曝光台 注意防骗
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1
4
= ρ δμΩ2 3
The elementary ‘induced’ component is
dHi = – dL(β cos ψ + φ sin ψ)
= – ( + / ) [ cos + ( / ) sin ] d 12
T
2
ρaU θ0 UP UT β ψ UP UT ψc r
= – [( + ) cos + ( + ) sin ] d 12
0 T
2
P T 0 P T P
ρa θ U U U β ψ θ U U U2 ψ c r
The mean value for b blades is
H b ac U U U
R
i
0 0
2
12
0 T
2
= – ( /2π) ρ [(θ + P T)β cos ψ
π ∫ ∫
+ ( 0 P T + P ) sin ]d d
θU U U2 ψψ r (3.37)
Mangler and Squire’s expression for the induced velocity can be used here but the
analysis becomes rather complicated and, since the terms involving the induced
velocity are fairly small, it is convenient to assume it to be constant. Then substituting
for UP, UT and β from eqns 3.22, 3.23, and 3.27, in which harmonic terms of higher
order than one have been ignored in β, and replacing λ′ by λ we find for H
No-feathering axis
d L cos φ + dD sin φ
sin β (dD sin φ
+
dL cos φ)
dD cos φ – dL sin φ
ψ
Fig. 3.23 Force components in plane of rotor
β
100 Bramwell’s Helicopter Dynamics
H abc R = 12
ρ Ω2 3
×
+ 1
3
+ 3
4
– 12
+ 1
4
– 16
+ 1
4
1 0 1 0 1
2
0 1
μδ θ λ μθ λ μ μ 2 0
2
a
a a a a b a (3.38)
We shall see in section 3.12 that, when the induced velocity is constant,
b
a
1
0
12
2 =
4 /3
1 +
μ
μ
Ignoring the small term in μ2 in the denominator, the last two terms in eqn 3.38
reduce to μa0
2 /36, which is very small compared with the other terms and may be
neglected. The coefficient form of the H-force can then be written finally as
h C s H
sA R
a
a
c H 2 2 a1 0 a1 0 a1
= / = = 2
2
+ 1
3
+ 3
4
– 12
+ 1
4
ρ
μδ θ λ μ λ θ μ
Ω 2
(3.39)
3.10 The rotor torque, Q
The torque dQ about the no-feathering axis on a blade element is, Fig. 3.23,
dQ = r(dD cos φ – dL sin φ)
or d = d – d 12
T
2 12
T
Q ρU δcr r ρU2 CLφcr r
The first term denotes the torque due to the profile drag. Calling this QP we have,
for b blades,
Q b U cr r
R
P
0 0
2
12
T
= ( /2π) ρ 2 δ dψ d
π ∫ ∫
= ( /2 ) 2 4 ( + sin ) d d
0
1
0
2
b π ρc R δx μ ψ 2xψ x
π
Ω ∫ ∫
= ρbcδΩ2R4(1 + μ2)/8 (3.40)
assuming the chord and drag coefficient to be constant.
The mean induced torque Qi is
Q b U C cr r
R
i L
0 0
2
12
T
= – ( /2π) ρ 2 φ dψ d
π ∫ ∫
= – ( /4 ) ( + ) d d
0 0
2
0 P T P
ρ π θ 2 ψ
π
abc U U U r r
R ∫ ∫
The integrand can be expanded and averaged as for the H-force but this is not
Rotor aerodynamics and dynamics in forward flight 101
necessary and it is more convenient and instructive to proceed as follows. The integrand,
including the scalar term 12
ρac, can be written
12
0 P T P
2 12
0 P T P
2
ρac(θ U U + U )r = (ρac/Ω)(θ U U + U )(UT – V cos αnf sin ψ)
since, from eqn 3.21,
Ωr = UT – V cos αnf sin ψ
or r = (UT – V cos αnf sin ψ)/Ω
Hence
12
0 P T P
ρac(θ U U +U2 )r dr
= 12
( + ) d – 12
cos
P ( + ) sin d
0 T
2
P T
nf
0 P T P
2 ρ
θ
ρ α
θ ψ acU
U UU r
acV
U U U r Ω Ω
= d – 12
cos
P nf ( + ) sin d
0 P T P
U 2
T
acV
U U U r Ω Ω
ρ α
θ ψ
But, from the previous sections,
12
0 P T P
2
i
12
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本文链接地址：
Bramwell’s Helicopter Dynamics(53)
