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total lift. The helicopter can be regarded as having a lift slope like a fixed wing
aircraft, but, as we saw in Chapter 3, the local blade lift is proportional to the term
UpUT and, since the vertical velocity increases UP by a constant amount, the change
of lift depends on the chordwise velocity component UT which is laterally asymmetric
and has a maximum on the advancing side of the disc. Because of the 90° phase lag,
this results in a backward tilt of the rotor disc producing a backward force component
and a nose up pitching moment. The effect increases roughly linearly with speed and
is zero in hovering flight. Thus, the helicopter rotor is unstable with respect to
incidence and becomes progressively more so as forward speed increases. A tailplane
is often fitted to provide positive incidence stability, but it is really effective only in
the upper half of the speed range.
5.1.3 Pitching angular velocity disturbance
Imagine the helicopter to be pitching with constant angular velocity q and that the
rotor is in equilibrium and pitching in space at the same rate. The rotor can then
be regarded as a gyroscope which will therefore be subjected to a precessing moment
tending to tilt it sideways. Because of the 90° response lag the rotor actually tilts
in the longitudinal plane, causing longitudinal forces and moments. Now we know
that a tilt of the rotor relative to the shaft produces cyclic pitch variations in the tip
path (disc) plane and a consequent aerodynamic moment. The rotor disc will therefore
tilt such that the aerodynamic moment is in equilibrium with the precessing moment.
It is found that the nose up pitching of the helicopter produces a nose down tilt of
the rotor. The corresponding nose down moment is in a favourable sense, i.e. it
opposes the disturbance. We find that in this angular motion the rotor force does not
remain perpendicular to the disc: the change of H-force is quite large and tilts the
resultant force vector in the opposite direction to that of the disc, thereby reducing
the moment. Under conditions of large inflow ratio, such as rapid climb, the sign
of the moment may even be reversed. The destabilising effect of the H-force is less
important when the moment is augmented by the presence of offset hinges or
hingeless blades.
In addition to the precessing moment mentioned above, the pitching motion causes
aerodynamic incidence changes which result in a lateral tilt of the rotor and therefore
of lateral forces and moments (section 1.6.3). Typically, the lateral tilt is about half
the longitudinal tilt. This is another example of the coupling between lateral and
longitudinal motion.
The above discussion applies equally well to rolling motion and we see that
rolling causes a moment in opposition to the motion, i.e. there is ‘damping in roll’,
as with a fixed wing aeroplane, but for a given size of aircraft the helicopter roll
damping is usually much weaker.
140 Bramwell’s Helicopter Dynamics
5.1.4 Sideslip disturbance
When the helicopter sideslips, i.e. when there is a component of wind relative to the
undisturbed flight direction, it appears to the rotor as if the relative wind speed were
unchanged but that it comes from a different direction. Thus the direction of maximum
flapping of the rotor is merely rotated through the angle of sideslip and, as we shall
see later, the change of sideways rotor flapping angle b1 depends directly on the
backward flapping angle a1 and vice-versa. Since a1 is usually much larger than b1,
the main result is a sideways tilt of the rotor away from the side wind. There is
therefore a rolling moment opposing the sideslip, corresponding to the dihedral effect
of the fixed wing aircraft. Further, the blades of the tailrotor experience a change of
incidence and the tailrotor acts like a fin producing favourable ‘weathercock’ stability.
5.1.5 Yawing disturbance
Yawing velocity causes a change of incidence at the tailrotor and, again, produces a
favourable ‘fin’ effect – or ‘damping in yaw’.
Summarising the above physical description of the effects of the disturbances, we
can expect that the nose up pitching moments which arise with forward and vertical
speed changes will result in longitudinal characteristics very different from those of
the fixed wing aeroplane. On the other hand it is rather remarkable that, although the
lateral force and moments arise in quite a different way from those of the conventional
aeroplane, they have similar signs and we shall see that the stability characteristics
are similar.
The dynamic stability analysis which now follows is formulated at the same level
as the equivalent fixed wing analysis in standard undergraduate texts, and is subject
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Bramwell’s Helicopter Dynamics(72)
