 Displacement-resonance threshold of a viscous oscillator

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.