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system centre of gravity to move off the axis of rotation and describe a circle, giving
rise to inertia forces which will subject the fuselage and its chassis to an oscillatory
force.
The mode of blade motion in which the direction of rotation of the centre of
gravity of the rotor system is in the opposite sense to the direction of rotation of the
rotor system is known as the ‘regressive’ lag mode, whereas the ‘progressive’ lag
mode refers to the case where the centre of gravity moves in the same direction as the
direction of rotor rotation.
The potential for instability occurs in the vicinity of a frequency coalescence
between the ‘regressive’ lag mode and a fuselage mode, provided that the fundamental
lag frequency is less than the rotational speed of the rotor. The phase relationships
between the couplings are such that a frequency coalescence when the lag mode
frequency is greater than the rotor speed does not produce an instability, neither does
a coalescence with the ‘progressive’ lag mode.
The important parameters with respect to ground resonance are blade lag mode
frequency and damping, fuselage frequency and damping, and fuselage mode shape.
Of lesser significance are flap mode stiffness and aerodynamic loads. Ground resonance
is basically a purely mechanical instability which could exist in vacuo.
In practice, a stability augmentation system can influence ground resonance, and
if a significant response of such a system can be anticipated at the frequencies
Aeroelastic and aeromechanical behaviour 343
associated with ground resonance, then its characteristics must be included in the
analysis. An alternative approach is to ensure that the feedback systems are filtered
in such a way so as not to respond at the frequencies of potential ground resonance
oscillations.
Let us calculate the displacement of the rotor centre of gravity for an arbitrary
motion of the blades. Let xkg and ykg be the co-ordinates of the centre of gravity of
the kth blade relative to the centre of the hub, Fig. 9.18. This figure can represent
either a hinged blade or a hingeless blade (as an offset rigid blade with hinge restraint,
Chapter 7).
We easily find that
xkg = – eR cos ψk – rg cos (ψk + ξk)
ykg = eR sin ψk + rg sin (ψk + ξk)
Since ξk is a small angle, these relationships can be written approximately as
xkg = – (eR + rg) cos ψk + rgξk sin ψk
ykg = (eR + rg) sin ψk + rgξk cos ψk
Summing over the b blades of the rotor the co-ordinates of the centre of gravity of
the rotor are
x
eR r
b
r
r k b
b
k k
b
g k k = –
+
cos + sin g
=0
–1
g
=1
–1
Σ ψ Σξ ψ
or
xr rbk
b
g k k = ( g / ) =0 sin
–1
Σξ ψ (9.29)
and, similarly,
yr rbk
b
g k k = ( g / ) =0 cos
–1
Σξ ψ (9.30)
x
y
eR
rg
ψk
ξk
Fig. 9.18 Blade displacement in lagging motion
344 Bramwell’s Helicopter Dynamics
Now, suppose that the blades oscillate in lagging motion with frequency κΩ such
that
ξk = ξ0 cos κψk
Then substituting in eqn 9.29
xr r bk
b
g k k = ( g 0 / ) =0 cos sin
–1
ξ Σ κψ ψ
= ( g 0 /2 ) =0 [sin ( + 1) – sin ( – 1) ]
–1
r b
k
b
ξ Σ κ ψk κ ψk
Since κ is arbitrary we find from eqn 9.29 that
x
r
b
b
b
b
rg b =
2
sin( + 1) + – 1 – sin( – 1) + – 1 g 0
1 –1
ξ
σ κ ψ π σ κ ψ π
(9.31)
where σ κ π
κ π σ κ π
1 = –1 κ π
sin ( + 1)
sin[( + 1) / ]
and =
sin ( – 1)
b sin[( – 1) /b]
Similarly we find that
y
r
b
b
b
b
rg b =
2
cos( + 1) + – 1 + cos( – 1) + – 1 g 0
1 –1
ξ
σ κ ψ π σ κ ψ π
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