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– 1
3
+ 1
8
2 4 ρ α β α ac R cR
cR
cR
Ω Ω Ω Ω
˙ ˙˙ ˙˙
2 2
The term c/R is the reciprocal of the aspect ratio of the blade and is typically about
1/20. Since the terms in α˙˙ and β˙˙ are usually negligible, the expression for MA
reduces approximately to
MA ac R C k B C k
= 1 2 4 2
8
–
d
d
( ) =
8
–
d
d
( ) ρ α βψ
γ α βψ
Ω Ω
Similarly, the aerodynamic pitching moment about the flexural axis is approximately
LA ac R ac R
= 1 2 2 2 2
32
d
d
= 1
32
d
d
ρ α +
ψ ρ θψ
β 3 3 Ω Ω
= 1
32
d
d
+
2
2
γ θψ
β B
A
Ω
Aeroelastic and aeromechanical behaviour 327
where A is the blade aspect ratio R/c. The equations of coupled pitch-flapping motion
are therefore
d
d
+ +
d
d
+ = – 1
32
d
d
+
2
2 1
2
2
2 2
θ
ψ
ν θ β
ψ
β γ θψ
F β
A
B
A
A
and
d
d
+ + d
d
+ =
8
–
d
d
( )
2
2 1
2
2
2
β
ψ
λ β θ
ψ
θ γ θ βψ
F
B
C k
where θ represents changes of α from the steady state.
If the displacement of the c.g. of a blade elemental strip σgc (positive when the c.g.
is behind the flexural axis) is constant along the span, it is easy to show that F =
– σgcrg M, where rg is the spanwise position of the c.g. of the blade; and, if we also
write A = Mk c A
2 2, the equations above become
d
d
+ 1
32
d
d
+ –
d
d
+ + 1
32
= 0
2
2 2 1
2 g g
2
2
2 2
θ
ψ
γ θψ
ν θ
σ β
ψ
β γ β B
A
r
ck
B
A A A A
d
d
+
8
( )
d
d
+ + d
d
+ –
8
( ) = 0
2
2 1
2
2
2
β
ψ
γ βψ
λ β θ
ψ
C k F θ γ θ
B
C k
The term proportional to F/B in the flapping equation and the second term involving
the aspect ratio in the pitching equation can be ignored. Then, writing for the flapping
moment of inertia B = MkBR
2 2, the final equations in hovering flight are
d
d
+
8
1
4
d
d
+ –
d
d
+ = 0
2
2 1
2 g g
2
2 θ
ψ
γ θψ
ν θ
σ β
ψ
β ⋅
k
k
r
ck
B
A A
2
2 (9.11)
d
d
+
8
( )
d
d
+ –
8
( ) = 0
2
2 1
2 β
ψ
γ βψ
C k λ β γ C k θ (9.12)
The frequency equation corresponding to the equations of motion 9.11 and 9.12 is
easily found to be
λ 4 γ λ
2
+ 3
8
( ) + 1
4
C k
k
k
B
A
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本文链接地址：
Bramwell’s Helicopter Dynamics(161)
