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0 T
2
ρac (θ U U + U ) sin ψ dr = – dΗ – ρac (θ U + UPUT)β cos ψ dr
= – dΗi – dTβ cos ψ
so that
12
0 P T P
ρac (θ U U + U2 )r dr
= d + (d cos + d )
P cos
i
U nf
T T H
V
Ω β ψ Ωα
= d sin – –
d
d
+ d
cos
nf i0 i
TV r H nf
V
Ω Ω Ω α βψ
α v 
   
(3.41)
since UP = V sin αnf – vi0 – Ωr dβ/dψ – V cos αnf β cos ψ.
Now
0
d d /d = A d /d
R
∫ r T β ψ M β ψ
and from eqn (1.2) for zero flapping hinge offset
MA = B Ω2 (d2βs/dψ2 + βs) = BΩ2(d2β/dψ2 + β)
in which βs is used temporarily to indicate that eqn (1.2) was derived for flapping
defined relative to a plane perpendicular to the shaft axis. The second equality is
easily derived using a1 = a1s + B1 and b1 = bis + A1.
Therefore the mean value of – d d /d
0
R
∫ r T β ψ with respect to azimuth is
102 Bramwell’s Helicopter Dynamics
–
2
d
d
+
d
d
d
2 2 2 BΩ
π
β
ψ
β βψ
ψ
π
0
∫ 2 
 

 
= –
4
d
d
d
d
d + d
d
( )d
2 2 2 2
BΩ 2
π ψ
βψ
ψ ψ β ψ
π π
0 0 ∫ ∫ 
 
 

 

 
= –
4
d
d
+
2 2
0
2
2
0
BΩ 2
π
βψ
β
π
π 
 
 

 

 
[ ]






= 0
Since the terms in the brackets are periodic and are therefore identical at the limits.
The other terms multiplying dT and dHi in eqn 3.41 are constants so that
Q TV H
V
i nf i0 i
= – ( sin – ) + nf
cos
Ω α Ωα v 


= – (Tλ + Hiμ)R
and the total torque is
Q = QP + Qi
= ρbcδΩ2 R4 (1 + μ2)/8 – (Tλ + Hiμ)R
= ρbcδΩ2R4(1 + 3μ2)/8 – (Tλ + Hμ)R (3.42)
The torque coefficient is
qc = Q/ρbcRΩ2R3 = Q/ρsAΩ2R3
= δ (1 + 3μ2)/8 – λtc – μhc (3.43)
The term – (λtc + μhc) might have been expected on physical grounds since it is
the scalar product, in non-dimensional form, of the resultant rotor force and resultant
flow through the rotor, i.e. it represents the work done by the rotor in producing the
rotor force. The first term of eqn 3.43 represents, of course, the torque required to
overcome the profile drag.
Equation 3.43 has been derived on the assumption that the spanwise velocity
component can be neglected. While this is a reasonable assumption for the calculation
of the lift, Bennett8 has argued that this does not apply to the drag which depends on
the resultant velocity over the blade. From Bennett’s calculations, the profile drag
contribution to the torque should be written as δ (1 + nμ2)/8, where n has the following
values
μ 0 0.3 0.6 1
n 4.5 4.58 4.66 4.67
Rotor aerodynamics and dynamics in forward flight 103
From similar calculations, Stepniewski9 proposes the expression δ (1 + 4.7μ2)/8.
We should also consider the increase of induced power due to the non-uniformity
of the induced velocity by introducing the factor k as discussed in Chapter 3. The
expression for the torque coefficient then becomes
qc = δ (1 + 4.7μ2)/8 – λtc – μhc + kλitc (3.44)
3.11 Blade flapping
The aerodynamic flapping moment dMA about the hinge due to the elementary lift is
d A = d = + d
12
T
2 P
T
M r L aU
U
U
cr r ρ θ


　
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