Past exam of the physics A course of the University of Cambridge / 2010 / Paper 1 / Section B / B6

Measure both vertical displacements upward from static equilibrium, so gravity cancels. The spring force changes by $k(x_1-x)$, while the damper is attached to a fixed surface and supplies $-b\dot{x}$. Newton's second law givesFor sinusoidal steady motion, substitution of $x=\mathrm{Re}(RA{e}^{i\omega t})$ gives the spring-base displacement response with a grounded damperA decaying homogeneous transient must be added unless initial conditions already select steady state. At low frequency, $R\to 1$ and the mass follows the slowly moving support; the first correction is $-ib\omega /k$. At high frequency, $R\sim -k/(m{\omega }^2)$: inertia suppresses motion and reverses its phase relative to the support. There is no $ib\omega$ in the numerator because the damper's other end does not move.

Minimize the squared denominator $D=(k-m{\omega }^2{)}^2+b^2{\omega }^2$. Its derivative is $2\omega [2m^2{\omega }^2-2mk+b^2]$, givingWrite ${\omega }_0=\sqrt{k/m}$ and $\zeta =b/(2m{\omega }_0)$. Then $|R|=[(1-r^2{)}^2+4{\zeta }^2{r}^2{]}^{-1/2}$ with $r=\omega /{\omega }_0$, and the finite-frequency peak is $1/[2\zeta \sqrt{1-{\zeta }^2}]$. Increasing light damping lowers and broadens the peak and shifts it left; the low-frequency limit remains1 and the high-frequency decay remains proportional to ${\omega }^{-2}$.

The resonance versus oscillatory damping thresholds differ. For $b^2\ge 2mk$, the maximum is at zero frequency and the curve decreases monotonically. This threshold is $\zeta =1/\sqrt{2}$, distinct from the free-oscillation critical damping threshold $\zeta =1$.