The Art of the Helicopter John Watkinson - Karatunov.net

98 The Art of the Helicopter
to provide thrust for forward flight as well as a fixed wing to unload the rotor at high
airspeeds. These ideas will be considered in Chapter 7.
3.25 Harmonic blade motion
So far the disc has been discussed as an entity since the disc attitude determines the
path of the machine. Now the motion of the blades within the disc will be explored.
In trimmed translational flight the application of cyclic feathering opposes the lift
asymmetry. Cyclic feathering changes the blade pitch angle sinusoidally. If the lift
asymmetry were also sinusoidal, the two effects would be in constant balance. Unfortunately
this is not the case. Sinusoidal cyclic feathering is not strictly what is needed,
but for practical reasons it is the most widely used solution. Cyclic feathering can only
make the average lift moment the same on both sides of the disc. It cannot keep the lift
of an individual blade constant at all angles of rotation.
Figure 3.30 shows what happens to a blade element as it rotates in forward flight
with cyclic pitch applied. The diagram assumes uniform inflow across the disc and
neglects coning for simplicity. Figure 3.30(a) shows the relative airspeed experienced
by a blade element. The rotational speed produces a constant component, but the
forward speed appears as a sinusoidal component to the revolving blade, adding speed
to the advancing blade and subtracting it from the retreating blade. The lift generated
by a blade is proportional to the square of the airspeed and this parameter is shown
in Figure 3.30(b). Note that the squaring process makes the function at 90 ◦ very much
greater than at 270 ◦ .
The coefficient of lift of the blade element is nearly proportional to the angle of
attack. The collective pitch setting results in a constant component of the angle of attack
and the cyclic pitch control causes this to become a shifted or offset sinusoid as shown
in Figure 3.30(c). Figure 3.30(d) shows the lift function which is the angle of attack
multiplied by the square of the airspeed. In other words waveform (d) is the product
of (b) and (c). The amplitude of the cyclic control in (c) has been chosen to make the
lift at 90 ◦ the same as the lift at 270 ◦ . Note that although the lift on the two sides of
the rotor has been made the same by the application of cyclic feathering, the lift is not
constant.
Unfortunately a sinusoidal function cannot cancel a sine-squared function. There is
a lift trough at about 270 ◦ where the square of the relative airspeed has become very
low and increasing the angle of attack fails completely to compensate. Lift symmetry
is only obtained because the same cyclic input also strongly reduces the angle of attack
at 90 ◦ . As a result there will be lift troughs at 90 ◦ and 270 ◦ .
The lift function contains a Fourier series of harmonics, or frequencies at integer
multiples of the fundamental, which is the rotor speed. As the lift troughs on the
two sides of the rotor are not the same shape, significant levels of odd harmonics will
exist, especially the third and fifth harmonics. The lift function excites the blades in
the flapping direction and they will respond according to the dynamic characteristics
of the blade and its damping. The result is that the blades do not describe a perfect
coning motion in forward flight, but as shown in Figure 3.31, they weave in and out
of the cone due to the harmonics. Figure 3.31(a) shows second harmonic flapping,
completing two cycles per revolution, whereas Figure 3.31(b) shows third harmonic
flapping. The flapping action directly modulates the axial thrust delivered to the mast
causing a hull motion variously described as ‘hopping’ or ‘plunging’.