Forced Oscillations, Resonance, and Quality Factor

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Forced oscillation

Oscillatory motion sustained by an external force, often one that varies periodically.

Driven, damped oscillator equation

For a mass–spring oscillator driven by a sinusoidal force, $m\ddot{x}+b\dot{x}+kx=F_0\sin(\omega t)$.

Transient response

The temporary part of a driven oscillator’s motion, which depends on initial conditions and fades when damping dissipates energy.

Steady-state response

The lasting periodic motion after transients fade; it occurs at the driving frequency with fixed amplitude and phase.

Quality factor of a lightly damped oscillator

A measure of resonance selectivity, approximately $Q\approx m\omega_0/b$ for a lightly damped mass–spring oscillator.

Quiz questions

1. Which equation describes a mass–spring oscillator with viscous damping and sinusoidal driving?Answer: $m\ddot{x}+b\dot{x}+kx=F_0\sin(\omega t)$
2. What happens to the resonance response when damping increases?Answer: The peak becomes lower and the response spreads across a broader frequency range.
3. In the ideal model, what happens under continuous driving at exact resonance when damping is zero?Answer: The oscillation amplitude grows without limit while the driving continues.
4. For a lightly damped mass–spring oscillator, which expression approximates the quality factor?Answer: $Q\approx\frac{m\omega_0}{b}$