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Damped & Driven Oscillations, Resonance

Source lecture(s): SC133 Lec 21

Intuition

Real oscillators run down: friction and drag bleed energy, and the sinusoid decays inside a shrinking envelope. Push back rhythmically and you can sustain the motion — and if you push at the oscillator's own frequency, tiny efforts build enormous swings. That amplification is resonance: the reason a child's swing responds to gentle, well-timed pushes, radios select stations, and soldiers break step on bridges.

Damped oscillations

Add a velocity-proportional drag \(-b\dot x\) to the SHM equation:

\[m\ddot x + b\dot x + kx = 0\]

For light damping the solution is a decaying sinusoid:

\[x(t) = A\,e^{-bt/2m}\cos(\omega' t + \phi), \qquad \omega' = \sqrt{\omega_0^2 - \left(\frac{b}{2m}\right)^2}\]
  • The amplitude envelope decays with time constant \(\tau = 2m/b\); energy (\(\propto A^2\)) decays twice as fast.
  • Frequency shifts down slightly — usually negligibly.

Three regimes:

Regime Condition Behavior
Underdamped \(b < 2\sqrt{km}\) oscillates, decays
Critically damped \(b = 2\sqrt{km}\) fastest return, no overshoot
Overdamped \(b > 2\sqrt{km}\) slow creep back

Car suspensions and door closers are tuned near critical damping — return quickly, don't bounce.

Driven oscillations & resonance

Drive with a periodic force \(F_0\cos\omega_d t\) and, after transients die, the system oscillates at the driving frequency with amplitude

\[A(\omega_d) = \frac{F_0/m}{\sqrt{(\omega_0^2 - \omega_d^2)^2 + (b\omega_d/m)^2}}\]
  • Peak near \(\omega_d \approx \omega_0\): resonance.
  • Peak height and sharpness are set by damping — the quality factor \(Q = \omega_0 m/b\) counts (roughly) the oscillations before the energy decays by \(e^{2\pi}\); high-\(Q\) systems ring long and resonate sharply.

Resonance in the wild: pushing a swing · tuning circuits (the same curve in SC134's RLC lab) · microwave ovens driving water molecules · opera singers and wine glasses · the Tacoma Narrows bridge (aeroelastic flutter — resonance's dramatic cousin) · MRI machines resonating nuclear spins.

Worked example: how many swings?

A pendulum with \(Q = 300\) (typical clock pendulum) loses what fraction of energy per cycle?

\[\frac{\Delta E}{E} \approx \frac{2\pi}{Q} = \frac{2\pi}{300} \approx 2\%\]

— which is exactly the energy the escapement mechanism must replace each swing to keep the clock running at constant amplitude.

Try it live

The oscillator lab has damping and driving sliders: sweep the drive frequency across \(\omega_0\) and watch the amplitude peak — then increase damping and watch the peak flatten.

Common mistakes

  • Expecting the driven system to move at its natural frequency. Steady state follows the drive; \(\omega_0\) only decides the amplitude.
  • "More damping always means less motion." Below resonance, damping barely matters; and critical damping returns fastest — overdamping is slower.
  • Confusing resonance with matching any frequency — the match must be to \(\omega_0\) (or, for swings pumped by kneeling, \(2\omega_0\) — parametric resonance).
  • Forgetting transients. Right after switch-on, the motion is a mix of natural and driven oscillations; the formulas above describe the long run.

Knowledge graph position

Prerequisites: SHM, Friction & drag. Leads to: Waves, AC circuits (SC134), every spectroscopy.

Quiz

Q1 (conceptual). Why do soldiers break step crossing a footbridge?

Answer

Marching in step delivers a periodic force; if its frequency lands near the bridge's \(\omega_0\), resonance pumps the amplitude cycle after cycle. Random steps spread the force over frequencies, starving the resonance.

Q2 (computational). An oscillator has \(m = 0.2\) kg, \(k = 80\) N/m, \(b = 0.4\) kg/s. Underdamped? Find \(Q\).

Answer

\(2\sqrt{km} = 2\sqrt{16} = 8 \gg 0.4\): strongly underdamped. \(\omega_0 = 20\,\text{rad/s}\), \(Q = \omega_0 m/b = 20(0.2)/0.4 = 10\).

Q3 (multiple choice). For the fastest return to equilibrium without overshoot, choose damping: (a) as small as possible (b) critical (c) as large as possible

Answer

(b). The definition of critical damping — and the design point of shock absorbers and analog meter needles.