The Art of the Helicopter John Watkinson - Karatunov.net

38 The Art of the Helicopter
Fig. 2.18 A body following a circular path is at constant acceleration and if this is calculated, it can be used
to derive the forces acting. (a) shows that after a short time δT, the velocity has changed direction. (b) The rate
of change of velocity is the acceleration. (c) shows that if δT is allowed to fall to zero, the acceleration points
to the centre of rotation.
becomes vanishingly small this vector will be seen to point to the axis of rotation as
can be seen in (c).
As the RPM increases, the length of the velocity vectors in Figure 2.18 increases in
proportion to RPM, as does the angle between them. As the resultant for small angles
is the product of the vector length multiplied by the angle between them, it is easy to
see that the acceleration or rate of change of velocity is proportional to the square of
the RPM. The advantage of the use of ω to measure angular velocity is that if this is
done the velocity at any radius r is simply ω × r and the acceleration is simply ω 2 × r.
There are no constants or conversion factors to remember which is a major advantage
of the MKS metric system.
It was shown above that F = m × a, it can be seen that the inward or centripetal
(Latin: centre seeking) force needed to accelerate an object in a circular path is simply:
F = m × ω 2 × r as ω = v/r then F = m × v 2 /r
When a mass moves along, or translates, it has kinetic energy. When an object rotates,
it also has kinetic energy, but the amount is less easy to calculate. This is because all
elements of a translating mass move at the same velocity, whereas in a rotating mass
different elements move at different velocities according to how far from the axis they
are. By adding up all of the kinetic energies of blade elements at different velocities, the
kinetic energy of the rotor can be found. If an elemental mass δm is rotating at radius r
with angular velocity ω, its velocity is ω × r and so its kinetic energy must be:
δm × ω 2 × r 2
2
An arbitrary body can be treated as a collection of such masses at various radii and
the integral of the kinetic energies of all of these gives the total kinetic energy. If this
is divided by the square of the angular velocity the result is the moment of inertia;
the rotational equivalent of mass. Any rotating body could have the same moment
of inertia if all of its mass instead were concentrated at one radius from the axis of
rotation. This is known as the radius of gyration.
The rotor blade requires an inward or centripetal force to accelerate it into a circular
path. Each element of the blade is at a different radius and so calculating the overall