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162 Bramwell’s Helicopter Dynamics
u0/θ0 = 0.0655
In dimensional terms this represents a subsidence in which there is a forward
speed change of 1 m/s for every 4.3° of nose up attitude. It is difficult to attach a
physical meaning to this mode.
The complex root λ = 0.165 + 0.65i gives
u0 /θ0 = – 0.036 + 0.117i
Reverting to dimensional values, this result shows that speed changes of about
0.5 m/s accompany attitude changes of 1°. The complex ratio u0/θ0 can be represented
by rotating vectors, and we see that the speed leads the attitude by about 107°. The
physical interpretation of this motion is as follows, Fig. 5.9. Imagine the hovering
helicopter to experience a small horizontal velocity disturbance Fig. 5.9(a). The
relative airspeed causes the rotor to tilt backwards and exert a nose up pitching
moment on the helicopter. A nose up attitude then begins to develop, and the backward
component of rotor thrust decelerates the helicopter until its forward motion is arrested.
At this point (b) the disc tilt and rotor moment vanish but the nose up attitude remains
so that backward motion begins, causing the rotor to tilt forwards and exert a nose
down moment (c). Following this, a nose down attitude is attained (d) which accelerates
the helicopter forward and returns it to the situation (a). The cycle then begins again
but, as we have found analytically, the motion is unstable and its amplitude increases
steadily.
The unstable motion described above is due entirely to the characteristic backward
flapping of the rotor with forward speed, although the rate of divergence is reduced
by the favourable damping in pitch mq. If it were possible for mu to be negative, i.e.
for the rotor to flap forward with speed, it follows that the last term of eqn 5.94 would
be negative, implying a positive real root and a pure divergence, which is even less
desirable. Zbrozek10 investigated the effects of configuration changes on the dynamic
stability but found that no reasonable departure from the conventional helicopter
shape would significantly improve the stability. In particular, the c.g. position, which
is of great importance in the stability of the fixed wing aircraft, has no effect on the
stability of the hovering helicopter. For the helicopter with zero offset hinges and
zero fuselage moment, the rotor force vector must pass through the c.g., as we saw
in Chapter 1, so that moving the c.g. merely has the effect of changing the fuselage
attitude without altering the pitching moments. When the flapping hinges are offset,
or if the blades are hingeless, a hub moment can be exerted and it is no longer
T T T T
M M
V (a) V = 0 (b)
V (c) V = 0 (d)
Fig. 5.9 Disturbed longitudinal motion of helicopter
Flight dynamics and control 163
necessary for the rotor force to pass through the c.g., so that changes of rotor force
can contribute to the pitching and rolling moments. But, as we saw earlier, in hovering
flight the thrust changes due to forward speed, ∂tc/∂μ, and to pitching rate, ∂tc/∂qˆ,
are both zero and, although ∂tc/∂wˆ ≠ 0, inspection of the coefficients of the quartic,
eqns 5.18 to 5.21, shows that when ˆV = 0 the mw derivative can make no contribution.
Thus, in hovering flight, it is true for all types of helicopter that movement of the c.g.
has no effect on the dynamic stability.
5.5.1 Forward flight
The roots of the stability quartic, eqn 5.17, have been calculated for the speed range
μ = 0 to μ = 0.35 and are shown in root-locus form in Fig. 5.10 for the two cases, c.g.
on the shaft (l = 0) and forward c.g. (l = 0.02). On the scale of this figure, the large
negative roots (corresponding to λ = – 1.26 of the hovering case) cannot be shown
but, since they represent the most stable mode, they are of least significance. The
most important roots are those representing the unstable oscillation and it can be seen
that, for the case l = 0, the destabilising effect of the positive mw becomes more
important as the speed increases; if higher speeds had been considered it would have
been found that the two complex branches of the curve would have met on the real
axis and then moved in opposite directions along this axis, implying two real roots
and at least one divergent mode.
The effect of setting the c.g. forward of the shaft is shown by the case l = 0.02
(lR = 13 cm). Here the moment generated by the rotor thrust is stabilising and
opposes the moment due to the rotor tilt. Although the aircraft remains unstable, the
λ im
Stable Unstable
0.4
0.3
0.05 0.1 0.2
0.6
　
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