The Art of the Helicopter John Watkinson - Karatunov.net

122 The Art of the Helicopter
helicopter in detail. For hingeless rotors use of the shaft axis has some advantages but
the tip path axis excels when discussing articulated and teetering heads because it is
easier to visualize what is taking place and the direction of the resultant rotor thrust is
clear. It is best used for flying training because it is most tangible to pilots. In practice it
is useful to be able to switch conceptually from one axis to another to obtain a proper
understanding.
The control axis was naturally used in the study of the autogyro and as the theory of
the helicopter was in many ways derived from work done on the autogyro much early
helicopter analysis also uses the control axis, including the classic work by Gessow
and Myers. 1 Although the authors make it perfectly clear that the control axis is being
used as a reference, not every reader understands what that means and there is scope
for misunderstanding if the axes become confused. One example is the occurrence of
numerous explanations that are seen claiming that flapping hinges prevent asymmetry
of lift in translational flight in gyroplanes and helicopters. This is complete nonsense
and comes about because the tip path and control axes have been confused. The blades
can flap with respect to the control axis without any flapping hinges or flexibility at all
simply because the control axis is not parallel to the tip path axis.
It is important to appreciate that all of these axes are hypothetical and the rotor
doesn’t care how we choose to observe it. Thus whatever axis is used for analysis the
result must always be consistent with the result of an analysis on another axis. To give
an example, as the blade pitch appears constant to an observer on the control axis,
no force is necessary to make the blades perform cyclic pitch changes other than to
overcome friction in the feathering bearings. If the same analysis is carried out on the
tip path axis, the blades appear to be oscillating along the feathering axis. The effect
shown in Figure 3.8 produces a restoring action tending to return the blades to flat
pitch. This causes the blades to have a torsional resonant frequency. It can be shown
that this is the same as the rotational frequency of the rotor. As a result the feathering
blades are in torsional resonance and so no force is required to operate them other than
to overcome friction. This is consistent with the analysis on the control axis.
If harmonic pitch control is used, then strictly there is no axis about which the blades
do not feather. However, the control axis could be redefined as the axis about which
there is no first harmonic feathering. In other words only the higher harmonics of
feathering would be seen with respect to the control axis. This is a reasonable approach
because the tip path plane is not really a plane because of flapping harmonics and yet
the concept is still useful. Interestingly with the correct application of harmonic pitch
control the feathering will see more harmonics so that the tip path may see fewer.
4.4 Flapping
Flapping is the movement of the blade CM along the shaft axis. It was shown in
section 2.14 that the fundamental resonant frequency of the flapping blade is almost the
same as the rotational frequency. With a zero-offset rotor the frequencies are identical
and this results in a 90 ◦ phase lag in the rotor. When flapping hinge offset is used, the
resonant flapping frequency rises slightly because the blades become a little shorter
without the diameter changing and the result is that the phase lag of the rotor becomes
a little less than 90 ◦ . This is compensated for in the design of the controls.
Flapping is self-damping because of the effect it has on relative airflow. When a
blade flaps down, its vertical velocity adds vectorially to the airflow to increase the
angle of attack, providing more lift to oppose the flapping. When a blade flaps up, the