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and establish relationships between a given set of flight conditions and the power
required to achieve them. It is also of interest to calculate such quantities as the
maximum speed, maximum rate of climb, etc., and for this purpose it is easier to
consider the maximum power available to the rotor and to solve an energy equation.
If the trim and energy equations are based on the same set of assumptions, it is, of
course, merely a matter of convenience as to which ones are used to obtain the
desired quantities.
The following analysis and discussion will be based on the comparatively simple
formulae derived in Chapter 3. The reader is reminded that these formulae are only
approximate and that their derivation has been made possible only by making a
number of simplifying assumptions, the chief of which are that the lift slope and
drag coefficient of the blade section are constant, and that blade stall does not occur.
The introduction of more complicated aerodynamic data, in which the lift and drag
are both arbitrary functions of incidence and Mach number, requires the use of
computational methods for solution of the trim equations and calculation of performance.
However, the simplified equations are often adequate for all but the most advanced
116 Bramwell’s Helicopter Dynamics
design work and have the important advantage that they enable a physical interpretation
of helicopter flight to be made easily.
4.2 Helicopter trim in forward flight
In Chapter 1 the longitudinal and lateral trim equations were derived. Because of the
asymmetry of the helicopter, e.g. the presence of a tailrotor in a single rotor helicopter,
the longitudinal and lateral equations should, strictly speaking, be solved simultaneously;
indeed, in a very thorough analysis of helicopter trim, Price1 finds that the various
trim parameters are related through no less than fourteen equations. However, the
complicated process of having to satisfy such a large number of equations simultaneously
is not necessary in practice, especially as the accuracy of some of the aerodynamic
data would hardly justify such detail. The longitudinal and lateral equations will
therefore be treated as separate groups and solved independently of one another.
Now it was stated in Chapter 1 that the resultant rotor force is almost perpendicular
to the rotor tip path plane, i.e. the H-force is small when the rotor force components
are referred to the tip path plane axes. Because of this fact these axes are very useful
for investigating helicopter trim, as it is much easier, when using the corresponding
force coefficients, to establish the rotor incidence αD and thence to obtain the other
parameters.
4.2.1 Longitudinal trim
Referring to Fig. 4.1, we can write the trim equations given in Chapter 1, eqns 1.41
and 1.42, with reference the tip path plane or disc axes.
Resolving forces vertically and horizontally we have
TD cos (αD + τc) – HD sin (αD + τc) = W + D sin τ (4.1)
TD sin (αD + τc) + HD cos (αD + τc) = – D cos τ (4.2)
Now the angle αD + τ is the inclination of the rotor-disc plane to the horizontal and
HD
V
αD
TD
a1–B1
D
Horizon
W
V
Fig. 4.1 Forces and moments in longitudinal plane
τc
Trim and performance in axial and forward flight 117
is usually a small angle in steady flight. Thus the usual small angle assumptions can
be applied to eqns 4.1 and 4.2 which then become, approximately,
TD ≈ T = W + D sin τc (4.3)
T(αD + τc) + HD = – D cos τc (4.4)
where the term HD sin (αD + τc) has been neglected. The angle of climb τc might not
be a small angle.
Let the helicopter fuselage drag D be written as
D VS = 12
2
ρ FP (4.5)
where SFP is the so-called equivalent flat plate area.
Then expressing eqns 4.3 and 4.4 in coefficient form by dividing through by
ρsAΩ2R2 gives
tc wc Vd
12
2
D 0 c = + ˆ sin τ (4.6)
and
tc D c hc V d
12
2
D D 0 c (α + τ) + = – ˆ cos τ (4.7)
where wc is a weight coefficient, d0 = SFP/sA, and Vˆ = V/ΩR.
Solving eqn 4.7 for αD gives
αD τ τ
12
2
= – ( 0 cos c + c ) / c – c D D
Vˆ d h t (4.8)
tcD having been obtained from eqn 4.6. Unless the rotor-disc incidence is unusually
large, it is usual to take tcD wc = . With this approximation, the only unknown quantity
in eqn 4.8 is hcD . Now, from numerical calculations, it appears that the most important
term of hcD is the first one of eqns 3.64 or 3.65, i.e. the term representing the rotor
　
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