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= c
4
2
3
(1 + 3 /2) + –
2 θ μ μα μ
or t as a
c 0
2
1 + nf
8
=
4
2
3
(1 + 3 /2) + μ θ μ μα
[ ]
Differentiating with respect to θ0 gives
∂
∂
t a
+ as
c
0
2
=
(1 + 3 /2)/6
θ 1 /8
μ
μ
With μ and solidity s known, the slope of the thrust coefficient with collective
pitch depends only on the blade lift slope a. Thus a can be determined from the slope
of a graph of thrust coefficient against collective pitch. Figure 3.25 shows such data
from Harris’s tests, from which we obtain ∂tc/∂θ0 = 0.523 and a = 5.5. The variation
of qc against collective pitch is shown in Fig. 3.26. When tc = 0 we have qc = δ/8
(since μhc is extremely small) and from the figure we find that δ/8 = 0.0018, giving
δ = 0.0144.
With a and δ deduced from the tests we can now calculate force, torque, and
o Experimental
— Theoretical
0.1
0.05
0 5° 10° 15°
θ 0.75
Fig. 3.25 Thrust coefficient as a function of collective pitch
t c
Rotor aerodynamics and dynamics in forward flight 111
0.01
0.005
0 5° 10° 15°
θ0.75
αnf = –1.4°
μ = 0.08
qc = 0.0018 + 1.13tcλ
Fig. 3.26 Torque coefficient as a function of collective pitch
flapping coefficients. These are shown in Figs 3.27 to 3.31, together with the
corresponding measured values.
It can be seen from Figs 3.27 and 3.26 that the agreement between theoretical and
measured values of thrust and torque is very good; the slight discrepancy at larger
values of collective pitch in the case of qc probably indicates that the value of the
profile drag coefficient δ should be higher in this region. The theoretical values of
hc, on the other hand, show less good agreement with experiment, Fig. 3.28, but it
should be noted that, unlike tc, hc represents only a small component of the resultant
rotor force and that the expression for hc does not take into account the effects of the
variable induced velocity since, as was mentioned in section 3.9, this would result in
a very complicated analysis. It is easy to account for the effect of the induced velocity
on the total power, however, since we need only apply a factor to the induced power,
as discussed earlier in this chapter. As can be seen from Fig. 3.29, the agreement
between the estimated and measured torque coefficients is excellent.
Agreement between the theoretical and experimental values of the flapping angles
is less satisfactory. The theoretical value of the coning angle a0, as one would expect,
is zero at the value of collective pitch angle for which the thrust also vanishes, and
it is not understood why the measured values show a significant coning angle at this
point or why the slope with collective-pitch angle is less than the theoretical value,
Fig. 3.30. Fortunately, since the coning angle plays very little part in performance
and stability estimations, the discrepancy is not serious.
Comparison between theoretical and experimental values of the backward flapping
angle a1, Fig. 3.31, displays a tendency previously observed20 in connection with
–8 –6 –4 –2 0 2 4 6
0.1
0.06
0.04
tc
αnf
μ = 0.08
θ0.75 = 8.97°
Fig. 3.27 Thrust coefficient as a function of shaft angle
qc
112 Bramwell’s Helicopter Dynamics
0.01
0.005
0 0.1 0.2 0.3
hc
μ
tc = 0.08
Fig. 3.28 H-force coefficient as a function of tip speed ratio
0.008
0.006
0.004
0.002
0 0.1 0.2 0.3
qc
μ
Fig. 3.29 Torque coefficient as a function of tip speed ratio
0 5° 10° 15°
5°
4°
3°
2°
1°
θ 0.75
a0
Fig. 3.30 Coning angle as a function of collective pitch
Rotor aerodynamics and dynamics in forward flight 113
μ
5°
4°
3°
2°
1°
0 0.1 0.2 0.3
a1
Squire’s results, namely, that the theoretical values underestimate the actual flapping
angle by about 10 to 20 per cent. This is probably due to the simplifying assumptions
made for the induced velocity distribution.
The longitudinal induced velocity distribution has a very pronounced influence on
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Bramwell’s Helicopter Dynamics(59)
