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frequency.
An interesting method of simplifying the ground resonance problem has been
given by Done14. In this method, details of which can be found in the original paper,
it has been assumed that, by a suitable choice of co-ordinates, the higher frequency
(progressive) mode branching from the point A in Fig. 9.20 can be neglected, so that
the simplified equations relate only to the chassis mode and to the lower frequency
rotor mode.
By means of this approximation and using eqns 9.36 to 9.38 Done arrives at a
characteristic quartic equation of the form
( + 2 + )[ + 2 + ( – ) ] +
( – )
2
2
c c c
2 2 2
c c
λ δκΩλ κΩ λ δκΩ Ω Γ2 μ Ω Γ λ4
Γ
+
( – )
2
c c = 0
2 3 μ Ω Γ δ κ λ
Γ
Ω 2 (9.47)
where Γ is the blade rotating lag frequency in the absence of Coriolis force coupling.
The roots λ = λre ± iλim, where λim ≡ κ, of eqn 9.47 can be plotted as shown in
Fig. 9.22. The lower diagram, showing the imaginary parts of the roots, corresponds
to the diagram of Fig. 9.20 but with the degrees of freedom coupled. As can be seen
the effect of coupling is to modify the shape of the diagram in the neighbourhood of
the (uncoupled) crossing points. If unstable oscillations occur, the branches fail to
meet at C since real roots occur in this region, as is shown by the small loop in the
λre diagram. The extent of the instability is indicated by the size of the gap between
the branches.
The amount of damping required which just gives harmonic oscillations can be
found by substituting λ = ± iκcΩ = ± i(Ω – Γ) into eqn 9.47. Done shows that this
leads to
δδc μκc κ κ
2
= /8 (1 – c)
Aeroelastic and aeromechanical behaviour 351
as given before in eqn 9.45 except that, since in Done’s analysis the blade is represented
by a point mass, the value of rg/lb is equal to unity.
The damping requirement to suppress ground resonance is strongly influenced by
the lag mode frequency.
Figure 9.23 indicates the variation of the damping requirement for a range of lag
mode frequencies. The closer the lag mode frequency is to 1Ω, the smaller the
amount of damping that is required to ensure stability.
As previously indicated, ground resonance is completely eliminated if the blade
lag frequency is greater than 1Ω. A rotor with this characteristic is termed ‘supercritical’.
This solution is found in the two-bladed teetering rotor and the Lockheed gyrocontrolled
multi-blade ‘rigid rotor‘ design.
However, unless the lag mode frequency is sufficiently greater than 1Ω, e.g. 1.5Ω,
there may be fatigue strength problems due to high amplification of the first harmonic
air and Coriolis loads. Flap–lag instability is also a possibility for a rotor with a
‘supercritical’ lag frequency, as described in section 9.7.
A significant point to note when designing for, and demonstrating in practice,
freedom from ground resonance is that the full available range of rotor lift must be
considered. This is because chassis geometry, oleo stiffness, tyre stiffness, and damping
rates are all functions of the reaction at the ground contact position. The non-linear
effects of oleo stiffness are very significant, and due to oleo ‘sticking’ when the load
becomes less than the ‘break-out’ load in the high lift condition, the oleo cannot
dissipate any energy. Hence, apart from the effects of stiffness change, the only
significant contribution to chassis mode damping in this condition is provided by the
hysteresis effects of the tyres.
Also to be considered are the full range of tyre pressures (deflated to overinflated),
λre
κ
Rotor speed Ω
Fig. 9.22 Effect of coupling on fuselage and whirling frequencies
352 Bramwell’s Helicopter Dynamics
tie-down situation and wheel orientation (particularly relevant to Naval helicopters),
wheel brake situation, and surface contact conditions.
The various defined loading states of the helicopter will also have to be considered,
since these will influence the chassis mode frequencies and shapes, e.g. the roll/
lateral mode frequency will be reduced by the presence of external stores.
Viscous damping in both the airframe and rotor is normally assumed in the
analysis, and once stability boundaries have been established, a conversion to the
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Bramwell’s Helicopter Dynamics(173)
