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1
2 2
1 1
β
ψ
γ β
k ψk λ β γ ψ ψ
k
E
F pˆ F qˆ
+ 2 cos – 2 sin +
d
d
sin +
d
d
2 cos
1
γ
γ ˆ ψ ˆ ψ ψ ψ ψ ψ
ˆ ˆ
p q
p q 
 
 
(9.48)
354 Bramwell’s Helicopter Dynamics
where E1, F1, γ1 and γ2 have been defined in section 7.5.
We now define the Coleman co-ordinates of flapping motion by
a
b
b
k b
b
k k k
b
1 =0 k k
–1
1 =0
–1
= – 2 Σ β cos ψ ; = – 2 Σ β sin ψ (9.49)
Then, adopting the same procedure as for the lagging equations, the result of summing
eqn 9.48 over all the blades leads to
′′ ′ ′ 
 
 
a a a b b
F
q F p
q
1 1 1
2
1 1 1
+ + ( – 1) + 2 + = – 2 1
2
– 2 +
d
d ν λ ν γ
ψ ˆ ˆ
ˆ
(9.50)
′′ ′ ′ 
 
 
b b b a a
F
p F q
p
1 1 1
2
1 1 1
+ + ( – 1) – 2 – = – 2 1
2
+ 2 +
d
d ν λ ν γ
ψ
ˆ ˆ
ˆ
(9.51)
where ν = γ2E1/2, F = γ2/γ1, and the dashes denote differentiation with respect
to ψ.
From eqn 7.106 of Chapter 7, the moment exerted on the hub by the flapping
deflection of the kth blade is
Mk = ( – 1) R k mxS ( x) dx 1
2 2
0
1
λ Ω 3β 1 ∫
Resolving about the rolling and pitching axes and summing over all the blades
gives for the rolling and pitching moments
L = b ac 2R4( – 1)b
1
2
ρ Ω λ 1/2γ1
M = b ac 2R4( – 1)a
1
2
ρΩ λ 1/2γ1
Including the thrust moment, the rolling and pitching equations of the helicopter
are
A
p
t
b ac R
b Thb
d
d
=
( – 1)
+
2 4
1
2
1
1 1
ρ λ
γ
Ω
2
(9.52)
B
q
t
b ac R a
Tha
d
d
=
( – 1)
+
2 4
1
2
1
1
1
ρ λ
γ
Ω
2
(9.53)
Non-dimensionalising eqns 9.52 and 9.53 by dividing by ρsAΩ2R3 gives
d
d
=
( – 1)
*
+
*
1
2
1
1
c
1
ˆp a
i
b
t h
i
b
ψ A A
λ
2μ γ μ
(9.54)
d
d
=
( – 1)
*
+
*
1
2
1
1
c
1
ˆ q a
i
a
t h
i
a
ψ B B
λ
2μ γ μ
(9.55)
where μ*, iA, and iB are the mass parameter and inertia coefficients defined in
Chapter 5.
Aeroelastic and aeromechanical behaviour 355
The rolling and pitching motions when the blade flapping acceleration is included
are represented by the four equations
′′ ′ ′ a a a b b Fp F
q
1 + 1 + 1 + 2 1 + 1 + 2 + Kq
d
d
ν χ ν ψ + = 0
ˆ
ˆ
ˆ (9.56)
– 2 – + + + +
d
d
+ – 2 = 0 1 1 1 1 1 ′ ′′ ′ a a b b b F
p ν ν χ ψ Kp Fq
ˆ
ˆ ˆ (9.57)
k b
p
A 1 –
d
d
= 0
ˆ
ψ (9.58)
k a
q
B 1 –
d
d
= 0
ˆ
ψ (9.59)
where
k
a
i
t h
i
k
a
i
t h
A i
A A
B
B B
=
( – 1)
*
+
*
; =
( – 1)
*
+
*
1
2
1
c 1
2
1
c λ
μ γ μ
λ
2 2μ γ μ
K = 2F1/2, = 1 – 1
γ χ λ2
The usual form for the solution, pˆ = pˆ0 e
λτ… etc., leads to a sextic characteristic
　
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本文鏈接地址：Bramwell’s Helicopter Dynamics(175)