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0.2
0.4
μ=0
0.25
0.35
–0.2
–0.6
–0.4
μ=0.1 0.15 0 0.3 0.35 0.2 0.4 0.6 λ re
–0.8 –0.6 –0.4 –0.2
l = 0.02
l = 0
Fig. 5.10 Root-locus plot for typical single rotor helicopter
164 Bramwell’s Helicopter Dynamics
deterioration of stability with increase of speed is much reduced. As we have noted
before, to obtain a moment contribution due to change of thrust requires a continuous
hub moment to be exerted on the fuselage in order for the rotor force vector to be
displaced from the c.g., and this can be achieved with offset hinges or hingeless
blades. But a constant moment exerted on the fuselage implies a fluctuating load on
the rotating hub, and problems of fatigue limit the amount that can be tolerated. Thus,
although setting the c.g. forward of the shaft with offset or hingeless rotors improves
the stability, the improvement is restricted by the necessity to keep loads low enough
to avoid fatigue failure.
5.5.2 The effect of a tailplane
The effect of a tailplane on the stability of the helicopter has been calculated by
considering the tailplane referred to in Chapter 4, i.e. one having a tail volume of 0.1
and a lift slope of 3.5. The derivatives (mw)′T and (mq)′T were calculated from eqns
5.91 and 5.92 respectively; for the purpose of illustration, (mu)T was taken as zero
since it depends on the rigging angle, which can be arbitrarily chosen, and (mw˙)′T
was neglected. The results for the case l = 0 are shown in Fig. 5.11, together with the
tailless case for comparison. It can be seen that in the upper half of the speed range
the beneficial effects of the tailplane become large enough to confer positive stability.
As the speed increases, the two negative roots coalesce to form two complex branches
whose values indicate a well-damped rapid oscillation.
λ im
–0.6 –0.4 –0.2 0 0.2 0.4 0.6 λ re
–0.2
–0.4
–0.6
0.2
0.4
0.2
0.8
0.35
0.3
0.2
0.1
μ = 0
0.6
0.3
0.35
Tailless, e = 0.04, l = 0
With tailplane
Fig. 5.11 Effect of tailplane on stability roots
Flight dynamics and control 165
5.5.3 The effect of hingeless rotors
The full analysis of the hingeless rotor will be made in Chapter 7, but to discuss its
effect on helicopter stability it need only be assumed that, like the rotor with offset
flapping hinges, the hingeless rotor can exert a longitudinal moment proportional to
the tilt of the disc a1s (relative to the shaft) and a lateral moment proportional to the
sideways tilt b1s. For illustration we shall take a hingeless rotor which, for a given
rotor tilt, can exert a hub moment five times greater than that of the 4 per cent offset
hinges of our example helicopter. In Chapter 7 it will be found that, under a given set
of conditions, the flapping of a hingeless rotor is almost identical to that of a hinged
one. Thus, all the rotor forces and flapping derivatives will be the same as before and
it is necessary only to increase terms such as Cms∂a1 /∂μ (eqn 5. 53) to 5 / Cms∂a1∂μ
to represent the hub moments of a hingeless rotor. This has been applied to the
moment derivatives calculated earlier (and shown in Fig. 5.8).
For hovering flight, the characteristic cubic of our example helicopter becomes
λ3 + 3.41λ2 + 0.11λ + 1.95 = 0
whose roots are
λ1 = – 3.54 and λ2,3 = 0.065 ± 0.74i
The subsidence is even more heavily damped than previously. Then divergence of
the oscillation is a little milder, and the period has decreased from 17.5 seconds to
15.4 seconds.
The roots of the quartic have been calculated for forward flight and are shown in
Fig. 5.12. Also shown is the same hingeless helicopter fitted with the tailplane of the
previous example. It can be seen that the instability, as might have been expected, is
intensified by the hingeless blades and that at the top speeds the unstable oscillation
degenerates into two purely divergent motions, indicated by the two positive real
roots. The tailplane reduces the severity of the instability but for our case is unable
to provide positive stability. Setting the c.g. forward would further improve the
stability, but we are again faced with the objection that this would involve the fluctuating
hub moments and associated fatigue problems.
The longitudinal dynamic stability of the hingeless rotor helicopter is therefore
generally inferior to that of the helicopter with flapping hinges of small offset, and
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