Environmental Engineering Reference
In-Depth Information
Kinematics of the Blade Motion
Velocity Analysis
We will now derive equations for the velocity of a point on the deformed blade in the
h-z-x
deformed blade system, starting in the
X-Y-Z
ground-based coordinate system and
then transforming the results first into
x
p
-y
p
-z
p
principal blade coordinates and finally into
the h-z-x coordinates. Referring to Figure 11-1, we can illustrate the required velocity
vector analysis as follows: Designate the
X-Y-Z
coordinates as the a reference frame. Call
the
x
p
-y
p
-z
p
principal blade coordinates, located at a point
B
on the elastic axis of the blade,
as the b reference frame. The velocity of the same point on the deformed blade in the a
system, here designated as
A
,
may then be written symbolically as
d
dt
r
A
/
B
(b) + W
x
r
A
/
B
V
A
(a) =
V
B
(a) +
(11-15)
where subscript
A/B
indicates a vector from point
B
to point
A.
Performing the computations
indicated by this equation and transforming the result, the result using Equation (11-1) and
then (11-2) gives
é
ê
ë
r
c
q
p
- f[
(
d
+ b
0
r
)
c
q
p
+
v
]
c
y - (
r
s
q
p
)
s
y
r
s
q
p
+
v
- [(
d
+ b
0
r
)
s
q
p
]
c
y + (
r
c
q
p
)
s
y
(
v
r
-
v
)
s
q
p
- [
d
+ (
v
-
v
r
)
c
q
p
]
s
y
f
f
f
V
A
(h - z - x ) =
R
W
(11-16a)
f
r
R
r
=
(11-16b)
f
f
(11-16c)
=
W
d
R
d
=
(11-16d)
v
R
v
=
(11-16e)
where
f = yaw angle of the rotor axis (rad)
d
= distance from the tower axis to the rotor hub (m)
and bold print designates a dimensionless variable. In addition, it has been assumed that the
order of magnitude for the various terms are as follows:
Order
1
variables
:
r
Order
ε
1/ 2
0
f
variables
:
Order
ε
0
variables
:
d
, b
0
,
v
,
v
,
v
,
H
/
R
Terms of order e
0
2
and higher have been neglected in Equation (11-16a).
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