# Resonance frequency in second-order system

## Main Question or Discussion Point

Suppose we have a second-order system with the following transfer function:

$G(s)= \frac{1}{s^{2} + 2ζω_{n}s +ω_{n}^{2}}$

To figure out its resonant frequency, obtain its frequency response function and then maximize it with respect to ω. You get:

$ω_{peak} = ω_{n}\sqrt{1-2ζ^{2}}$

So it appears that $ω_{peak} ≥ 0$ for $ζ ≤ \frac{\sqrt{2}}{2}$

But what happens if the damping ratio is greater than that, but still less than 1? Then does the system simply oscillate but not resonate at any particular frequency? This bit confuses me. I'd appreciate any help in clearing this up.

Refrigerator

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tiny-tim
So it appears that $ω_{peak} ≥ 0$ for $ζ ≤ \frac{\sqrt{2}}{2}$