Damped oscillationsAQA A-Level Further Maths: Revision notes

Real oscillations lose energy to resistance. In the simplest model the damping force is proportional to the velocity and opposes it, so for a particle of mass $m$ with restoring force $kx$ and resistance $cx'$: $mx''=-kx-cx'.$ Dividing by $m$ gives a second order equation of the form $x''+2px'+qx=0$ with $p,q>0$. The $x'$ term is the damping: if it is absent the motion is simple harmonic. The type of motion depends on the roots of the auxiliary equation $m^2+2pm+q=0$, that is $m=-p\pm\sqrt{p^2-q}$, and so on the sign of $p^2-q$ (equivalently the discriminant of the auxiliary equation).

If $p^2<q$ the roots are complex, $m=-p\pm i\beta$ with $\beta=\sqrt{q-p^2}$, and $x=e^{-pt}(A\cos\beta t+B\sin\beta t).$ This is an oscillation whose amplitude decays like $e^{-pt}$. The particle passes through $O$ repeatedly, with the time between successive passes in the same direction equal to $\frac{2\pi}{\beta}$. This is longer than the undamped period $\frac{2\pi}{\sqrt q}$, because the damping slows the motion. Light damping occurs when the resistance is small compared with the restoring force.

If $p^2=q$ there is one repeated real root $m=-p$, and $x=(A+Bt)e^{-pt}.$ This is the borderline case: the particle returns to equilibrium as quickly as possible without oscillating (it can pass through $O$ at most once). Critical damping is used in car suspensions and door closers so that things settle fast and without bouncing.

If $p^2>q$ there are two distinct real roots $m_1,m_2$, both negative, and $x=Ae^{m_1t}+Be^{m_2t}.$ There is no oscillation. The particle creeps back to $O$ more slowly than with critical damping (it passes through $O$ at most once). Heavy damping occurs when the resistance is large, for example a door closing slowly in thick oil.

Find the discriminant of $m^2+2pm+q=0$, which is $4(p^2-q)$:

• negative: light (oscillates, decaying amplitude)
• zero: critical (fastest return, no oscillation)
• positive: heavy (slow return, no oscillation). Example. A 1 kg particle has restoring force $9x$ and resistance $kv$. Then $x''+kx'+9x=0$ and the discriminant is $k^2-36$. Critical damping needs $k=6$; light damping needs $0<k<6$; heavy damping needs $k>6$. In every case the exponential factor forces $x\to0$ as $t\to\infty$.