When examining the amplitude response of a dynamic system, we often use the magnitude ratio. The magnitude ratio is defined as the amplitude of the periodic response of a system at the steady state to the ideal amplitude response.
The periodic response is given by Equation 3 from the Forced Vib Eqn Solution and is repeated here:
(1)
The ideal amplitude response is linear with amplitude of the forcing response. In this case, the ideal response is KA.
The magnitude ratio, M(
), is given as:
The magnitude ratio depends on the frequency of the input signal relative to
the natural frequency of the system (
) and the damping ratio, 
.
The graph at the right shows that as the damping ratio decreases, the amplitude of the peak increases dramatically. For a damping ratio of 0, the magnitude ratio would be infinitely large. The peak amplitude occurs when the forcing frequency is equal to the resonant frequency; this occurs at
slightly greater than 1. The resonant frequency is given by:
(3)
Tip: Operate a second order system at frequencies less than 30% of the resonant frequency ( < 0.30) to prevent large amplitude errors.
