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Resonance and amplification

When a harmonic force has the same frequency as the system's natural frequency (resonance), the amplitude becomes much larger than the static deflection $F_0/k$ alone would give. The amplification at resonance is limited only by the damping: less damping means a higher peak.

X_{res} = \dfrac{F_0/k}{2\zeta}

amplitude at resonance

Symbols

$X_{res}$	amplitude at resonance	m
$F_0$	force amplitude	N
$\zeta$	damping ratio

Example

$F_0 = 500$ N, $k = 50$ kN/m, $\zeta = 0.05$ :

$X_{res} = (500/50\,000)/(2\cdot 0.05) = 0.1$ m.

With light damping the amplification at resonance is roughly

1/(2\zeta)

– halve the damping, double the peak.

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Part of Machine Dynamics and Vibrations: Forced vibrations and isolation.