Environmental Engineering Reference
In-Depth Information
The necessary intermediate coordinate transformations for components in the
x-y-z
,
x
r
-y
r
-z
r
,
and
x
h
-y
h
-z
h
coordinate systems shown in Figure 11-1 can be obtained as follows:
-- set q
p
=
0 for transforming to the
x-y-z
system
-- set q
p
= b
0
= 0 for transforming to the
x
r
-y
r
-z
r
system
-- set q
p
=
b
0
= y = 0 for transforming to the
x
h
-y
h
-z
h
system
Moment-Curvature Relationship
The HAWT blade is assumed to be a long, slender beam so that the normal strength-of-
materials assumptions concerning the bending deformations are valid. Figure 11-2 shows
an infinitesimal element of the deformed blade. It is assumed that the blade bends only
about its weakest principal axis of inertia, which is about the x
p
-axis in the figure. No other
deformations are considered in this analysis. The strength-of-materials model for elastic
bending assumes a one-dimensional form for Hooke's Law that neglects all stresses except
the longitudinal (in this case
spanwise
)
bending stress. It results in the following
differential equation relating bending moment to curvature in our blade:
M
x
,
p
= -
EI
x
,
p
d
2
v
dz
p
-
EI
x
,
p
v
(11-3)
where
M
x
,
p
=
bending moment about the blade principal axis of inertia,
x
p
(N-m)
[ ]ยข =
d[ ]/dz
p
,
etc.
E =
modulus of elasticity (N/m
2
)
I
x
,
p
= area moment of inertia of the airfoil section about the
x
p
-axis (m
4
)
v
= displacement in the
y
p
(flap) direction (m)
Figure 11-2. Deformed blade element showing forces and moments, all acting in a
positive sense.
[Wright
et al.
1988]
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