The Art of the Helicopter John Watkinson - Karatunov.net

Technical background 51
centripetal force in the absence of acceleration, a new virtual force of equal magnitude
but opposite direction has to be imagined to act. This is called the centrifugal force. In
the real world centrifugal force does not exist, it is a mathematical convenience needed
to allow an apparent state of equilibrium in a rotating frame of reference.
In a fixed frame of reference, consider a turning rotor. Figure 2.31(b) shows that if
one of the blades should move upwards, the centre of mass of the blade will move closer
to the axis of rotation, reducing the moment of inertia. Conservation of momentum
requires that the reduced moment of inertia be balanced by an increase in velocity.
Consequently the blade will accelerate in the direction of rotation as it moves up. Naval
gunners discovered a related effect which is that when a shell is fired a distance of several
miles, the shell moves significantly with respect to the earth’s axis in a rotating frame
of reference and so it will also experience conservation of momentum which takes it to
the left or right of its intended target. These are examples of the Coriolis effect, named
after the mathematician who derived the mechanics of rotating frames of reference.
Figure 2.31(c) shows that when the frame of reference rotates at the same angular
velocity as the component of interest, the rotation apparently disappears. However, the
conservation of momentum effect is still occurring and the blade appears to accelerate
with respect to the frame of reference. In order to account for this acceleration, a new
virtual force is imagined to act. This is called the Coriolis force. In the real world the
Coriolis force does not exist, like centrifugal force, it is a mathematical convenience
needed to allow the use of a rotating frame of reference.
The uninformed will often make reference to both centrifugal and Coriolis forces as
though they really existed, usually as a result of reading an advanced textbook without
understanding it. Such observations must be treated with suspicion. Centrifugal force
and Coriolis force are the technical equivalent of the unicorn and the mermaid. They
are in the imagination and the literature but they don’t exist.
There is a further consequence of the use of a rotating frame of reference that affects
vibration frequencies created in the rotor and transmitted to the hull. The frequencies
generated in the rotor may have the rotational frequency of the rotor both added and
subtracted. The phenomenon is known as heterodyning and was treated in section 2.11.
2.16 Rotating masses and precession
The rotors of a helicopter at flight RPM contain a large amount of stored energy and
this alters the way they respond to forces. In order to control the helicopter, the entire
rotor has to be tilted in the direction the pilot wishes to go. This will not be achieved by
applying a couple in that direction. Figure 2.32(a) shows that if one blade is considered,
when a couple is applied to the rotor, as it rotates the couple will alternately try to move
the blade up and down. In fact in a rotating frame of reference the blade experiences a
sinusoidal forcing function at the rotational frequency of the rotor. The way in which
the blade responds to this is non-intuitive.
Figure 2.32(b) is drawn in a rotating frame of reference with respect to which the
blade is stationary and it becomes correct to refer to centrifugal force. The figure
shows that if it is assumed that the blade is displaced from its normal plane of rotation
and released, there will be a component of centrifugal force trying to return the blade
to the original plane. For small angles this force is proportional to the displacement,
so the condition for SHM is met. The blade will oscillate like a pendulum about the
normal plane at some natural frequency. This frequency can be calculated from the
mass of the blade and the centrifugal force. It turns out that the frequency is nearly
identical to the rotational frequency.