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in blade mass would decrease the rotational moment of inertia of the total rotor
system, thus adversely influencing the autorotational characteristics of the rotor and,
in the case of partial power failure, the ‘fly-away’ manoeuvre.
Compared to the classical bending torsion flutter of aircraft wings, the motion of
Aeroelastic and aeromechanical behaviour 325
the helicopter blade is modified by the powerful centrifugal action which effectively
increases the stiffness of the flapping motion.
The mechanism of pitch-flap flutter is that, as the blade flaps, inertial and aerodynamic
moments arise which twist the blade and, in turn, modify the aerodynamic flapping
moment. The inertial effect of pitching on the flapping motion is very small.
Let us derive the flapping and torsional equations of motion. It will be sufficient
to consider a rigid blade hinged at the root but with arbitrary flapping and torsional
frequencies. We shall take the elastic axis of the blade as one of our reference axes
which, since it will not in general be a principal axis of the blade, requires the
equations of motion to be derived with a little care. If the axes are fixed in the blade,
Fig. 9.5 the angular velocity components about these axes when the blade flaps and
twists are found to be
ω1 = θ˙ + Ω sin β ≈ θ˙ + Ωβ
ω2 = – β cos θ + Ω sin θ cos β ≈ β˙ + Ωθ
ω3 = β sin θ + Ω cos θ cos β ≈ Ω
Since the blade is very thin in relation to its chord and span, the products of inertia
D and E are negligible, but F, which represents the chordwise distribution of mass
relative to the elastic axis, should be retained. Then, from eqn A.1.7 the components
of angular momentum are
h1 = Aω1 – Fω2
h2 = Bω2 – Fω1
h3 = Cω3
Differentiating with respect to time and retaining only first-order terms, we obtain
h˙1 A˙˙ A F˙˙ F
= θ + Ω2θ + β + Ω2β
h˙2 B˙˙ B F˙˙ F
= – β – Ω2θ – θ – Ω2θ
z
x
Ω
θ
β
θ
β
˙θ
˙β
y
θ
Fig. 9.5 Blade deflection in combined flapping and torsion
326 Bramwell’s Helicopter Dynamics
The above expressions represent the inertia terms of the torsional and flapping
motion. If the elastic stiffnesses about these axes are included, the equations of
motion can be written
d
d
+ +
d
d
+ =
2
2 1
2
2
2 2
θ
ψ
ν θ β
ψ
F β
A
L
A
 A
 

 
Ω
(9.9)
d
d
+ + d
d
+ =
2
2 1
2
2
2 2
β
ψ
λ β θ
ψ
F θ
B
M
B
 A
 

 
Ω
(9.10)
where ν1Ω and λ1Ω are the uncoupled torsional and flapping frequencies and LA and
MA are the aerodynamic torsional and flapping moments.
In calculating the flapping and feathering moments, we should consider the unsteady
aerodynamic coefficients discussed in Chapter 6 and represented by the function
C(k). It will be assumed that the flexural axis coincides with the aerodynamic centre.
Then, from eqn 6.26, with x = – 12
and replacing the theoretical lift slope 2π by a
general value a, the spanwise thrust distribution is
d
d
= 12
– + 12
( ) + 1
8
– + 1
4
2 2 2 Tr
ac r cr
ρ Ω α Ωβ αΩC k ρac Ωrα rβ cα
˙ ˙ ˙ ˙˙ ˙˙

 

 
[ ]
where α˙ is the component of angular velocity about the blade span, which includes
the component of shaft angular velocity due to flapping, i.e. ˙ α = ˙ θ + Ωβ and
z = – rβ. The flapping moment MA is found by integrating rdT/dr over the span,
which gives
M ac R cR
A C k
= 1 2 4
8
– + 2
3
ρ Ω α Ωβ αΩ ( )
 ˙ ˙
 

 
+ 1
8
1
3
　
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本文鏈接地址：Bramwell’s Helicopter Dynamics(160)