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d
d
ζ ′′ 1 ψ = 0 (9.60)
– + –
d
d
1 = 0
hH
i
k a
q
B
B
ˆ ˆ
η ′′ ψ (9.61)
ζ′′ δζ′ κ ζ η′ δη ′ ψ + 2 + ( – 1) + 2 + 2 + – –
d
d
2 = 0
La1 Lb1 Hh
pˆ
(9.62)
η′′ δη′ κ η ζ′ δζ ′ ψ + 2 + ( – 1) – 2 – 2 + + +
d
d
2 = 0
La1 Lb1 Hh
qˆ
(9.63)
The derivation of these equations is given in Bramwell’s paper15. As can be seen,
we now have six second order differential equations which lead to a polynomial
characteristic equation of the tenth degree when the usual method of solution is
adopted. Lytwyn et al.16 have considered a mathematical model with eighteen degrees
of freedom which includes flexibility of the fuselage and of the feathering mechanism.
The problem is too complicated for simple results to be obtained as was found
possible with ground resonance. Numerical methods have to be used for calculating
the roots of the equations for ranges of values of the parameters occurring in a
particular case.
The results of such calculations, though, have been rather inconsistent. There
appears to be no ‘resonance’ of the kind occurring in the ground resonance problem,
i.e. there is no sudden deterioration of damping near the coincidence of blade lagging
and fuselage rolling mode frequencies. There may be an unstable lagging motion17
when only the natural air damping (about 2 per cent of critical) and structural damping
are present, and this instability remains fairly constant over a wide range of rotor
speeds. This may be due to the fact that the flap damping is very high, which would
tend to ‘flatten out’ the response curve so that resonance is not apparent. Increasing
the lag damping to about 5 per cent of critical seems to place this mode in the stable
region. Typical roots from such calculations for the case of 2 per cent damping are
λ1,2 = + 0.008 ± 0.37i (slow)
λ3,4 = – 0.03 ± 1.7i (fast)
the unit of time being that corresponding to one radian of rotor revolution.
358 Bramwell’s Helicopter Dynamics
Figure 9.24 indicates the real part of the slow lag mode root as a function of rotor
speed for a range of values of fundamental lag mode frequency and 2 per cent critical
lag damping.
Since the lag natural frequency for semi-rigid and hingeless rotor systems is
typically greater than 0.6Ω, it is seen that stability is ensured for values of rotor speed
throughout the normal range and well beyond.
Because the fuselage ‘stiffness’ term arises from blade flapping, which is
aerodynamically heavily damped, the effective damping in the ‘slow gyroscopic’
modes is high. However, we have noted that the important parameter in the suppression
of ground resonance is the product of the blade lag mode and chassis mode dampings.
Indeed, analysis shows that with high fuselage damping and low lag damping the
width of the unstable region is large17.
Thus, the possibility of air resonance remains if the lag damping is very low. On
the other hand, analysis shows that the amount of lag damping required to suppress
the instability is quite small. Consequently air resonance will only be a potential
problem for rotors with relatively high flap stiffness and low lag damping, a possible
combination for both semi-rigid and hingeless rotor systems.
However, the amount of lag damping required is much less than is typically required
to suppress ground resonance. Therefore, a helicopter which is stable under the
conditions most likely to give rise to ground resonance is unlikely to be subject to an
air resonance problem.
Reference 19 provides useful additional information on aspects of air resonance.
References
1. Loewy, R. G., ‘Review of rotary-wing V/STOL dynamic and aeroelastic problems’, J. Amer.
Helicopter Soc., July 1969.
2. Hansford, R. E., and Simons, I. A., ‘Torsion–flap–lag coupling on helicopter rotor blades’, J.
Amer. Helicopter Soc., October 1973.
0.02
0.01
– 0.01
– 0.02
0.8
0.9 1.0 1.1
1.2
0.6
0.4
0.2
ωb
Ω
0.8
Ω
Slow lag mode
Fig. 9.24 Real part of slow lag mode root, 2 per cent critical lag damping
Aeroelastic and aeromechanical behaviour 359
3. Pei, Chi Chou, ‘Pitch-lag instability of helicopter rotor blades’, J. Amer. Helicopter Soc.,
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