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2
2 1
1 1 Lr
Z
r
r r F Y
r
r r F
r
r
y
r
r
∫ ∫ z
= – d
d
+ d
d
2
2 A
2
2 A
Z
r
N Y
r
M (9.2)
since the integrals are the external lagging and flapping moments respectively. But,
if EIy and EIz are the lagwise and flapwise blade stiffnesses, we also have
d2Z/dr2 = MA/EIz and d2Y/dr2 = NA/EIy
so that eqn 9.2 can be written
d
d
= 1 – 1
A A
Lr
M N
EIy EIz

 

 
(9.3)
It is clear that the torque due to blade deflection will be zero if EIy = EIz at every
point of the blade. If this is satisfied the blade is called a ‘matched-stiffness’ blade.
For a hingeless blade the structural element near the root, which allows most of the
lag bending of the blade, can be ‘matched’ relatively easily to achieve almost zero
Z
X
r
r1
P
Z
Z1
(Perpendicular
to paper)
dFy
Y
X
Y
Y1
dFz
(Perpendicular
to paper)
r
r1
P
Fig. 9.2 Projection of the deformed blade onto the
XY plane
Fig. 9.3 Projection of the deformed blade onto the
XZ plane
322 Bramwell’s Helicopter Dynamics
torsional moment along the entire blade. It is obvious that the torque exerted on a
hinged blade will also be zero if the feathering hinge lies outboard of the lagging and
flapping hinges, Fig. 9.4, for then the axis about which the torque is calculated
follows the blade when it lags and flaps.
Hansford and Simons2 have shown that by neglecting the pitching inertia of the
blade, which is justifiable because of the high torsional stiffness, it is possible to
write the torsional deflection θ as
θ ψ β ψ ξ ψ
λ κ
ν
β
θ
( ) = ( ) ( )
1 – +
1
2
1
2
1
2 I
I
(9.4)
where Iβ and Iθ are the blade flapping and pitching moments of inertia, and λ1Ω, κ1Ω
and v1Ω are the rotating blade uncoupled natural frequencies in flap, lag and torsion
respectively.
If, for example, the flapping frequency is 1.1Ω, the required lagging frequency for
zero twist is found to be 0.458Ω. Thus, a condition of zero or very small blade twist
can be achieved by suitably choosing the flapping and lagging frequencies. It can be
seen from eqn 9.4 that the relationship is non-linear in β and ξ, but for small variations
about steady flapping and lagging angles β0 and ξ0 the twist can be expressed as
Δθ
λ κ
ν
β ζ ζ β β
θ
=
1 – +
( + )
1
2 0 0
1
2
1
2
⋅
I
I
(9.5)
This relationship between the pitch angle Δθ and the flapping and lagging angles
can be expressed as
Δθ = kββ + kξξ (9.6)
The same relationship applies to the hinged blade. The coefficient kβ represents the
δ3 hinge effect and kξ the α2 hinge effect, as mentioned at the beginning of this
section. The inclusion of these terms merely adds – γ (kββ + kξξ)/8 to the left-hand
side of the appropriate equation of motion (see eqn 9.25). The usual solution procedure
leading to a quartic characteristic equation may be followed, and the stability can be
discussed in terms of the roots of the equation. This has been done by Pei3, who finds
for an approximate stability criterion that
Z
X
β
Fig. 9.4 Outboard position of feathering hinge
Aeroelastic and aeromechanical behaviour 323
F
k
k
ξ˙ / C
ξ
β θ β
β
θ +
2
1 – ( )
> 0
0 0 0
⋅ 0
2
Ω (9.7)
where F˙ξ is the lag damping coefficient in the equation of motion involving lagging
motion (see, for example eqn 9.26) and C is the moment of inertia in lagging motion.
However, Pei also shows that the above criterion can be deduced from simple physical
arguments. Since the lagging motion is generally of much lower frequency than that
of the natural flapping motion, the flapping response can be calculated as if the
lagging excitation at any instant were being steadily applied. Therefore, the change
of lift moment due to the lagging and flapping motion can be written as
　
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本文鏈接地址：Bramwell’s Helicopter Dynamics(158)