Structural dynamics of wind turbines - Wind Power Generation and Wind Turbine Design - page 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