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Assuming constant lift slope a, we can write
CL = aα
as in Chapter 2, where a generally has a value of about 5.7.
The elementary thrust is therefore
d = ( + / ) d 12
T
2
T aρU θ UPUTc r
= ( + ) d 12
T
2
aρ θU UPUT c r
= [ ( + sin ) + ( – d /d – cos ) 12
aρΩ2R2θx μ ψ2 λ′ xβ ψ μβ ψ
× (x + μ sin ψ)]c dr (3.26)
Before eqn 3.26 can be integrated, β and λ′ must first be expressed as functions of
ψ. Assuming steady flight conditions, β can be expressed as
β = a0 – a1 cos ψ – b1 sin ψ – a2 cos 2ψ – b2 sin 2ψ – ... (3.27)
so that
dβ /dψ = a1 sin ψ – b1 cos ψ + 2a2 sin 2ψ – 2b2 cos 2ψ – ... (3.28)
Let us take Mangler and Squire’s series for the induced velocity, i.e. let
vi vi0
12
= 4 [c0 + =1 cn(x, D)cos n ]
n
Σ ∞
α ψ (3.29)
In this, we assume that the expression holds equally for the no-feathering plane as for
the plane it actually applies to, which most nearly corresponds to the tip path plane.
To find the total rotor thrust we calculate the average thrust of a blade taken round
the disc and multiply by the number of blades. To do this it is easier first to average
the elementary thrust, given by eqn 3.26, with respect to azimuth and then integrate
along the blade. The average value of dT over the azimuth range 0 < ψ < 360° is
found to be
d = [ ( + ) + sin – 2 + ]d 12
2 3 12
2
nf 0 i
12
2
T ρacΩRθx2 μ xVˆ α xcλ μb2 x (3.30)
where λ
i = vi0/ΩR, vi0 being the mean induced velocity, and ˆV = V/ΩR.
Assuming for simplicity that the chord c and pitch angle θ are constant along the
blade, integration of eqn 3.30 along the blade gives
98 Bramwell’s Helicopter Dynamics
T = 1ac R[ ( + 3 /2) + V sin – + b]
4
2 3 2
3 0
2
nf i
1
4
2
ρ Ω θ 1 μ ˆα λ μ 2
since, from eqn 3.14
0
1
0
∫ c x dx = (15/8) ∫ (1 – 2)x dx
0
1
η η
= (15/8) 3 1 – 2 d
0
1 ∫x x x
= 1
4
It is easy to show that we could have obtained exactly the same result if eqns 3.1
or 3.4 had been used for the induced velocity. Then defining
λ = sin αnf – λi
ˆV
(3.31)
the thrust for b blades becomes
T = 1b ac R [ ( + 3 /2) + ]
4
2 3 2
3 0
ρ Ω θ 1 μ2 λ (3.32)
where we have neglected the very small term in b2.
Defining a thrust coefficient by
tc = T/ρsAΩ2R2
eqn 3.42 gives
t a
c 0
= 2
4
2
3
[ θ(1 + 3μ/2) +λ] (3.33)
Current blades are usually without taper so that it is justifiable to integrate
eqn 3.30 on the assumption that the chord is constant. Most blades have considerable
twist, however, but in Chapter 2 we found that for linear twist the thrust equation in
hovering flight was still valid provided the pitch angle θ0 at 3
4 radius was taken.
Since the thrust equation in forward flight involves only a small extra term in μ2,
taking the 3
4 -radius pitch angle for θ0 should be a very good approximation in this
case too. For a tapering blade the chord can also be taken as that at 3
4 radius when
defining the rotor solidity.
3.9 The in-plane H-force
Referring to Fig. 3.23, the force component dH perpendicular to the no-feathering
axis and in the rearward direction is
dH = (dD cos φ – dL sin φ) sin ψ – (dL cos φ + dD sin φ) sin β cos ψ (3.34)
Rotor aerodynamics and dynamics in forward flight 99
For small φ and β, eqn 3.34 becomes
dH = dD sin ψ – dL(β cos ψ + φ sin ψ) (3.35)
The first term of eqn 3.35 can be regarded as the profile drag contribution to H,
while the bracketed term can be regarded as the ‘induced’ component arising from
the inclination of the lift vector.
The profile drag term HP, found by averaging the first term of eqn 3.35 round the
disc, is
H b U c r
R
P
0 0
2
12
T
= ( /2π) ρ 2 δ sin ψ dψ d
π ∫ ∫ (3.36)
Assuming the profile drag coefficient δ and chord c to be constant, we find
HP bc R
　
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