Environmental Engineering Reference
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where
f
w
is the warping function of the cross section,
u, v,
w
are the local
displacements of the beam axis,
q
is the torsion angle and
S
2
is the transformation
matrix between [O;
xyz
] and [O
x h z
]. By keeping the terms up to second order
(with the exception of some higher order terms that are not always negligible), it
follows that:
′;
⎡
⎛
2
⎞
⎛
2
⎞
⎤
u
′
u
′
ˆ
ˆ
cos(
qq
+−
uw
′′
) 1
−
u
′
sin(
qq
+− ⋅ −
uw
′′
)
1
⎢
⎥
⎜
⎟
⎜
⎟
t
t
⎝
2
⎠
⎝
2
⎠
⎢
⎥
⎢
⎥
2
2
uw
′
′
⎢
ˆ
ˆ
ˆ
ˆ
⎥
S
=−
u
′
cos(
qq
++
)
w
′
sin(
qq
+ − − −
)
1
(
u
′
sin(
qq
++
)
w
′
cos(
qq
+
))
( 35 )
2
⎢
t
t
t
t
⎥
22
⎢
⎥
⎛
2
⎞
⎛
2
⎞
u
′
w
′
⎢
⎥
ˆ
ˆ
−
sin(
qq
+ −
uw
′′
) 1
−
w
′
cos(
qq
+ ⋅ −
)
1
⎜
⎟
⎜
⎟
⎢
t
t
⎥
⎝
2
⎠
⎝
2
⎠
⎣
⎦
where
q
t
is the local twist angle and
y
ˆ
∫
qq
=+ ′′ ′
uw y
d
( 36)
0
Even if warping is neglected the complication of the model is clear. There are
several non-linear terms which will render the overall code implementation hard
to follow especially in the multi-body context. For example, the expression for the
tension force will become:
⎛
2
2
⎞
⎛
2
⎞
uw
FEv
′
′
q
xzqq
′
2
2
=⋅
′
+ + + +
E
(
)
′
+
⎜
⎟
⎜
⎟
h
t
22
2
⎝
⎠
⎝
⎠
(37)
(
) (
)
−
E
z
cos(
qq z
+
)
+
E
sin(
qq
+
)
u
′′
−
E
z
cos(
qq z
+
)
−
E
sin(
qq
+
)
w
′′
t
t
t
t
Note that the loads will be defi ned in the [O
;
x h z
] system while the dynamic
equations are defi ned with respect to [O;
xyz
]. Therefore a further transformation
is needed, also using
S
2
, this time for the loads. The transformed loads are next
introduced into the equilibrium equations and so the fi nal system is obtained.
′
8 Structural analysis and engineering practice
In this last section, we discuss the kind of information structural and aeroelastic
analysis can provide and how this is linked to the design of wind turbines. Due to
the approximate character of all the tools we use, there is always need for valida-
tion which is carried out on the basis of measured data. Note that “absolutely”
reliable measurements are not always easy to obtain. Full scale tests are subjected
to a number of uncertainties mostly related to the uncontrolled infl ow conditions.
Besides that quite often there is lack of detailed structural input especially as
regards the drive train. On the other hand laboratory testing on scaled models is
subjected to similarity incompatibilities. Therefore comparisons must be carried
out carefully, taking into account of all possible uncertainties.
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