= Solution
The <Q factor> measures stored <energy> relative to loss: $Q=2\pi E/(\text{energy lost per cycle})$, or $Q=\omega_0 E/P_{\rm loss}$ in the weak-loss limit. For viscous <damping> $m\ddot x+b\dot x+kx=0$, write $\gamma=b/(2m)$, $\omega_0=\sqrt{k/m}$ and $\omega_d=\sqrt{\omega_0^2-\gamma^2}$. The conventional oscillator parameter is
$$
\boxed{Q=\frac{m\omega_0}{b}=\frac{\omega_0}{2\gamma}.}
$$
In free ringdown, fit the amplitude envelope $A(t)=A_0e^{-\gamma t}$. Its decay time is $\tau_A=1/\gamma$, whereas the cycle-averaged <energy> envelope decays on $\tau_E=1/(2\gamma)$; thus $Q=\omega_0\tau_A/2=\omega_0\tau_E$. Equivalently the <logarithmic decrement> between same-sign peaks is $\delta=\ln(A_n/A_{n+1})=2\pi\gamma/\omega_d$. It gives \b[$Q\simeq\pi/\delta$ for weak <damping>], or the exact parameter relation $Q=\sqrt{1/4+(\pi/\delta)^2}$. Fit multiple peaks above the noise floor, measure the damped period, and verify an exponential envelope rather than assuming all amplitude loss is viscous.
For a driven measurement, allow transients to settle and scan <frequency> at fixed <force> amplitude. Near <resonance> and at high $Q$, the response is Lorentzian in <displacement> <power>, with half-power points at <displacement> amplitudes $1/\sqrt2$ of the maximum. Their angular-frequency separation gives \b[$Q\simeq\omega_0/\Delta\omega$]. Halving the amplitude is not a half-power criterion.
The <absorbed-power bandwidth of a viscously damped oscillator> can be found exactly. Since
$$
\overline P(\Omega)=\frac{bF_0^2\Omega^2}{2[(k-m\Omega^2)^2+b^2\Omega^2]},
$$
its maximum is at $\Omega=\omega_0$, and the half-power <frequencies> satisfy $|k-m\Omega^2|=b\Omega$. Hence
$$
\Omega_\pm=\frac{\sqrt{b^2+4mk}\pm b}{2m},\qquad
\boxed{\Omega_+-\Omega_-=b/m,\quad Q=\omega_0/(\Omega_+-\Omega_-).}
$$
For broad <resonances>, <displacement> and absorbed-power peaks differ, so the measured observable must be specified. Frequency-dependent drive, backgrounds or several overlapping modes can otherwise masquerade as a changed <Q factor>.
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