= Resonance versus oscillatory damping thresholds
{title2=$\zeta_{\rm peak}=1/\sqrt2,\quad\zeta_{\rm critical}=1$}
For a constant-force-amplitude displacement response, the frequency-domain magnitude has a nonzero-frequency peak only for damping ratio $\zeta<1/\sqrt2$. The free time-domain motion remains underdamped for all $\zeta<1$. Thus an oscillator can have complex characteristic roots but no nonzero-frequency displacement resonance peak. The distinction depends on the measured response and drive convention, not merely on the existence of damping.
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