Environmental Engineering Reference
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equal amount, as can be seen by both predictions crossing the x-axis at the same
time. This is because the added mass was negligible as explained below.
However, the wind tunnel tests involve simplifications that rarely occur in
practice, so that accurately predicting yaw behaviour of an operating turbine is
more complex than suggested by Fig.
8.6
. For example, the linear second order
yaw behaviour of a delta wing is strictly valid only for a constant wind speed, but
as mentioned above, it was not possible in practice to determine the effects of non-
constant U. The actual tail fin was tested with the rotor immobilized (to remove
complications to the yaw behaviour arising from power extraction), with the air
density q = 1.18 kg/m
3
. Table
8.2
gives the other relevant parameters.
Before considering the yaw behaviour, it is worth pointing out that the oper-
ational Re range of this tail fin, from the blades being stationary at U
0
= 3 m/s to
rated power at 10 m/s, is approximately 2-6 9 10
5
, which is much less than the
range in Re of the operating blades.
Example 8.1 Determine the relationships between the quasi-static and USB nat-
ural frequency and damping ratio for a delta wing obeying (
8.6
) when the added
mass can be ignored.
Answer Substituting A = bc/2 for a delta wing and (
8.6
)in(
8.5a
) gives the
approximate version of (
8.9a
). Thus both theories predict the same natural fre-
quency. Making similar substitutions in (
8.5b
) and using (
8.9b
), the ratio of
damping ratios is
f
USB
f
QS
¼
1
þ
c
=
3
ðÞ
Þ
2
ð
so that USB always predicts higher damping, but the difference reduces with
increasing tail arm length.
Example 8.2 Determine the USB and quasi-static natural frequency and damping
ratio for the tail fin detailed in Table
8.2
.
Answer Using the tabulated values, K
1
/U
2
= 9 p 9 1.18 9 0.44
2
9 0.96 =
0.1722 kg, from Eq.
8.8
. Similarly K
2
/U = 9 p 9 1.18 9 0.44
2
9 (0.62 ?
0.51) = 0.2291 kg m, and K
3
= 9 p 9 1.18 9 0.44
2
9 0.51 9 (0.51
2
/5 ?
0.62
2
/3 ? 0.51 9 0.62/2) = 0.0311 kg m
2
. Thus the added mass is negligible in
comparison to the inertia of the turbine. From Eq.
8.9a
, x
n
/U & H(0.1772/
Table 8.2 Parameters for
tail fin test data in Fig.
8.7
Parameter
Value
r = x ? 2c/3
0.96 m
0.112 m
2
A (area of tail fin)
x (tail boom length)
0.62 m
c (tail fin chord)
0.51 m
b (tail fin span)
0.44 m
2.74 kg m
2
I (moment of inertia of tail
fin and boom)
K (slope of linear portion of lift curve)
2.71/rad from Eq.
8.6
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