Damped and forced oscillations, resonance

Physics A2 · Oscillations. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.

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The idea

A resistive force such as friction or air resistance takes energy from an oscillating system, so its amplitude falls with time. This is damping. With light damping the system still oscillates, with an amplitude that gets smaller each cycle. With critical damping it returns to equilibrium in the shortest time without oscillating. With heavy damping it also does not oscillate, but returns much more slowly.

A system left alone oscillates at its natural frequency. If a periodic force drives it, it is in forced oscillation and vibrates at the driving frequency.

Resonance happens when the driving frequency equals the natural frequency: the amplitude of the driven system is then a maximum.

Rules to remember

Damping: a resistive force removes energy from the oscillation, so the amplitude decreases
Light damping: oscillations continue with decreasing amplitude; the period stays almost the same
Critical damping: no oscillation; displacement falls to zero in the shortest possible time
Heavy damping: no oscillation; displacement falls to zero slowly
Forced oscillations take place at the frequency of the driver, not at the natural frequency
Resonance: driving frequency = natural frequency → maximum amplitude
More damping: resonance peak lower and broader, with the maximum at a slightly lower frequency. Energy ∝ amplitude²

Common mistake

Students mix up critical and heavy damping. Both have no oscillation, but critical damping is the fastest return to equilibrium, while heavy damping is slow.

Worked example

A metal panel on a machine has a natural frequency of 20 Hz. The motor speed is raised steadily from 5 to 40 revolutions per second, and the motor shakes the panel once per revolution. How does the amplitude of the panel's vibration change?

The motor is the driver, so the driving frequency rises from 5 Hz to 40 Hz and the panel vibrates at the driving frequency.
Below 20 Hz the driving frequency is far from the natural frequency, so the amplitude is small but grows as 20 Hz is approached.
At 20 revolutions per second the driving frequency equals the natural frequency of 20 Hz: resonance, so the amplitude is a maximum.
Above 20 Hz the driving frequency moves away from the natural frequency, so the amplitude falls again.

Answer: The amplitude rises to a maximum at 20 revolutions per second (resonance) and then decreases.

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