Structural dynamics of wind turbines - Wind Power Generation and Wind Turbine Design - page 136

Figure 7: Basic notations on a blade section.

FL =− sin( j −Θ + ) D cos( j −Θ )

x y y

FL = cos( j −Θ + D

MM

) sin( j −Θ )

z y y

(25)

=

y

In case the blade element momentum (BEM) aerodynamic model is used:

(1 −⋅ a ) ( UW − cos Θ− Θ U sin )

U
U

wz y y

(26 )

eff , z

tan j = =

(1 +⋅ a ) ( ′ U + Θ− Θ W sin U cos )

eff , x wx y y

denote the axial and circumferential induction factors, UW denote
the lag and fl ap deformation velocities and U xy , U wz the components of the rela-
tive wind infl ow velocity which includes the wind infl ow and the blade rotational
speed (in the axial case for example the x -component will be the rotational speed
of the blade while the z -component will be the wind infl ow).
The lift L , drag D and pitching moment M of the section in eqn (25) will depend
on the effective angle of attack

where a and a ′

ajq =− −Θ

and the relative fl ow velocity

eff w y

2 2 2

eff can substantially vary not only because the mean wind
speed has a wide range, but also because of the dynamic response of the blades.
According to (13), the blade velocity is subjected to its own deformation velocities

WU U = + . α

eff eff , x eff , z

and the dynamics of the confi guration through R and T leading to the following
expression for the local blade velocity:

( )

T T T

U = (,, ) UVW = T r = T R + T ( r + Su ) + Su

( 27)
The coupling introduced by (25) and (27), is quite complicated and certainly non-
linear. One complication is connected to its unsteady character which requires the use
of unsteady aerodynamic modelling. Models of this type provide the sectional lift,
drag and pitching moment coeffi cients as functions of the sectional steady polars and
the dynamic infl ow characteristics, namely the pitching and heaving motion of the
section. So a link between the aerodynamic loads and the elastic response is estab-
lished through explicit functional relations. Of wide use in wind turbine aeroelasticity
are the Beddoes-Leishman [19] and the ONERA models [20]. They both are appli-
cable over a wide range of angle of attack covering both attached and stalled condi-
tions which explains why they are usually referred to as dynamic stall models.

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