The Art of the Helicopter John Watkinson - Karatunov.net

Technical background 41
Figure 2.20(b) shows that the same characteristic is obtained if the mass is supported
on a spring. The mass supported by a spring is found widely in engineering because
structures have mass and can never be infinitely rigid. An ideal spring produces a restoring
force proportional to the displacement. The constant of proportionality is called
the stiffness and is the reciprocal of compliance. When such a system is displaced there
is sustained resonance. Not surprisingly the displacement is sinusoidal and is called
simple harmonic motion or SHM and has all of the characteristics of one dimension of
a rotation as shown in Figure 2.17.
The only difference between the mass on a string and the mass on a spring is that
when more energy is put into the system, the mass on a string goes faster because the
displacement cannot increase but more tension can be produced. The mass on the spring
oscillates at the same frequency but the amplitude has to increase so that the restoring
force can be greater.
The velocity of a moving component is often more important than the displacement.
The vertical component of velocity is obtained by differentiating the displacement.
As the displacement is a sine wave, the velocity will be a cosine wave whose amplitude
is proportional to frequency. In other words the displacement and velocity are
in quadrature with the velocity lagging. This is consistent with the velocity reaching
a minimum as the displacement reaches a maximum and vice versa. Figure 2.21
Fig. 2.21 The displacement, velocity and acceleration of a body executing simple harmonic motion (SHM).