Tong W. Wind Power Generation and Wind Turbine Design
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Aerodynamics and Aeroelastics of Wind Turbines 115
large input–output area ratio, whilst being relatively short. A permanent danger is the
early separation of air fl ow through the diffuser as can be seen from Fig. 28 [ 41 , 47 ],
where a relatively short diffuser was designed for an area ratio of approximately 3.
The author does not agree that a fi nal theory has been formulated in which the
specifi c problem has been solved. It may be necessary to extend the work of Loth
and Mc Coy [ 70 ] who discussed an array of disks as a model of an actuator volume
to a real continuum theory. This is an important task because a safe upper limit,
such as Betz’ law, is important for all investigations. A successful commercial
design has also not been produced so far.
8 Summary and outlook
The history of wind turbine aerodynamics spans approximately 150 years. The
most important infl uence has been given by modelling fl ow properties which then
serve as tools for predicting loads on the structure. As a consequence safe and
competitive wind turbines up to rated power of 6 MW have been constructed so
far. First investigations of even larger blades feasible for rated power up to 10 MW
are under consideration [ 29 ].
Further progress can be expected when CFD when taken in the strict sense of the
expression as solving the Navier-Stokes equation without any further assumptions
enters the usual design process in companies. Nevertheless full-scale experiments in
wind tunnels and outdoors are equally as important as analytic theories.
Figure 28 : Sketch of the separation of fl ow through a short diffuser .
116 Wind Power Generation and Wind Turbine Design
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120 Wind Power Generation and Wind Turbine Design
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CHAPTER 4
Structural dynamics of wind turbines
Spyros G. Voutsinas
National Technical University of Athens, Greece.
Structural dynamics of wind turbines aims at assessing the integrity and reliability
of the complete construction against varying external loading over the targeted
life time. Since wind induced excitation is the most important, structural dynam-
ics is closely connected to aeroelasticity. Most of structural analysis is based on
modelling while tests are reserved for prototype assessment. To this end elastic
modelling at component level and dynamic modelling of the complete system are
required. The aim of the present chapter is to provide an overview on the theory
and approximation tools of structural dynamic analysis of modern wind turbines
on these two aspects. At component level beam models of varying complexity are
discussed while at system level the multi-body approach is defi ned and its use in
formulating the dynamic model of the complete wind turbine is explained. Next,
after a short introduction to aeroelastic coupling, the issue of stability is outlined.
Finally some indicative results show the kind of information structural analysis
can provide. The focus of the presentation is on the procedures rather than the
details of derivations for which appropriate pointers to the literature are given.
1 Wind turbines from a structural stand point
A typical wind turbine confi guration includes as main components the rotor blades,
the drive train and the tower, all considered as fl exible structures. When considering
the wind turbine as a system, all components are modelled as slender beams sub-
jected to combined bending, torsion and tension with distributed material prop-
erties like stiffness, damping and density. At this level, the elastic behaviour of
the different components is described by continuous displacement and rotation
fi elds along the span of each component. In the discrete context the displacements
and rotations will correspond to the elastic degrees of freedom (DOF) . The beam
like modelling associated to structural fl exibility is discussed in Sections 3 and 7.
122 Wind Power Generation and Wind Turbine Design
Clearly assessment of local stresses and strains especially when dealing with
composites, requires more advanced modelling that account for the internal
material structure.
Solely the elastic DOF cannot describe the complete dynamics of a wind tur-
bine. There are also rigid DOF while the connections among the components fur-
ther complicate the dynamics of the system. The rotor blades are connected to the
hub (the one end of the drive train) allowing rotation with respect to the drive-train
axis while the nacelle which houses the drive train, is connected to the tower top
also allowing at least rotation in yaw. In addition to blade rotation and yaw there is
also the blade pitch. Pitch can vary either collectively or cyclically or even inde-
pendently for each blade. Finally in the case of two-bladed rotors, the rotor system
as a whole is allowed to teeter. At the connection points, specifi c compatibility
conditions are required which transfer the deformation state from one connected
component to the other. This calls for a multi-component analysis which we detail
in Section 4.
Most of the rigid body DOF are connected to the control system. At least in the
more recent designs there is clear preference for variable pitch and variable speed
control concepts. So when the wind turbine operates in closed loop, the variation
of the pitch and the rotor speed will satisfy the control equations which therefore
become part of the dynamic description of the system. In certain cases, there is
need to also include concentrated properties like point masses, springs or dampers.
For example, the yaw gear and the gear box in the drive train are modelled as com-
bined point masses and springs while the hub is modelled as concentrated mass.
Concentrated properties are associated to specifi c DOF for which we must formulate
appropriate dynamic equations as in the case of the control variables.
Although reference has been made to the formulation of dynamic equations we
did not specify the way this is done. The general framework is presented in Section 2.
One way to proceed is to base the derivation on Newton’s second law which
requires the dynamic balance of forces and moments. It has the advantage of a
direct interpretation in relatively simple systems but becomes cumbersome when
dealing with complicated systems. Alternatively energy or variational principles,
like the principle of virtual work or Hamilton’s principle offer a general and sys-
tematic way for deriving the dynamic equations of even very complicated mechan-
ical systems. It is true that depending on the complexity of the system, the
derivations can become quite lengthy. However, the current availability of sym-
bolic mathematics software has removed former reluctance in using variational
principles.
Once the dynamic equations of the complete wind turbine are formulated, the
next step is to include the aerodynamic loading on the blades. The main diffi culty
in this respect is that aerodynamic loads depend on the dynamics of the blades in
a non-linear manner. A further complication is due to the onset of stall which because
of the aeroelastic coupling becomes dynamic with clear lagging behaviour. A brief
account on aeroelastic coupling is given in Section 5.
The usual way to solve the coupled fi nal aeroelastic equations is to integrate
them in time. Due to their non-linear character in principle we need an implicit
Structural Dynamics of Wind Turbines 123
iterative solver. Depending on the non-linearities considered and the level of detail
we wish to include in the model, the computational effort can become high espe-
cially for industrial purposes. So, in order to reduce the computational cost, quite
often linearization is introduced at various levels. There are both structural as well
as aerodynamic non-linearities involved. Structural non-linearities are connected
to large displacements and rotations, so a usual simplifi cation is to consider the
case of small deformations with respect to either the un-deformed state or a refer-
ence equilibrium deformed state. In this context signifi cant computational cost
reduction can be accomplished by introducing the structural modes of the system.
A similar procedure can be followed for the non-linearities of aerodynamic origin.
However, as already mentioned the onset of stall and the resulting load hysteresis
complicates the whole procedure and linearization of the unsteady aerodynamic
loading must be done carefully.
Linearized models are not only introduced for computational cost reduction.
They are also used in carrying out aeroelastic stability analysis. Aeroelastic insta-
bilities including fl utter will appear whenever the system lacks suffi cient damping.
In systems with strong aeroelastic coupling the damping has two components; one
linked to the material characteristics of the structure and the other contributed by
the unsteady aerodynamics. Both damping contributions are subjected to changes.
Because blades are made of composite materials, structural damping depends on
the ambient temperature while it is known that ageing will degrade their damping.
On the other hand, aerodynamic damping depends on the fl ow conditions and will
decrease or even become negative as the fl ow approaches or enters stall. This
means that close to nominal operating conditions, the effective angles of attack
along the blade will approach their maximum and therefore the less favourable
situation will occur. Wind speed fl uctuations due to turbulence will force the blade
to enter stall so if there is no suffi cient structural damping the blades will fl utter. In
this connection linearization of the complete servo-aeroelastic system offers a
direct and cost-effective way to assess stability. The option of using the non-linear
form of the dynamic equations is also possible but at a much higher cost. Besides
that the response recorded by a non-linear simulation depends on the excitation
applied and in principle will excite all the modes of the system. In order to focus
on a specifi c mode an appropriate excitation must be applied which is not always
possible. Stability is discussed in Section 6.
Finally in Section 8 a brief presentation is given on the kind of information
structural analysis can provide and what is their use in the design process. The
presentation relies on simulations carried out in practice and whenever possible
measured data are also included so as to have some insight into the quality of
predictions current analysis can achieve.
2 Formulation of the dynamic equations
Structural dynamics is based on dynamic equilibrium (Newton’s second law)
stating that a solid volume V bounded by ∂ V will accelerate as a result of volume
124 Wind Power Generation and Wind Turbine Design
and surface loading. The balance of forces and moments leads to the following
equations[ 1 ]:
dd d, d d d
VVV V V V
gV t S rV g RV t RS r RVrrr r
∂∂
+= ×+×=×
∫∫∫ ∫ ∫ ∫
(1 )
In eqn ( 1 ), the current position
0
(,)rr t
of a material point originally at
0
r
, includes
the elastic displacement and any rigid body motion while
R
denotes the distance
from the point with respect to which moments are taken. (An overhead arrow
will denote geometric vectors while matrices will be in bold.) Usually the volume
loading term is due to gravity. Then as regards surface loading, when V is part of a
solid body there will be two terms: the aerodynamic loading defi ned on the part of
∂ V in contact with air, and the internal loading defi ned on the rest of ∂ V in contact
with the remaining of the solid body. Internal surface loading is directly connected
to the stress tensor s :
T
tns=
with n
denoting the outward unit normal, which
together with Green’s theorem allows transforming the surface integrals into vol-
ume integrals and thus deriving the dynamic equations in differential form. Elastic
models are next introduced which relate stresses with strains so that at the end the
equations are formulated with respect to the displacements and rotations contained
in the defi nition of r
(for further reading the reader can consult [ 14 ] out of the
long list of modern textbooks on structural mechanics).
In structural analysis the differential form of the equations is seldom used.
Instead the equations are reformulated in weak form based on the principle of
virtual work. Weak formulations are the starting point of fi nite element discrete
models of the Galerkin type ([ 2 ] and Section 3.2). An alternative and defi nitely
more powerful way to formulate dynamic equations is to use Hamilton’s principle
[ 3 ]. Let T and U denote the kinetic and strain energies, respectively, defi ned by a
set of displacements and rotations collectively denoted as u ( t ) = (u
1
, u
2
,…)
T
.
Assuming the presence of non-conservative loads F
i
connected to u
i
then
d(T ) (T )
0,
dt
i
ii
UU
Fi
uu
⎛⎞
∂− ∂−
−++=∀
⎜⎟
∂∂
⎝⎠
(2)
This is also known as Lagrange equations. Each equation corresponds to balance
of either forces or moments depending on whether the associated u
i
is a displace-
ment or a rotation. Among the various advantages Hamilton’s principle has, of par-
ticular importance is that the fi nal form of the dynamic equations is easily obtained
using symbolic mathematical software [ 4 , 5 ].
3 Beam theory and FEM approximations
3.1 Basic assumptions and equation derivation
Beam theory applies when one of the dimensions of the structure is much larger than
the others [ 6 , 7 ]. Using asymptotic analysis, it is possible to eliminate the depen-
dence on the two shortest dimensions by appropriate averaging of the distributed