Quality factor and viscous damping regimes ($Q={\omega }_0/(2\beta )$)

For a linear oscillator with viscous drag, this parameter puts the underdamped/critical boundary at $Q=1/2$. The energy-per-cycle interpretation and narrow displacement linewidth are most useful at high quality factor; critical and overdamped free motion do not complete oscillation cycles. Amplitude and averaged-energy ringdown times differ by a factor of two. The damping classification is not the same as the condition for a nonzero-frequency displacement maximum.

Minimizing $({\omega }_0^2-{\omega }^2{)}^2+4{\beta }^2{\omega }^2$ gives a displacement maximum at $\sqrt{{\omega }_0^2-2{\beta }^2}$ only when $\beta <{\omega }_0/\sqrt{2}$. An underdamped oscillator with $1/2<Q\le 1/\sqrt{2}$ consequently has no nonzero-frequency displacement peak. The absorbed-power response instead peaks at ${\omega }_0$, with exact half-power bandwidth $2\beta$ for this model. Comparing bandwidth definitions requires naming the measured response.