Damping, forced oscillations and resonance

What the exam asks

• Definitions: free oscillation, forced oscillation, natural frequency, driving frequency, damping, resonance, and light, critical and heavy damping.
• Sketches: displacement–time graphs for each type of damping, and amplitude–driving frequency graphs for light and heavier damping on the same axes.
• Explanations: why damping lowers and broadens the resonance peak, and where the energy goes.
• Phase: between driver and oscillator below, at and above resonance (AQA, OCR A, Edexcel).
• Applications: car suspension, earthquake-resistant buildings, footbridges, musical instruments, radio tuning and MRI.
• Numbers: amplitude ratios per cycle, the fraction of energy lost per cycle, and resonance conditions (driving frequency = $\frac{1}{2\pi}\sqrt{k/m}$).

Core ideas

Free and forced oscillations

A free oscillation happens when a system is displaced and left alone. It oscillates at its natural frequency $f_0$, which is set by the system itself: $f_0 = \frac{1}{2\pi}\sqrt{k/m}$ for a mass–spring system. A oscillation happens when a periodic driving force keeps acting. Once the start-up transients die away, the system oscillates at the , not at its natural frequency.

Damping

Damping is the loss of energy from an oscillating system to its surroundings by resistive forces such as friction, air resistance or viscous drag. It reduces the amplitude over time.

The energy of an oscillator is proportional to amplitude squared, $E \propto A^2$. If the amplitude falls to 90% each cycle, the energy falls to $0.90^2 = 81\%$, so 19% is lost per cycle.

Resonance

When the driving frequency equals the natural frequency, energy is transferred from the driver to the oscillator most efficiently. The amplitude builds up to a maximum. This is resonance. With very light damping the amplitude can become large enough to damage the system.

The amplitude–driving frequency graph:

• At very low driving frequency, the oscillator follows the driver, so its amplitude is roughly the driver’s amplitude.
• The amplitude rises to a sharp peak at (or very close to) $f_0$.
• At high driving frequency, the amplitude falls towards zero because the oscillator cannot keep up.

Increasing the damping:

1. Lowers the peak amplitude at every frequency near resonance.
2. Broadens the peak, so resonance is less sharp.
3. Moves the peak to a slightly lower frequency.

Far from resonance, the curves for different damping almost coincide.

Phase

Barton’s pendulums show this well. Several paper-cone pendulums of different lengths hang from a string driven by one heavy pendulum. The pendulum whose length equals the driver’s has the largest amplitude and lags it by a quarter cycle. Shorter pendulums swing nearly in phase with small amplitude, and longer ones swing nearly in antiphase with small amplitude. Every pendulum swings at the driver’s frequency.

Useful and harmful resonance

• Useful: musical instruments (air columns and strings resonate at their natural frequencies, as stationary waves), radio tuning (an LC circuit resonates with the chosen station), MRI (nuclei resonate with the applied radio-frequency field), and clocks (quartz crystals).
• Harmful: footbridges that sway when pedestrians’ footfall matches a natural frequency; buildings in earthquakes; car parts, washing machines and engines vibrating at particular speeds.
• Solutions: change the natural frequency (stiffen or change the mass), add damping (tuned mass dampers, viscous dampers, rubber mounts), or avoid the driving frequencies.

Worked examples

Exam technique

• Sketch both curves on the same axes. The more-damped curve must lie below the lightly damped one near $f_0$, with its peak slightly to the left, and both must meet at low frequency.
• Definitions carry the marks: resonance is “when the driving frequency equals the natural frequency, giving maximum amplitude”. Mention both parts.
• Energy language: say energy is “transferred to the surroundings as thermal energy (internal energy)” or “dissipated”. Never say it “disappears”.
• Applications 6-markers: use the chain “driving frequency matches natural frequency → resonance → large amplitude → damage”, then give the fix and explain how it changes $f_0$ or adds damping.

Common mistakes

Quick recap

• Free oscillation: at the natural frequency $f_0$. Forced oscillation: at the driving frequency.
• Light damping: exponential decay (constant ratio per cycle). Critical: fastest return without oscillation. Heavy: slow return, no oscillation.
• $E \propto A^2$, so energy is lost faster than amplitude.
• Resonance: driving frequency = natural frequency, maximum amplitude, maximum energy transfer.
• More damping: a lower, broader peak at a slightly lower frequency.
• Phase: in phase below $$, lag at , antiphase above.