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Le e
2 e = – [ – ( – ) sin ] –
2
( – ) sin 2
ξ κ ρ κ κ θ
βρ
Δξ ξ β Δκξ κβ θ (7.69)
where
Δ = 1 + (1 – )
+
e e sin
ρ ρ 2
κ κ
κ κ θ ξ β
ξ β
(7.70)
κ
κ κ
κ κ κ
κ κ
β κ κ
β β
β β
ξ
β ξ
ξ ξ
=
( + )
, =
( + )
B H
B H
B H
B H
and ρe is the degree of elastic coupling defined by
ρe = κβ/κβ = κξ/κξ B B (7.71)
When ρe = 0, the hinge system is contained entirely at the hub, i.e. the outboard
springs are infinitely stiff, and no elastic coupling is possible. When ρe = 1, all the
blade flexibility exists outboard of the feathering hinge and there is full elastic
coupling.
To illustrate the effect of elastic coupling using the above model, let us consider
a fully coupled hingeless blade (ρe = 1) with a given flapping stiffness but whose lag
stiffness is to be varied. Let the collective-pitch angle be 15°.
Suppose that the uncoupled flapping and lagging frequencies ωβ and ωξ are given
by the following Southwell formulae:
ωβ κβ β
2 = ( /I ) + 1.12Ω2 (7.72)
ω κ ξ ξ ξ
2 = ( /I ) + 0.23Ω2 (7.73)
where I is the (same) moment of inertia in flapping and lagging and Ω is the rotor
angular velocity. Let κβ /I be fixed and equal to 0.13 Ω2. If, for simplicity, we suppose
that the coning angle is zero, so that the Coriolis moments can be ignored, and also
that the change of lag stiffness κξ does not alter the constant in eqn 7.73 then, using
eqns 7.68 and 7.69, the flapping and lagging equations are
d
d
+ 1.185 + 0.067 + 1
4
– 0.13 = 0
2
2 2 2
β
ψ
κ
β
κ
ξ ξ ξ
IΩ IΩ
(7.74)
and
d
d
+ 0.239 + 0.933 + 1
4
– 0.13 = 0
2
2 2 2
ξ
ψ
κ
ξ
κ
β ξ ξ
IΩ IΩ
(7.75)
Substituting the assumed solutions β = β0 eλψ and ξ = ξ0 eλψ leads to the characteristic
equation
λ
κ
λ
2 ξ κξ
2
2
2 + 1.185 + 0.067 + 0.239 + 0.933
IΩ IΩ
– 1
16
– 0.13 = 0 2
κ ξ 2
IΩ
(7.76)
266 Bramwell’s Helicopter Dynamics
Solution of this equation for a range of lag stiffnesses κξ gives the flap and lag
frequencies shown in Fig. 7.22. The figure shows an interesting phenomenon. When
the blades are elastically uncoupled, the curves of flap and lag frequencies cross
over as the lag stiffness is increased. However, when elastic coupling exists, the
frequencies lie close to the uncoupled values when the frequencies are fairly widely
separated, but as the cross-over point is approached the curves begin to diverge
from one another and each becomes asymptotic to the frequency curve of the other
mode.
It is instructive to examine the flap–lag ratio of each mode as the lag stiffness is
varied. The flap–lag ratio can be obtained from either of eqns 7.74 and 7.75. Thus,
from eqn 7.74, after substituting β = β0 eλψ and ξ = ξ0eλψ, we get
β
ξ
κ
λ κ
ξ
ξ
0
0
2 2
2 2 =
( / – 0.13)
16[ + (1.185 + 0.067 / )]
I
I
Ω
Ω
Inserting values of λ corresponding to the lower curve shows that β0/ξ0 is very
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Bramwell’s Helicopter Dynamics(133)
