Damped and forced oscillations
A bell fades, a swing grows
- Strike a bell and it rings, then slowly fades — that fading is damping.
- Push a swing at just the right rhythm and it grows huge — that is resonance.
- Both come from how an oscillator handles energy.
Damping
- A resistive force (friction, drag) removes energy, so the amplitude shrinks over time.
- The lost energy turns into heat.
Damping of an oscillation is caused by:
Friction or drag carries energy away as heat, so each swing is smaller than the last.
Three kinds of damping
- Light: amplitude dies away slowly, still oscillating.
- Critical: returns to rest fastest, with no overshoot (a meter needle).
- Heavy: returns slowly with no oscillation (a strong door closer).
Match each type of damping to its behaviour.
Critical damping returns to equilibrium in the shortest time without overshooting; heavy damping is slower still.
Forced oscillations
- A driving force of frequency $f_{\text{d}}$ pushes the system.
- The system then oscillates at the driving frequency, not its own natural one.
A forced oscillator vibrates at:
It follows the driver — it oscillates at the driving frequency $f_{\text{d}}$.
Resonance
- Resonance: when $f_{\text{d}}$ equals the natural frequency $f_0$, the amplitude is largest.
- Lighter damping → a sharper, higher peak. (A swing, a wine glass, a building in an earthquake.)
Resonance happens when the driving frequency equals the ____ frequency.
At $f_{\text{d}} = f_0$ the driver feeds energy in most effectively and the amplitude peaks.
Lighter damping gives a sharper, higher resonance peak.
Less energy is lost each cycle, so the amplitude builds higher and the peak is narrower.
Select all the examples of resonance.
Each of the first three matches a driving frequency to a natural frequency. A rolling ball is not a driven oscillation.
You've got it
- damping = a resistive force shrinking the amplitude (energy → heat)
- light (slow decay), critical (fastest, no overshoot), heavy (slow, no oscillation)
- resonance: driving frequency = natural frequency → biggest amplitude (sharper if lightly damped)