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equation is solved for λc, giving the required rate of climb.
4.3.3 Autorotative forward flight
Autorotation is defined as self-sustained rotation of the rotor in the absence of applied
torque, i.e. when Q = qc = 0. The work to be done to overcome the rotor and fuselage
drag must be obtained at the expense of the potential energy of the helicopter. Level
flight autorotation is impossible, and steady flight can be achieved only by descending.
To find the rate of descent at a given forward speed we simply put qc = 0 in eqn 4.21
and solve for λc at the appropriate value of μ. Thus
λ c δ μ μ μ
c
2 12
c 12
3 0
c
= –
8
(1 + 3 ) + (1 + ) +
t
k
st d
t
(4.23)
and the rate of descent Vdes is given by
Vdes = – λcΩR
The angle of descent τdes is clearly
τdes = tan–1 (Vdes/V) (4.24)
The rate and angle of descent of our example helicopter is shown in Figs 4.16 and
4.17.
It can be seen from eqns 4.21 and 4.23 that the rate of descent is proportional to
the torque coefficient in level flight; in fact, the rate of descent curve, Fig. 4.16, is
Trim and performance in axial and forward flight 133
250
200
150
100
50
0 0.1 0.2 0.3
μ
Fig. 4.16 Rate of descent in autorotation
0
–10°
–20°
–30°
–40°
–50°
–60°
Angle of descent
0.1 0.2 0.3
μ
Fig. 4.17 Angle of descent in autorotation
merely the power curve, Fig. 4.14, drawn to a different scale. Thus, the minimum rate
of descent occurs at the same speed as the minimum power in level flight.
From eqn 4.24 the condition for least angle of descent is given by
d
d
=
cos d
d
des – = 0
2
des
2
des
des
τ τ
V V
V
V
V
V 


i.e. dVdes/dV = Vdes/V
Except at low speeds, when the disc angle may be quite large, this condition can
be written as
dλc/dμ = λc /μ
(- ------ indicates tangent to curve,
providing least angle of descent)
Rate of descent m/s
134 Bramwell’s Helicopter Dynamics
and the solution can be found by the point at which the line drawn from the origin
makes a tangent to the curve of λc against μ or of Vdes against μ, as shown in
Fig. 4.16.
In autorotation there must be a flow up through the rotor disc so that the total
moment, or torque, of the blade forces is zero. Figure 4.18 shows the forces on a
blade section with the resultant force dR perpendicular, or nearly perpendicular, to
the plane of rotation. It can be seen that the resultant velocity vector W must be
inclined upwards relative to the plane of rotation in order that there should be a
component of lift to balance the blade drag.
It is clear that in autorotation the collective pitch will be lower than in forward
flight. To find the collective pitch angle to trim it is best to use eqn 3.66, putting
qc = 0 and neglecting the small term in μhcD, giving
λD δ μ
2
= (1 + 3 )/8 cD t (4.25)
Since tcD wc D = , λ can easily be calculated from eqn 4.25 and then substituted in
eqn 3.63 to obtain θ0. The collective pitch variation with μ is shown in Fig. 4.19. The
fact that it is practically constant follows from the need for an almost constant flow
through the rotor to maintain zero torque, as can be seen from an inspection of
eqn 4.25.
4.3.4 General remarks on performance estimation
The performance estimations discussed in this chapter have been based on very
simple assumptions, particularly with regard to the aerodynamic properties of the
Fig. 4.18 Forces on aerofoil in autorotation
dD
dL dR
Plane of rotation
W
5°
4°
3°
2°
1°
0
θ0
0.1 0.2 0.3
μ
Fig. 4.19 Collective pitch to trim in autorotation
Trim and performance in axial and forward flight 135
blades. One of the most important, and which has allowed a particularly simple
analysis, is the assumption of constant blade section drag coefficient even though, as
we shall discuss in detail in Chapter 6, the local incidence may vary over a wide
range and enter the stall region.
An early attempt to consider the dependence of the drag coefficient δ on the
incidence α was that of Bailey (1941)4, who assumed that
δ = 0.0087 – 0.0216α + 0.4α2
　
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