The Art of the Helicopter John Watkinson - Karatunov.net

Technical background 43
force whereas above it is inversely proportional to the square of the frequency and is
anti-phase to the force.
In the vicinity of resonance the amplitude is a function of the damping and is said
to be resistance controlled. With no damping the Q-factor is high and the amplitude
at resonance tends to infinity, resulting in a sharp peak in the response. Increasing the
damping lowers and broadens the peak so that with high damping the velocity is nearly
independent of frequency. Figure 2.23 shows the effect of different damping factors
on the step response, i.e. the response to a sudden shock. The underdamped system
enters a decaying oscillation. The overdamped system takes a considerable time to
return to rest. The critically damped system returns to rest in the shortest time possible
subject to not overshooting.
Below resonance the displacement of the spring is proportional to the force. Here
force and displacement are in phase. Above resonance the acceleration of the mass
is proportional to the force. Here force and acceleration are in phase. It will be seen
from Figure 2.22 that the velocity leads the displacement but lags the acceleration.
Consequently below resonance the velocity leads the applied force whereas above resonance
it lags. Around resonance there is a phase reversal so that at the precise resonant
frequency there is no phase shift at all. Figure 2.24 shows that the rate of phase change
in the vicinity of resonance is a function of the damping.
Fig. 2.23 The effect of different damping factors on the response to a step input.
Fig. 2.24 Damping factor affects the rate of phase change near resonance.