Environmental Engineering Reference
In-Depth Information
In order to incorporate Equations (11-57) in a finite element procedure for subsequent
complex eigenvalue analysis
, they must be recast in a pseudo time domain form for develop-
ing contributions to the finite element mass, stiffness and damping matrices. This can be
accomplished by leaving the Theodorsen function as is, but using the explicit w's in the equa-
tions to construct time derivatives, producing the following equations:
b
2
a
2
U
2
a
C
(
k
)
U
b
2
U
a +
b
2
U
2
h
-
h
+
C
(
k
)a +
[
1 +
C
(
k
)(1 - 2
a
)
]
L
= 2pr
U
2
b
(11-59a)
æ
C
(
k
)
U
b
2
U
˙
a +
d
2
b
2
U
˙
a
h
+
C
(
k
)a + [1 +
C
(
k
)(1
-
2
a
)]
d
1
ç
ç
M
= 2pr
U
2
b
(11-59b)
í
+
ab
2
b
3
2
U
2
1
8
+
a
2
ç
2
U
2
h
-
a
è
Now, using Equations (11-59) with the principle of virtual work, contributions to the finite
element stiffness, mass and damping matrices can be developed that include the complex-
valued Theodorsen function.
However, before that is done, it is noted that the HAWT blade does not conform to the
configuration of an infinitely long uniform wing to which the above equations apply. Rather,
such quantities as the semi-chord,
b
, and the resultant velocity,
U
, vary significantly along the
span of the blade. Moreover, the lift curve slope, which in the above equations is assigned the
theoretical value of 2p corresponding to a flat plate, varies with blade span. These variations
can be approximated by assembling a conglomerate of uniform blades, wherein the above
equations are assumed to be applicable incrementally. Specifically, the quantities mentioned
above are represented by a linear variation over the length of the element and included within
the integral over the element length associated with the principle of virtual work.
Significant detail can be incorporated in this analysis by refining some of the variable
quantities mentioned above. For example, inboard stall for a twisted blade turning in still
air might be accommodated by reducing the lift curve slope in that region of the blade.
More complicated inflow variations can also be accommodated if they are known
a priori
.
However, in comparison with other approximations that have been made in this analysis,
these types of refinements are deemed excessive. The simple aeroelastic stability analysis
presented here is meant to serve primarily as a common sense check for use during the blade
design process.
In matrix notation, the finite element equation to be used for investigating the aeroelastic
stability of the wind turbine blade is as follows:
[
M
+
M
a
(W)]{
u
} + [
C
C
(W) +
C
a
(w, W)]{
u
}
+[
K
(
u
0
, W) +
K
tc
+
K
cs
(W) +
K
a
(w, W)]{
u
} = 0
(11-60)
where
M
is the conventional mass matrix and
K
(u
0
, W) is the stiffness matrix, centrifugally
stiffened commensurate with displacements,
u
0
, resulting from centrifugal loads correspond-
ing to the rotor rotational speed, W. The displacement,
u
, and its time derivatives represent
motion about this centrifugally loaded state. The a and
h
degrees of freedom of Equations
(11-59) are included in
u
.
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