= Solution
For $m\ddot x+b\dot x+kx=F_0\cos\omega t$, define $\omega_0=\sqrt{k/m}$ and $\beta=b/(2m)$. The parameter relating <quality factor and viscous damping regimes> is
$$
\boxed{Q=\frac{\omega_0}{2\beta}=\frac{m\omega_0}{b}}.
$$
At weak damping it also equals $2\pi$ times stored <energy> divided by <energy> lost per cycle. The stored-energy convention and <frequency> matter: this cycle-based interpretation should not be extended to free motions that never complete a cycle.
For $\beta<\omega_0$ the free motion is $e^{-\beta t}[A\cos\omega_dt+B\sin\omega_dt]$, where $\omega_d=\sqrt{\omega_0^2-\beta^2}$. Thus \b[<underdamping> means $Q>1/2$]. High $Q$ gives many oscillations per amplitude-decay time, narrow <resonance> and strong <frequency> selectivity; a lightly loaded tuning fork, optical cavity or low-loss electrical <resonator> provides examples. The amplitude decay time is $\tau_A=1/\beta=2Q/\omega_0$, while the slowly averaged <energy> decays with $\tau_E=1/(2\beta)=Q/\omega_0$. These times differ by a factor of two.
At <critical damping>, $\beta=\omega_0$ and $Q=1/2$, with $x=(A+Bt)e^{-\omega_0t}$. There is no ringing; a simple critically damped position indicator can settle without oscillation. For $\beta>\omega_0$ and $Q<1/2$, there are two real negative decay rates $-\beta\pm\sqrt{\beta^2-\omega_0^2}$. Increasing very strong drag slows the long-time return, whose rate approaches $\omega_0^2/(2\beta)$. A strongly viscously damped mechanical indicator illustrates this regime. These classifications concern the linear homogeneous equation, not every possible nonlinear damping law.
The driven displacement amplitude is
$$
A(\omega)=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+4\beta^2\omega^2}}.
$$
The <displacement-resonance threshold of a viscous oscillator> requires $\beta<\omega_0/\sqrt2$, or \b[$Q>1/\sqrt2$]; it lies at $\sqrt{\omega_0^2-2\beta^2}$. Some underdamped systems therefore have no such maximum. Average absorbed <power> instead peaks at $\omega_0$ for this viscous model. Its half-power frequencies are $\sqrt{\omega_0^2+\beta^2}\pm\beta$, so their separation is $2\beta$ and $Q=\omega_0/\Delta\omega$. For broad response this ratio remains a useful parameter but not a claim of many free oscillations. The familiar displacement-linewidth version is a high-$Q$ approximation.
A <series RLC circuit> has $\omega_0=1/\sqrt{LC}$ and $Q=\omega_0L/R$ for the charge oscillation. Resistance dissipates <energy>, inductance provides the inertial analogue and capacitance the restoring analogue. Ringdown and a calibrated half-power scan independently estimate loss; parasitic resistance, loading and radiation change the measured value. A large $Q$ increases <frequency> selectivity but also lengthens settling time, so it is not universally the best instrumental choice.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2012/paper-2-damping-regimes.png]
{title=Original free decay and forced-displacement curves distinguishing underdamping, critical damping, overdamping and the resonance threshold}
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