency (top) and damping ration (bottom) over the
operational wind speed range of the first edgewise blade mode for the
reference, and flap–twist coupled blades with coupling coefficients of
γy=±0.1. The frequency of the edgewise mode is not influenced by
the flap–twist coupling. Damping for the coupled blades varies around the
reference blade, but it remains close to the damping of the reference blade.
Aeroelastic frequency (a) and damping ratio (b)
over
the operational wind speed range of the first edgewise blade-only
mode for the reference and flap–twist coupled blades with coupling
coefficients γy=±0.1.
The grey solid lines indicate other blade modes of the reference blade.
The grey dashed line indicates the pretwisting reference speed of 8 m s-1.
Aeroelastic mode shapes
Amplitudes (a) and phase angles (b) of the first
aeroelastic edgewise mode at 8 m s-1 wind speed for the reference
(blue), and edge–twist to feather (green) and stall (red) coupled DTU 10 MW
RWT blades. Coupling coefficients are γx=±0.1 constant along the
blade. Amplitudes are normalized to 1.0 m tip deflection in the edgewise
direction, and phase angles are relative to the edgewise tip deflection.
Amplitudes (a) and phase angles (b) of the first
aeroelastic edgewise mode at 16 m s-1 wind speed for the reference
(blue), and edge–twist to feather (green) and stall (red) coupled DTU 10 MW
RWT blades. Coupling coefficients are γx=±0.1 constant along the
blade. Amplitudes are normalized to 1.0 m tip deflection in the edgewise
direction, and phase angles are relative to the edgewise tip deflection.
Amplitudes (a) and phase angles (b) of the first
aeroelastic flapwise mode at 8 m s-1 wind speed for the reference
(blue), flap–twist to feather (green), and flap–twist to stall (red) coupled DTU 10 MW
RWT blade. Coupling coefficient is γy=±0.1 constant along the
blade. Amplitudes are normalized to 1.0 m tip deflection in the flapwise
direction, and phase angles are relative to the flapwise tip deflection.
Aeroelastic frequency (a) and damping ratios (b)
over the tip
speed for the first edgewise mode in a runaway scenario.
Coupling coefficients are γx/y=±0.1 constant along the blade.
Next, the modes shapes are investigated to identify the cause of the changes
in aeroelastic damping. Figure shows the amplitudes and
phase angles of the first edgewise aeroelastic mode at 8.0 m s-1 wind
speed for the reference and edge–twist to feather and stall coupled blades
with coupling coefficients of γx=±0.1. The amplitudes are
normalized to 1.0 m tip deflection in the edgewise direction, and the phase
angles are relative to the edgewise tip deflection. The phase angle of the
edgewise component is close to zero along the blade. The flapwise components
at the tip and phase angles in the outer part are similar for all blades:
0.27 m and -60∘ for the reference, 0.24 m and -40∘ for
the edge–twist to feather, and 0.33 m and -70∘ for the edge–twist
to stall coupled blade. The torsional component at the tip of the reference
blade is 0.18∘ and has a phase angle of 170∘. Thus, in
contrast to the structural mode shape, the reference blade tip twists towards
feather (instead of stall) for edgewise deflection towards the leading edge.
This twist to feather coupling is caused by the nonlinear geometric coupling
when the blade is bending downwind due to the mean aerodynamic forces. The
torsional components and phase angles of the coupled blades are 0.68 and
-180∘ for edge–twist to feather, and 0.37 and 15∘ for
edge–twist to stall. In an earlier publication
it was shown that edgewise damping is
dominated by the work of the lift, which reduces when the lift is ahead of the
flapwise component. As the flapwise components of the coupled blades are in
the same order, it is sufficient to focus on the difference in amplitude and
phase angle of the torsional component. For the edge–twist to feather coupled
blade, torsion is lagging 140∘ behind the flapwise component (i.e.
the lift induced by torsion is almost in counterphase with the flapwise
velocity) which increases the work of the lift force and the damping. The
reference blade has a similar phase angle but a lower torsional amplitude,
resulting in lower damping compared to the edge–twist to feather coupled
blade. For the edge–twist to stall coupled blade, torsion is 85∘
ahead of the flapwise component and the work of the lift (and the damping) is
reduced.
Amplitudes (a) and phase angles (b) of the first
aeroelastic edgewise mode as indicated in Fig. at approx.
140 m s-1 tip speed for the reference (blue), and edge–twist to
feather (green) and stall (red) coupled DTU 10 MW RWT blade. Coupling
coefficients are γx=±0.1 constant along the blade. Amplitudes are
normalized to 1.0 m tip deflection in the edgewise direction, and phase
angles are relative to the edgewise tip deflection.
Aeroelastic frequency (a) and damping ratio (b)
over the operational wind speed range of the first edgewise turbine
modes for reference and edge–twist coupled blades with coupling
coefficients γx=±0.1.
The grey lines indicate the remaining turbine modes (tower side–side,
fore–aft, first flap backward whirling, symmetric, forward whirling, second flap backward whirling, symmetric, forward whirling
from low to high frequency) of the reference blade.
Figure shows the amplitudes and phase angles of the
first edgewise aeroelastic mode at 16.0 m s-1 wind speed for the
reference and edge–twist to feather and stall coupled blades with coupling
coefficients of γx=±0.1. The flapwise components of the reference
and edge–twist to feather coupled blade reduce compared to the edgewise mode
at 8 m s-1 wind speed. The flapwise components at the tip and phase
angles in the outer part are 0.13 m and -60∘ for the reference,
0.07 m and 5∘ for the edge–twist to feather, and 0.28 m and
-80∘ for the edge–twist to stall coupled blade. The torsional
components and phase angles are 0.09 and -130∘ for the reference,
0.55 and -170∘ for the edge–twist to feather, and 0.52 and
0∘ for the edge–twist to stall coupled blade. The lower damping for
edge–twist to feather compared to edge–twist to stall coupled blades can be
explained by the flapwise amplitude, which is 3 times larger for the
edge–twist to stall coupled blade. The increased amplitude results in a
larger damping due to the direct coupling of the angle of attack/lift and the
flapwise velocity. The torsional component of the edge–twist to feather
coupled blade is close to counterphase with the flapwise velocities and
therefore has little influence on the damping. The torsional component of the
edge–twist to stall coupled blade is nearly in phase with the flapwise
velocities, which reduces the damping.
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-1.
The coupled blades have a constant coefficient of γy=±0.1.
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-1.
The coupled blades have a constant coefficient of γy=±0.1.
Figure shows the amplitudes and phase angles of the first
flapwise mode at 8.0 m s-1 wind speed for the reference and flap–twist
to feather and stall coupled blades with coupling coefficients of
γy=±0.1. The amplitudes are normalized to 1.0 m tip deflection in
the flapwise direction, and the phase angles are relative to the flapwise tip
deflection. The phase angle of the flapwise component is about 10∘
in the outer part of the blade. The edgewise components at the tip are around
0.17 m and in phase with the flapwise blade tip deflections for all three
blades. The torsional components and phase angles are 0.36 and
-120∘ for the reference, 0.68 and -160∘ for the
edge–twist to feather, and 0.40 and -35∘ for the edge–twist to
stall coupled blade. As for the edgewise mode, the flapwise damping is
dominated by the work of the lift. For the flap–twist to feather coupled
blade torsion is close to counterphase with the flapwise component resulting
in a torsional component that contributes little to the work of the lift and
the damping. The torsional phase angle of the flap–twist to stall coupled
blade, however, is lagging the flapwise component by about
45∘, which results in an increased work of the lift. Together with
the reduced frequency the increased lift work results in higher damping.
Runaway analysis
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 0∘ and the generator torque is zero. The stability analysis
has been conducted with four different coupling coefficients
γx=±0.1 and γy=±0.1 along the whole blade span.
Figure shows aeroelastic frequency and damping over the
tip speed for the lowest damped mode, which is the first edgewise mode in such
a runaway scenario. The eigenvalue analysis shows that the reference blade
becomes unstable at a tip speed of about 180 m s-1.
report an edgewise instability at approximately
22 rpm or 205 m s-1 at the tip using nonlinear time domain analysis.
The edge–twist to feather coupled blade has a slightly lower frequency and a
lower damping than the reference blade. The lower damping of the edge to
feather coupled blade results in instability at a much lower tip speed of
130 m s-1. The edge to stall coupled blade shows a slightly higher
frequency and a higher damping than the uncoupled blade. Frequency and
damping of the flap–twist to feather coupled blade is very close to the
reference. The flap–twist to stall coupled blade is close to the reference
until a tip speed of about 140 m s-1, where the frequency starts to
reduce and, as a result, damping increases. This behaviour is
characteristic for a mode that approaches divergence instability.
The mode shapes of the reference and edge–twist coupled blades at a tip speed
of about 140 m s-1 of the runaway scenario are shown in
Fig. . The amplitudes are normalized to 1.0 m tip
deflection in the edgewise direction, and the phase angles are relative to
the edgewise tip deflection. The phase angle of the edgewise component is
close to zero along the blade. The flapwise components at the tip and phase
angles in the outer part are 0.06 m and 55∘ for the reference,
0.29 m and 75∘ for the edge–twist to feather, and 0.18 m and
-80∘ for the edge–twist to stall coupled blade. The torsional
components and phase angles are 0.46 and -160∘ for the reference,
0.93 and -170∘ for the edge–twist to feather, and 0.21 and
-35∘ for the edge–twist to stall coupled blade. As for the previous
mode shapes, the difference in damping can be explained by observing the
amplitudes and phase angles between the torsional and flapwise components.
The edge–twist to feather coupled blade has the largest torsional amplitude
and a phase angle that is closest to the flapwise velocities (which are
90∘ ahead of the flapwise component), which reduces the damping. For
the reference and edge–twist to feather coupled blades the torsional
amplitudes decrease and the phase angle moves away from the flapwise
velocity,
which has a positive effect on the modal damping.
Turbine modal properties
The blade-only analysis has shown that the damping of the edgewise mode is
sensitive to the pitch angle (see Fig. ) because the
pitching affects the mode shape relative to the inflow. For a stability
analysis of the edgewise mode, all factors that could influence the mode
shape should therefore be considered. The effect of turbine dynamics on the
modal properties of the edgewise mode has been investigated in
Fig. . The plot shows modal frequency and damping of
the backward whirling (BW), forward whirling (FW), and symmetric edgewise
modes for the reference and edge–twist coupled blades (γx=±0.1).
The grey lines indicate the remaining turbine modes of the reference blade,
which, counting from low to high frequency, are tower side–side, tower
fore–aft, and the first and second backward whirling, symmetric, and forward
whirling flapwise modes. The frequency of the edgewise modes changes little
with the coupling. Between 6 and 11 m s-1 wind speed damping of the
edge–twist to feather coupled blade increases for all three edgewise turbine
modes (backward and forward whirling and symmetric) as predicted by the
blade-only analysis. For edge–twist to stall the damping reduces. Above rated
wind speed, the damping of the backward and forward whirling modes drop for
all blades. Damping of the symmetric modes increases. Above rated wind speed,
edge–twist to feather coupling reduces the damping of the symmetric and
backward whirling modes. The damping of the backward whirling mode is very
similar to the aeroelastic damping of the edgewise blade mode (see
Fig. ) and for the edge–twist to feather coupled
blade the backward whirling mode becomes the lowest damped mode.
Flap–twist to feather coupled blades have been reported to reduce fatigue
loads of the flapwise blade root bending moment of the blade
. To investigate the load
alleviation in frequency domain, the frequency response of the flapwise blade
root bending moment to mean wind speed variation between 0 and 2 Hz for
steady-state operation at mean wind speeds of 5, 10, 15, and 20 m s-1
is shown in Fig. . The coupling coefficient is
γy=±0.1 constant along the blade. The flap–twist to feather
coupled blade shows a reduced magnitude for wind speed variations below
0.5 Hz. The magnitude increases for flap–twist to stall coupled blades.
Above 0.5 Hz the frequency response is similar for all blades. The dip in
the frequency response around 0.25 Hz is caused by antiresonance due to interference with the tower
fore–aft mode.
Figure shows the frequency response of the tower bottom
for-aft moment to mean wind speed variation between 0 and 0.5 Hz for steady
state operation at mean wind speeds of 5, 10, 15, and 20 m s-1.
Flap–twist to feather coupling tends to reduce the frequency response for all
operational points while the response increases for flap–twist to stall
coupled blades.
Discussion
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. and
) and reduces for edge–twist to stall coupling.
For wind speeds where the blade is pitched, the damping of the edgewise mode
reduces for edge–twist to feather and increases for edge–twist to stall
coupling. Increased damping for edge–twist to feather and reduced damping for
edge–twist to stall coupling is also observed by
and .
, however, observe reduced damping for
edge–twist to feather and increased damping for edge–twist to stall coupling
if the direction of the edgewise vibration is between the inflow and the
rotor plane. The qualitative differences of the edge–twist coupling effect on
damping reported in previous studies, and the observed change over the
operational wind speed range (see Fig. ) show that
damping of the edgewise mode can be sensitive to changes in the mode shape.
The effect of turbine dynamics have therefore been investigated (see
Fig. ). The results show that the effects of edge–twist
coupling on the edgewise turbine modes are similar to the blade-only mode
(i.e. edge–twist to feather coupling increases damping if the pitch angle is
close to zero and reduces damping if the blade is pitched). Analysis of the
edgewise mode shape further shows that geometric coupling due to prebending
and load deflection has a significant influence on the edgewise mode shape.
An effect of blade deflection on the edgewise mode shape has also been
observed by .
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 and
.
Flap–twist to feather coupling increases the frequency and reduces the
damping of the first flapwise blade mode (see Fig. ).
Flap-twist to stall coupling reduces the frequency and increases the damping. Similar observations have been made in previous studies
.
Flap–twist coupling has little influence on the damping of the edgewise mode
(see Fig. ). A similar observation has been made by
for swept blades. The reduced frequency
response to mean wind speed variations of the blade root flapwise moment (see
Fig. ) concurs with reduced fatigue loads observed by
, , , and
. Flap–twist to feather coupling also
reduces the tower bottom fore–aft moment (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.
Conclusions
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.
Acknowledgements
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
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