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calculated and then used in eqns 2.25 and 2.38 to obtain the thrust and torque.
Further, θ – φ is the local blade incidence and a(θ – φ) the local blade lift coefficient.
As an example of the use of eqn 2.52, let us consider a three-bladed rotor whose
pitch angle at the blade root is 12° and whose blades have a washout* of 5°. The
blade has a radius of 25 ft (7.6 m) and a constant chord of 1.5 ft (0.46 m). The lift
slope of the blade section is assumed to be 5.7. Table 2.1 shows how the required
quantities vary along the span, φ being obtained as the solution of eqn 2.52.
From eqn 2.36 it can be seen that we can calculate the thrust coefficient by the
integration of x2CL, which is proportional to the blade aerodynamic loading. The
variation of x2CL along the blade span is shown in Fig. 2.12. On integration we find
that tc = 0.0639.
Let us compare this value with the thrust coefficient calculated from eqns 2.30 and
2.31. Eliminating λ
i gives the following quadratic in tc
1/2 :
tc = (a/4)[2θ0 /3 – √(stc/2)] (2.53)
and the pitch angle to be used is the value of θ at 3
4 R, i.e. 7.5°as discussed in
* A twisted rotor blade or wing is said to have ‘washout’ when the incidence of the tip section is
less than that of the root.
Table 2.1 Variation of φ, α and CL with blade section radius
x = r/R 0.3 0.5 0.7 0.8 0.9 1
θ rad 0.178 0.158 0.136 0.126 0.115 0.105
σ 0.191 0.114 0.082 0.0715 0.0636 0.0573
φ 0.102 0.0795 0.0639 0.0585 0.0531 0.0483
α = (θ – φ)° 4.36 4.49 4.13 3.86 3.54 3.24
CL = a(θ – φ) 0.434 0.447 0.411 0.385 0.353 0.324
54 Bramwell’s Helicopter Dynamics
section 2.4. Solving eqn 2.53, with s = 0.0573, gives tc = 0.0638, which agrees
extremely well with the previous result and shows that the simple analysis gives an
accuracy well within that of the assumed value for the lift slope.
2.7 The optimum rotor
It was stated in the last section that the lowest induced power occurs when the
induced velocity is uniform over the disc. The optimum rotor would be one designed
so that this state was achieved and, in addition, the angle of attack would be chosen
so that the section would be operating at the most efficient lift coefficient, which is
not necessarily at the highest CL/CD ratio.
In hovering flight, the pitch angle of a blade element is
θ = α + vi /Ωr
= α + λ
i/x (2.54)
where vi is constant. The angle of attack α is also the constant value chosen as the
most efficient. Thus the pitch angle can be considered as consisting of a constant part
and a part which varies inversely with blade radius.
Now the thrust on an annulus of the rotor from the blade element theory is
d = d 12
T ρΩ2r2 aαc r
and from momentum theory
dT = 4 i
πρrv2 dr
Equating these differential thrusts shows that to ensure constant induced velocity
the chord must vary inversely with the radius. Thus, the optimum rotor must be
twisted in accordance with eqn 2.54 and tapered inversely as the radius. The latter
requirement would result in an unusual blade shape and one that would be difficult
to construct. Departures from the optimum blade, which usually means only that the
0.4
0.3
0.2
0.1
0 0.2 0.4 0.6 0.8 1.0
x
x2CL
Fig. 2.12 Non-dimensional blade loading as a function of span
Rotor aerodynamics in axial flight 55
chord is kept constant, do not result in a serious loss of efficiency; usually the amount
is about 2 to 3 per cent more power for a given thrust. The subject is dealt with in
some detail by Gessow and Myers12. The reader is recommended to compare an
optimum rotor with one of the same solidity having, say, constant chord and twist
differing from the optimum.
The equivalent chord ce of a rotor on a thrust basis is defined as
c
cx x
x x
e
0
1
2
0
1
2
=
d
d
∫
∫
= 3 d
0
1
2 ∫ cx x (2.55)
and on a torque basis
ce cx x
0
1
= 4∫ 3 d (2.56)
These are the values of the chord for which constant-chord blades would yield the
same thrust and torque as a tapered blade, for the same radius and incidence distribution.
2.8 The efficiency of a rotor
The efficiency of any device should indicate the measure of the success with which
that device performs its duty. It is reasonable to want a hovering rotor to produce the
most thrust for the least power; that is, to make the ratio T/P as large as possible. This
　
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