Resonance versus oscillatory damping thresholds (${\zeta }_{\mathrm{peak}}=1/\sqrt{2},{\zeta }_{\mathrm{critical}}=1$)

For a constant-force-amplitude displacement response, the frequency-domain magnitude has a nonzero-frequency peak only for damping ratio $\zeta <1/\sqrt{2}$. 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.