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particular form of damping employed in the rotor can be made. For example, a
common design of blade hydraulic damper utilises a very high rate of ‘V2’ damping
followed by a constant force cut-off. The conversion of the viscous damping
requirement to another form of damping is based on equating the energy dissipation
per cycle of oscillation. This leads to the concept of an allowable blade ‘swing
angle’ in the ground resonance mode, above which the motion of the helicopter will
be divergent. Disturbances of the helicopter producing blade ‘swing angles’ below
this value will subside.
This situation implies that it is necessary to know the levels of hub acceleration in
the plane of the rotor which will be experienced in service, and which may force the
blades to oscillate in the ground resonance mode.
In order to ensure stability over the full range of the many variables involved, it
has become the practice to define a chassis mode case (which may be realistic or
artificial) that will lead to the worst possible instability.
It can be shown that more blade lag damping is required as the chassis mode frequency
increases (provided that the frequency coalescence still occurs below the maximum
possible rotor speed). Thus an assumption is often made in the design stage that the
frequency coalescence occurs at the maximum operating rotor speed, and sufficient
60
40
20
Fuselage damping, % critical
Lag frequency
0.2Ω
0.3Ω
0.4Ω
0.7Ω 0.6Ω 0.5Ω
0 20 40 60
Lag damping, % critical
Fig. 9.23 Damping requirement as a function of lag mode frequency
Aeroelastic and aeromechanical behaviour 353
damping is then provided to ensure stability in this worst case. Thus, freedom from
ground resonance will be established for the full range of operating conditions.
9.10 Air resonance
In Chapter 5, the dynamic stability of the helicopter was discussed on the assumption
that the motion was ‘quasi-steady’, i.e. the acceleration of the helicopter could be
ignored and the response of the rotor depended only in the instantaneous translational
and angular velocities of the helicopter. As we shall see, in making this assumption
a certain mode of motion is suppressed which is of little significance as a conventional
stability mode, but may, in the case of hingeless rotors, couple with the blade ‘regressive’
lag mode to produce an instability in flight which is closely related to ground resonance.
For such rotor systems, the regressive cyclic flapping mode can couple with the
fuselage roll and pitch motions to produce ‘slow gyroscopic’ modes (sometimes also
referred to as ‘pendulum’ modes) of the helicopter at frequencies which are close to
the ‘regressive’ lag mode frequency at normal operating rotor speed.
Let us suppose that the hingeless helicopter is pitching and rolling with a frequency
high enough to prevent significant translational velocities. As we shall be concerned only
with the first flapping mode, let the displacement of a point of the blade be given by
Z = RS1(x)β (ψ)
where we write β (ψ) as the azimuth co-ordinate of the blade by analogy with the
flapping angle β of the rigid blade, as in section 7.5. The mode response equation is,
by eqn 7.86.
d
d
+ = 1
(1)
( )d
2
2 2 2 1
β
ψ
λ β 1
2
0
1
Ω R f
F
x
∫ S x x ∂
∂
where λ1Ω is the flapping frequency and S1(x) is the first mode bending shape.
For pitching and rolling motion only, the blade loading ∂F/∂x can easily be shown
to be
∂
∂
F
x
= ac 2R – xS xp xq
d
d
+ sin + cos 12
3
1 ρ βψ
Ω ˆ ψ ˆ ψ
+ 2 ( cos – sin ) +
d
sin +
d
m 2R2 xp xq m 2R2 x cos
p
d
x
q
d Ω ˆ ˆ Ω
ˆ ˆ
ψ ψ ψ ψ ψ ψ
The first bracket represents the change of aerodynamic incidence due to the angular
motion and flapping; the second and third brackets denote the gyroscopic and angular
acceleration inertia forces. The flapping equation of the kth blade can then be written
d
d
+
2
d
d
+ =
2
( sin + cos )
2
2
2 1
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Bramwell’s Helicopter Dynamics(174)
