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+ (V cos αnf cos ψ sin β – V sin αnf cos β)k
To complete the calculation of the velocity components we must add the contributions
due to flapping rβ˙ j and the relative wind due to the induced velocity – vik. Thus, the
components of relative wind along and perpendicular to the blade section are:
i direction: V cos αnf cos ψ cos β + V sin αnf sin β (3.17)
j direction: –V cos αnf sin ψ – Ωr (3.18)
k direction: –V cos αnf cos ψ sin β + V sin αnf cos β – r˙β – vi (3.19)
It is generally assumed that the spanwise component of velocity in the i direction
can be neglected. It is also usual to denote the component that is tangential to the
plane of no feathering, or j direction (eqn 3.18), by UT, taken as positive when it
blows from leading to trailing edge, and the k direction or perpendicular component
in eqn 3.19 by UP. Then for small values of β, UP and UT can be written
Rotor aerodynamics and dynamics in forward flight 95
UP = – Vβ cos αnf cos ψ + V sin αnf – r˙β – vi (3.20)
UT = V cos αnf sin ψ + Ωr (3.21)
We now define
λ′ = (V sin αnf – vi)/ΩR
where vi may be a function of azimuth and radius, and
μ = (V cos αnf)/ΩR
UP and UT can then be written as
UP = ΩR (λ′ – xβ˙/Ω – μ cosβ ψ)
= ΩR(λ′ – x dβ/dψ – μβ cos ψ) (3.22)
and
UT = ΩR (x + μ sin ψ) (3.23)
Since UT lies in the no-feathering plane, the local incidence α of the blade can be
written (Fig. 3.20)
α = θ + φ
where
φ = tan–1(UP/UT) ≈ UP/UT (3.24)
since φ is a small angle except, perhaps, near the blade root.
When 0 < ψ < 180° the blade is said to be ‘advancing’, and the half of the rotor
disc defined by this range of azimuth angle is referred to as the ‘advancing’ side of
the rotor disc; in this region the relative wind due to the rotational speed of the blade
is increased by a component of the forward speed. Similarly, in the azimuth range
180° < ψ < 360° the blade is said to be ‘retreating’ and to lie in the ‘retreating’ half
of the disc; in this region the forward speed component reduces the relative chordwise
wind.
It is clear that over some part of the retreating blade the forward speed component
will be greater than that due to the rotational speed, i.e. the relative flow will be from
the trailing edge to the leading edge. Referring to eqn 3.23, this occurs when
Fig. 3.20 Velocity components at a blade section
UT
UP
W
φ
θ
96 Bramwell’s Helicopter Dynamics
x + μ sin ψ < 0 and this inequality defines a circular region whose diameter is μR,
Fig. 3.21. This region is known as the ‘reverse flow region’. Since the incidence of
the blade is defined in relation to leading edge to trailing edge flow, it is obvious that
the calculation of the lift and flapping moment in this region must be treated with
some care. Fortunately, the contribution from this region is usually very small, e.g. for
μ = 0.3 the area of the reverse flow region is only 21
4 per cent of the total rotor area
and the velocities there will also be small. The dynamic pressure in the reverse flow
region is low; thus, below values of μ of about 0.4, and bearing in mind that an
advance ratio of 0.5 represents a practical maximum for current helicopters, the
effect of the reverse flow on rotor thrust and other performance contributors may be
neglected, as is done in the following sections.
180°
270°
0°
μR
90° UT
UP
Forward
ψ
Fig. 3.21 Reversed flow region
θ
3.8 Calculation of the rotor thrust
The lift dL on an element of the blade is given by
d = d 12
L ρW2CLc r (3.25)
and, if the thrust dT is taken as the component of the resultant force along the nofeathering
axis (Fig. 3.22),
Fig. 3.22 Lift and drag components at a blade section
UP
dD
w
dL
UT θ
φ
Rotor aerodynamics and dynamics in forward flight 97
dT = (dL cos φ + dD sin φ) cos β
≈ dL
Also, having ignored the spanwise component of the induced velocity,
W2 U U U
P
2
T
2
T
= + ≈ 2
since UP is usually much smaller than UT.
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