The Art of the Helicopter John Watkinson - Karatunov.net

42 The Art of the Helicopter
shows the displacement, velocity and acceleration waveforms of a body executing
SHM (simple harmonic motion). Note that the acceleration and the displacement
are always anti-phase.
Eventually the resonance of a mass on a spring dies away. The faster energy is taken
out of the system, the greater the rate of decay. Any mechanism that removes energy
from a resonant system is called damping.
The motion of a rigid body can be completely determined by the mass, the stiffness
and the damping factor. It is important to consider what happens when resonant
systems are excited at different frequencies. Figure 2.22 shows the velocity and displacement
of a mass-stiffness-damping system excited by a sinusoidal force of constant
amplitude acting on the mass at various frequencies. Below resonance, the frequency
of excitation is low and little force is needed to accelerate the mass. The force needed
to deflect the spring is greater and so the system is said to be stiffness controlled. The
amplitude is independent of frequency, described as constant amplitude operation, and
so the velocity rises proportionally to frequency below resonance.
Above resonance the inertia of the mass is greater than the stiffness of the spring and
the response of the system is described as mass controlled. With a constant force there
is constant acceleration yet as frequency rises there is less time for the acceleration to
act. Thus velocity is inversely proportional to frequency. As the displacement is the
integral of the velocity the displacement curve is tilted by an amount proportional to
frequency so that below resonance the displacement is constant and in-phase with the
Fig. 2.22 The behaviour of a mass-stiffness-damping system: (a) amplitude, (b) velocity, (c) acceleration.