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induced velocity distribution were given in Fig. 3.16. Thus, the contribution of the
induced power to the torque coefficient can be expressed more accurately by
(1 + k)λ
i tcD . Further, we have yet to include the torque which must be provided to
the tailrotor. The tailrotor is driven by a shaft geared to the main rotor, but the torque
supplied to the shaft depends on the inclination of the tailrotor axis to the fuselage.
Thus, for example, it is possible to incline the axis so that the tailrotor autorotates and
for no power to be necessary at the tailrotor, causing a drag force which, in turn,
would require a forward tilt of the main rotor to trim it, with a corresponding increase
of power to be developed at the main rotor shaft. It can easily be verified that the
amount of power required at the tailrotor shaft, plus the work which must be done to
overcome the tailrotor force, is independent of the tailrotor shaft angle. As we have
found with the main rotor, the power absorbed can be expressed simply as that which
would be needed to overcome the profile drag of the blades and the induced power.
Hence, the power Pt required for the tailrotor is
Pt t s A R
t
t
2
it c t t t
3 =
8
(1 + 3 ) + ( ) t
δ μ λ ρ
Ω
and the effective increment to the mainrotor torque coefficient is
Trim and performance in axial and forward flight 129
q t
s A R
sA R c
t
t
2
it c
t t t
2
t t 3 =
8
(1 + 3 ) +
( )
( )
δ μ λ
Ω
Ω
Now it is reasonable to assume that, as the tip speeds of the tailrotor and the main
rotor are usually equal, the terms in the square bracket have roughly the same values
as those of the main rotor, although, as we saw earlier, λit may be rather higher than
in hovering flight. Hence, the power to be attributed to the tailrotor is, to a good
approximation, stAt/sA times that of the main rotor. Thus, a simple way to calculate
the tailrotor power is merely to increase the mainrotor power by the fraction stAt/sA
whose value is typically about 0.06. As a percentage of the total power, the tailrotor
power varies from about 6 per cent in hovering to about 3 per cent at high speed.
The torque coefficient can finally be written as
q kt
s A
sA
c i w V V d
2
c
t t
c c
12
3
= 0
8
(1 + 3 ) + (1 + ) 1 + + sin + D
δ μ λ τ
ˆ ˆ (4.20)
The required power P is calculated from ΩQ = qcρsAΩ3R3 and is shown for the
example helicopter in level flight in Fig. 4.14. The four contributions to the power are
shown by the broken lines, the value of the induced power factor k being taken as
0.17.
Suppose the maximum installed power of our example helicopter is 900 kW. It can
be seen from Fig. 4.14 that the maximum excess power occurs at μ = 0.154 (32 m/s)
and is 496 kW. This gives a maximum rate of climb of 11 m/s.
The maximum forward speed occurs when the installed power and the required
power are equal; the intersection of the two curves in Fig. 4.14 occurs at μ = 0.358,
i.e. at 74.8 m/s.
Fig. 4.14 Variation of power with forward speed
800
600
400
200
kW
Power
Maximum installed power
Total
Parasite
Blade profile
Induced
Tailrotor
0 0.1 0.2 0.3 μ 0.4
130 Bramwell’s Helicopter Dynamics
4.3.1 Fuselage parasite drag
The figure for the parasite drag of our example helicopter is a value, typical for its
weight, of production helicopters. The flat plate parasite drag of a number of helicopters
as a plot of equivalent flat plate area is shown against gross weight in Fig. 4.15. The
points define fairly well a typical curve of drag against weight. A second curve is
shown which is based on aerodynamically clean experimental helicopters. This latter
curve represents the lowest drag which can reasonably be achieved in helicopter
design, although it falls far short of best fixed wing practice. It is clear that the
particular basic shape which must be adopted by helicopter fuselages, and the fact
that the helicopter is normally expected to fulfil a variety of roles, means that it is
unable to reach the degree of aerodynamic refinement which is possible in fixed wing
practice. In fact, both helicopter drag curves are roughly proportional to W1/2, instead
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