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d
( – 1)
12
1
2 πρ α α ρ γ ξ
ξ
˙ ˙
∞ ∫ √
+ πρb2(α˙V + ˙z˙ – αbx)
(6.24)
The first term is the ‘quasi-steady’ lift which alone would occur if the motion were
steady, the second term represents the lift due to the vortex wake, and the third term
is the lift due to the fluid inertia or ‘apparent mass’. Denoting the quasi-steady lift by
Lq we have
Lq Vb V z b x
12
= 2πρ [α + ˙ + α˙ ( – )]
=
+ 1
( – 1)
– 1 d
1
2
∞ ∫ √
ξ
ξ
γ ξ
on eliminating Γ and A0 A
12
+ 1 from eqns 6.13, 6.18, and 6.21. It then follows that
eqn 6.24 can be written
L = L b V z bx
d
( – 1)
d
( – 1)
+
d
( – 1)
q + ( + – )
1
2
1
2
1
2
⋅ √
√ √
∞
∞ ∞
∫
∫ ∫
γ ξ ξ
ξ
γ ξ ξ
ξ
γ ξ
ξ
πρ 2 α˙ ˙˙ α˙˙ (6.25)
For sinusoidal motion, the lift can be expressed as L = L0 eiω t and the vorticity in
the wake, as was discussed earlier, must be of the form
γ (ξ, t) = γ0 eiω (t–bξ/v)
= γ0 eiω t e–ikξ
where k = ωb/V is the ‘reduced frequency’.
The ratio of integrals occurring in eqn 6.25 then becomes
Rotor aerodynamics in forward flight 211
1
2
1
2
1
2
1
2
1
d
( – 1)
d
( – 1)
+
d
( – 1)
=
d
( – 1)
+ 1
– 1
∞
∞ ∞
∞
∞
∫
∫ ∫
∫
∫
√
√ √
√
γ ξ ξ
ξ
γξ ξ
ξ
γ ξ
ξ
γ ξ ξ
ξ
ξ
ξ
=
( – 1)
e d
+ 1
– 1
e d
1
2
– i
1
– i
∞
∞
∫
∫
√
ξ
ξ
ξ
ξ
ξ
ξ
ξ
ξ
k
k
= C(k), Theodorsen’s function, referred to earlier.
The lift can now be written
L = C(k) Lq + πρb2 (α˙V + ˙z˙ – α˙˙bx) (6.26)
The pitching moment about the point of rotation x = x is found to be
M bC k x L b xz V x b x = ( )( + ) + [ – ( – ) – ( + )] 12
q
12
1
8
πρ 3 ˙˙ α˙ α˙˙ 2 (6.27)
In particular, the lift in purely rotational harmonic motion about the mid-chord
can be written as
L = πρVb2α˙ + 2πρV2b(1 + ik/2)C(k)α (6.28)
and for heaving motion as
L = πρb2 ˙z˙ + 2πρVbC(k )z˙ (6.29)
In both cases the term proportional to C(k) is due to the bound circulation.
If L0 is the steady lift at the instantaneous incidence the values of L/L0 can easily
be shown to reduce to the forms of eqns 6.10 and 6.11.
It is interesting now to consider the velocity component v3/4 at the 3
4 -chord point
due to the aerofoil motion. For aerofoil rotations about the mid-chord, we have
v3/4
12
= αV + bα˙
For harmonic motion, α = α0 eiωt, and
v 3/4 = α˙V(1 + ik/2)
If Lb denotes the lift due to the bound circulation, i.e. that part of the lift proportional
to C(k), we see from eqn 6.28 that
Lb = 2πρVbC(k)v3/4
For the heaving motion we have simply
v3/4 = ˙z
212 Bramwell’s Helicopter Dynamics
and, again, from eqn 6.29,
Lb = 2πρVbC(k)v3/4
Thus, in both cases, it appears that the part of the lift arising from the bound circulation
is proportional to the ‘downwash’ at the 3
4 -chord point. For this reason, the 3
4 -chord
point is known as the ‘rear aerodynamic centre’ and is the appropriate chordwise
position to consider when lifting-line techniques are used in cases such as those
described above, where the chordwise variation of velocity may be important.
The effect of shed vorticity in the special case of hovering flight has been investigated
by Loewy8 and Jones9. Although hovering flight is usually regarded as a symmetrical
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Bramwell’s Helicopter Dynamics(107)
