Simple harmonic motion and damped oscillationsEdexcel A-Level Further Maths: Revision notes

A particle moves with simple harmonic motion (SHM) when its acceleration is proportional to its displacement from a fixed point and directed towards it: $\ddot{x}=-\omega^2x.$ The auxiliary equation $m^2+\omega^2=0$ has roots $m=\pm\omega\mathrm{i}$, so the general solution is $x=A\cos\omega t+B\sin\omega t,$ which can be written $x=R\cos(\omega t-\varepsilon)$ with amplitude $R=\sqrt{A^2+B^2}$. For the spring-mass system with mass $m$ and stiffness $k$, Newton's second law gives $m\ddot{x}=-kx$, so $\omega^2=\frac km$.

• Period $T=\frac{2\pi}{\omega}$; frequency $f=\frac1T=\frac{\omega}{2\pi}$.
• Maximum speed $=R\omega$ (at the centre of oscillation); maximum acceleration $=R\omega^2$ (at the extremes). Example: $\ddot{x}=-16x$, released from rest at $x=0.5$: $\omega=4$, $x=0.5\cos4t$, period $\frac\pi2$ s, maximum speed $0.5\times4=2$ m s$^{-1}$, and at $t=\frac\pi6$, $x=0.5\cos\frac{2\pi}3=-0.25$ m. Example: $\ddot{x}+25x=0$, $x(0)=0.2$, $\dot{x}(0)=1.5$: $x=0.2\cos5t+0.3\sin5t$ and the amplitude is $\sqrt{0.13}=0.361$ m.

A resistive force proportional to velocity adds a term in $\dot{x}$: $m\ddot{x}=-kx-c\dot{x}$, which gives an equation of the form $\ddot{x}+2\lambda\dot{x}+\omega^2x=0\qquad(\lambda>0).$ The discriminant of $m^2+2\lambda m+\omega^2=0$ gives three types of motion:

• Light damping ($\lambda<\omega$): complex roots $-\lambda\pm q\mathrm{i}$, $x=e^{-\lambda t}(A\cos qt+B\sin qt)$. Oscillations with exponentially decaying amplitude.
• Critical damping ($\lambda=\omega$): repeated root, $x=(A+Bt)e^{-\lambda t}$. The fastest return to equilibrium without oscillation.
• Heavy damping ($\lambda>\omega$): two negative real roots, $x=Ae^{m_1t}+Be^{m_2t}$. Slow return, no oscillation.

Example: $\ddot{x}+2\dot{x}+5x=0$ with $x(0)=1$, $\dot{x}(0)=3$. $m=-1\pm2\mathrm{i}$, so $x=e^{-t}(\cos2t+2\sin2t)$.

• The factor $e^{-t}$ is the envelope: the amplitude decays exponentially.
• The period of the oscillation is $\pi$, and $x(t+\pi)=e^{-\pi}x(t)$: each cycle the displacement is multiplied by $e^{-\pi}\approx0.043$.
• For $\ddot{x}+c\dot{x}+5x=0$ the motion is not oscillatory when $c^2\ge20$, that is $c\ge2\sqrt5$. As $t\to\infty$, $x\to0$ whatever the initial conditions. Comment on models: real damping may not be exactly proportional to velocity.

If an external driving force is added, the equation becomes $\ddot{x}+2\lambda\dot{x}+\omega^2x=F\cos\Omega t$ (or $F\sin\Omega t$). The general solution is complementary function + particular integral.

• The CF is the damped solution above, which decays: the transient.
• The PI, of the form $a\cos\Omega t+b\sin\Omega t$, persists: the steady state, oscillating at the driving frequency. For large $t$ the motion is the steady state. With no damping and $\Omega=\omega$ the PI contains $t\sin\omega t$, so the amplitude grows without bound (resonance).