Past exam of the physics A course of the University of Cambridge / 2015 / Paper 2 / Section B / B6

Take $m,k,b>0$, and define ${\omega }_0=\sqrt{k/m}$ and $\gamma =b/(2m)$. Substitution of $e^{rt}$ into the unforced equation gives $r^2+2\gamma r+{\omega }_0^2=0$. The roots determine whether the return is oscillatory.

For return after a displacement in this ideal model, critical damping, case (b), is the usual comfort choice: it removes repeated bouncing while avoiding the slow relaxation mode of strong overdamping. For release from rest it gives a monotone return with the fastest asymptotic decay available without oscillation. This criterion does not prove optimal comfort for every road profile or initial velocity; actual comfort also depends on acceleration, road forcing and suspension travel.

For the forced response, set $\Omega =2\pi u/d$ and use $x_{\mathrm{ss}}=\mathrm{Re}(\hat{x}{e}^{i\Omega t})$. ThenThe general forced solution adds the appropriate homogeneous transient from (a), (b) or (c). Initial data were not specified, so its constants cannot be fixed. Positive damping removes this transient at long times. This solution uses the particular driving force in the supplied equation; it does not silently replace it by a different moving-support model.

Maximising $X$ is minimising its squared denominator $D$. Differentiation givesA nonzero displacement resonance exists only if $b^2<2mk$. Here ${\Omega }_r^2=0.175{\mathrm{s}}^{-2}$, and thereforeThe low-frequency limit is $X(0)=A$, whereas $X\to 0$ at large $\Omega$. The parameters give damping ratio $\zeta =b/(2\sqrt{mk})=0.456$, consistent with an underdamped system and the existence of this amplitude maximum.

For $b^2>4mk$, define $\beta =\sqrt{{\gamma }^2-{\omega }_0^2}$. The two characteristic roots are real and negative:This is overdamping: there is no repeated oscillation, and the slow root controls late-time return. A generic initial velocity can still cause a single crossing of equilibrium, so “nonoscillatory” need not mean monotone for all initial data. For large $b$, the slow root approaches $-k/b$.

For $b^2=4mk$, the repeated characteristic root requires a second linearly independent solution $t{e}^{-\gamma t}$:This is critical damping. If $x(0)=x_0$ and $\dot{x}(0)=0$, then $x=x_0(1+\gamma t)e^{-\gamma t}$, demonstrating the monotone return used in the suspension discussion.

For $b^2<4mk$, put ${\omega }_d=\sqrt{{\omega }_0^2-{\gamma }^2}$. Complex-conjugate roots give the real solutionThis is underdamping: oscillation at ${\omega }_d$ persists within a decaying exponential envelope. damping reduces the oscillation frequency below ${\omega }_0$. Constants in all three cases are determined by two initial conditions.