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Flight dynamics and control 173
Now we found for the longitudinal hovering case that, in addition to the speed
being zero, some of the longitudinal derivatives were also zero, and this led to a great
simplification. Strictly speaking this is not so for the lateral case, because the yawing
and rolling motions are coupled by the tailrotor, and this coupling is represented by
the derivatives lr and np. However, if we assume that lr is negligible – if, for example,
the tailrotor shaft were on the roll axis – the resulting motion would be analogous to
the longitudinal case, with the corresponding characteristic equation
(λ – nr)[λ3 – (yv + lp)λ2 + yvlpλ – lvwc] = 0 (5.133)
The root λ = nr indicates that the yawing motion is independent of the sideways
and rolling motion. The cubic is analogous to the longitudinal equation, eqn 5.94, but
we should note that the moment of inertia in roll is much lower than in pitch, with a
consequent increase in the numerical values of the coefficients.
The numerical values of the lateral derivatives are
nr = – 0.25, yv = – 0.052
lp′ = mq′ = – 0.099, lp = –3.0
l v′ = – mu′ = – 0.016, lv = –23
The characteristic cubic is
λ3 + 3.05λ2 + 0.16λ + 1.96 = 0
with roots
λ = –3.19 and λ = 0.07 ± 0.78i
Substitution of the real root λ = – 3.19 back into the equations of motion shows
that it corresponds to an almost pure rolling motion, i.e. it can be regarded as the
‘damping-in-roll’ root. This motion is heavily damped with a time to half amplitude
of less than half a second. The complex root represents a divergent oscillation of
period 14.8 seconds which doubles amplitude in 18 seconds. The yawing motion
previously mentioned has a time to half amplitude of about 5 seconds.
5.7.1 Forward flight
The quartic eqn 5.101 has been solved for a range of μ, and the roots are shown in
the root-locus plot of Fig. 5.16. It can be seen that the mildly unstable oscillation in
hovering flight very soon becomes stable and becomes progressively more so as the
speed increases. The increase of the imaginary part of the root indicates that the
period of the oscillation steadily becomes shorter. The mode shape of the oscillation
at the higher speeds shows that ψ ≈ – βss = – v ˆ/μ, i.e. the helicopter ‘weathercocks’
with very little sideways translation, and the motion is similar to the ‘Dutch roll’
oscillation of the conventional aeroplane. By neglecting the rolling that occurs and
using the above approximation, it follows from eqn 5.100 that the time of oscillation
of the ‘Dutch roll’ oscillation is approximately
T = 2πtˆ/√(μnv )
174 Bramwell’s Helicopter Dynamics
S
μ =0
0.3
0.2
0.1
–0.5 –0.4 –0.3 –0.2 –0.1 0 0.1 0.2
4% offset hinge
Hingeless
r
0.3
0.2
0.1
–1
2
3
1
0.35
Spiral roots
Fig. 5.16 Root-locus plot of lateral stability
We also find that the quartic has a small root given approximately by
λ = – E2/D2
which corresponds to the ‘spiral root’ and, as we have already seen, a large negative
root which corresponds to almost pure damping in roll. Thus, except for fairly low
speeds, where the characteristic hover mode is dominant, the lateral stability modes
of the helicopter are similar to those of the conventional aircraft. This is rather
remarkable when it is considered that, apart from the tailrotor, which would be
expected to act like a fin, the forces and moments from a helicopter rotor arise in
quite a different manner to those of the aerodynamic surfaces of a fixed wing aircraft.
5.7.2 The effect of hingeless rotors
As with the longitudinal case, we represent the hingeless rotor by increasing the hub
moment of the 4 per cent offset hinge of our example helicopter by a factor of five.
This increases the moment derivatives lp and iv, since they arise almost entirely from
the main rotor, but not the ‘weathercock’ derivative nv.
The characteristic equation for hovering flight becomes
λ3 + 11.2λ2 + 0.59λ + 6.48 = 0
whose roots are
λ1 = – 11.2 and λ2,3 = ± 0.76i
The subsidence is now very heavily damped and the previously unstable oscillation
has now become neutrally stable with a period of 15.2 seconds, which is almost
identical to the period of the longitudinal motion.
　
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