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for linearisation to apply. For the particular case of Mangler and Squire’s loading, we
find
Pi Ti0 x x x
0
1
= (225/8) v ∫ (1 – ) d 5 2
= (75/64)Tvi0 = 1.172Tvi0
= (1 + k)Pi0
92 Bramwell’s Helicopter Dynamics
say, where Pi0 is the induced power for constant induced velocity. This value
agrees with that given earlier. It should be noted that the rotor disc loading is
proportional to x2√(1 – x2) which implies a blade thrust loading proportional to
x3√(1 – x2).
If a similar analysis for hovering flight is made, i.e. if we assume that Δp = 2ρv i
2
instead of 2ρVcvi as in the calculations above, we find that Mangler and Squire’s
distribution gives 1 + k = 1.11. Hence, the induced power is 11 per cent higher
than the ideal power in hovering flight, rising to about 17 per cent at high forward
speed.
If Δp = Cxn (implying a blade thrust loading proportional to xn+1), we easily find
that
1 + = (1 + ) / (1 + ) when = 0 12
3/2 3
4 k n n μ
and
1 + = (1 + ) /(1 + ) at high 12
k n2 n μ
These relationships are shown in Fig. 3.16.
1.4
1.3
1.2
1.1
1
Mangler-hovering
Mangler-forward flight
Forward flight
Hovering
0 1 2
n
1 + k
3.7 Velocity components at the blade
Before being able to calculate the forces and moments on a blade, it is necessary to
know the velocity components of the air relative to any point of the blade. In the
following sections the blade will be assumed to be a rigid beam with a flapping
hinge, and only simple ideas of induced velocity and aerofoil characteristics will be
used. The analysis follows closely the classical work of Glauert1 and Lock7, and,
because the mathematical development is eased, the no-feathering axis system will
be used.
We take as our final reference axes a set of right-hand axes fixed in the blade as
Fig. 3.16 Induced power factor as a function of radial pressure distribution exponent
Rotor aerodynamics and dynamics in forward flight 93
was shown in Fig. 1.18. It is sufficient to assume for the calculation of the aerodynamic
forces that the flapping hinge offset is zero. The only velocity component affected by
the flapping hinge offset is that due to blade flapping, but, since the hinge offset is
usually only a few per cent of the blade radius, the error in assuming it to be zero is
negligible. Taking the upward direction as positive, Fig. 3.17, and taking unit vectors
i1, j1, k1, with k1 along the no-feathering axis and i1 sideways, the forward velocity
V of the helicopter can be expressed as
i1
j1
v
k1
No-feathering axis
Fig. 3.17 Blade axes
αnf
ψ
Forward
j1 j2
i1
Fig. 3.18 View in plane of no feathering
Ω
V = V cos αnf j1 – V sin αnf k1
The blade is itself rotating with angular velocity Ω and it lies at an azimuth angle
ψ with respect to the rearward direction of the helicopter, Fig. 3.18. The blade, at
present, lies in the no-feathering plane. Taking a new unit axes system i2, j2, k2, with
i2 and j2 defined as in Fig. 3.18, the helicopter’s forward velocity can now be written
as
V = –V cos αnf cos ψi2 + V cos αnf sin ψj2 – V sin αnfk2
since k1 = k2.
i2
94 Bramwell’s Helicopter Dynamics
The rotational motion of the blade will add a velocity component Ωr at a blade
section radius r, in the direction j2; then, writing W for the total velocity vector at this
section, we have
W = –V cos αnf cos ψ i2 + (V cos αnf sin ψ + Ωr) j2 – V sin αnfk2
Now let the blade flap through angle β about j2 into the final blade position
represented by the axes i, j, k, Fig. 3.19. The relationship between the sets of axes i2,
j2, k2 and i, j, k is the same as in eqn 1.24, i.e.
k
i
i2
k2
N.F.A.
Fig. 3.19 Flapped blade
β
=
cos 0 sin
0 1 0
– sin 0 cos
2
2
2
i
j
k
i
j
k
β β
β β
Using this transformation, W can be written
W = – (V cos αnf cos ψ cos β + V sin αnf sin β)i + (V cos αnf sin ψ + Ωr) j
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