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t
2 1
3/2 w
l s l s
l
Γ
4
1 2
π (6.2)
where l2 l l l
1
2
2
2
3
= + + 2
Then, measuring ds in the direction away from the blade,
ds1 = –dx = –R(x1 sin φ + μD)dφ
ds2 = – dy = – x1R cos φ dφ
From Figs 6.5 and 6.6 we see that, relative to the blade from which the vortex
element originated,
l1 = r cos ψ – r1 cos φ + μDR(ψ – φ)
y
r1
x
z
V
–αD
φ
ds
r
l
Γ
Figs 6.5 and 6.6 Plan and side views of vortex filament
ψ
200 Bramwell’s Helicopter Dynamics
l2 = –R(x sin ψ – x1 sin φ)
l3 = λD R(ψ – φ)
The blade in question will also be affected by the trailing vortices of the preceding
blades. The above equations still apply provided we allow for the appropriate phase
shift in the translational terms of l1, l2, l3. Thus, more generally,
l1 = R[x cos ψ – x1 cos φ + μD(ψ – φ + (2π/b)n)]
l2 = –R(x sin ψ – x1 sin φ) (6.3)
l3 = λDR(ψ – φ + (2π/b)n)
where n = 0, 1, 2, …, b – 1, and b is the number of blades.
Now the appropriate value of Γ of the trailing vortex filament is the change of
circulation (Fig. 6.2) over an element of the span dr, i.e. (∂Γ/∂r)dr. Then, integrating
along the trailing vortices from infinity up to the blade and along the span and
summing for all the blades, we have
w
R
x xx x x
n D
b n
t =0 0
1
–
1 1 D 1
3/2 = 1
4
+ cos( – ) + ( sin – sin )
π
ψ ψ φ μ ψ φ
Σ ∫ ∫ ∞
+
{ – – (2 ) } cos
D d d
3/2
1
1
μ ψ φ π φ
φ /b n
D x
x
∂
∂
Γ (6.4)
where ψn = (2π/b)n, n = 0, 1, 2, …, b – 1
and D = x2 + x – 2xx cos ( – ) + ( + ){ – + (2 /b)n}
1
2
1 D
2
D
ψ φ μ λ2ψ φ π 2
+ 2μD{ψ – φ – (2π/b)n}(x cos ψ – x1 cos φ)
To calculate the induced velocity wb from the bound vortices of each blade, consider
two blades separated by angle ψn, as shown in Fig. 6.7.
The co-ordinates of a bound vortex element Γn are
x1 = – r1 cos (ψ + ψn), y1 = r1 sin (ψ + ψn)
Hence, for the elementary line
ds1 = – dr1 cos (ψ + ψn), ds2 = dr1 sin (ψ + ψn)
y
l
Γn
r1
ψn
ds
ψ
x
P
Fig. 6.7 Effect of bound circulation on succeeding blade
r
Rotor aerodynamics in forward flight 201
and the components of the line joining the line element to a point on the reference
blade are
l1 = – r cos ψ + r1 cos (ψ + ψn)
l2 = r sin ψ – r1 sin (ψ + ψn)
Then, noting that this calculation does not include the reference blade, n = 0, the
total contribution of the bound vorticity to the downwash at P is
w
R
x x
x x xx
x
n
b
n n
n
b = 1
– 1
0
1
1
2
1
2
1
= 1 3/2 1
4
sin ( )
( + – 2 cos )
d π
ψ
ψ
Σ ∫ Γ
(6.5)
According to Miller2, the contribution of the bound vortices is usually extremely
small.
Finally, calculating the induced velocity component ws for the shed part of the
wake, we have, see Fig. 6.8,
ds1 = – dr1 cos ψ0, ds2 = dr1 sin ψ0
The values of l1, l2, l3 are the same as those of eqn 6.3; hence, integrating in both
the azimuthwise and spanwise directions gives
w
R
x R
D
x
n
b
n
n
s = 0
– 1
0
1
–
D
= 1 3/2 1
4
sin( – ) – ( – ) sin
d d π
ψ φ μ ψ φ φ
φ
φ
ψ
Σ ∫ ∫∞
∂
∂
Γ (6.6)
The upper limit of the inner integral in eqn 6.6, must be treated with care. In
integral eqn 6.4, the azimuthwise integration is taken up to the span axis of the blade
since, in calculating the effect of the trailing vortices, it is reasonable to neglect
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Bramwell’s Helicopter Dynamics(102)
