AcademyOscillations
Academy
Driven Oscillations and Resonance
Level 1 - Physics topic page in Oscillations.
Principle
Resonance occurs when periodic driving transfers energy faster than damping removes it.
This page adds an external periodic force to the damped oscillator model.
Notation
\(F_0\)
driving-force amplitude
\(\mathrm{N}\)
\(\omega\)
driving angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(\omega_0\)
natural angular frequency
\(\mathrm{rad\,s^{-1}}\)
\(b\)
linear damping coefficient
\(\mathrm{kg\,s^{-1}}\)
\(A\)
steady-state amplitude
\(\mathrm{m}\)
\(\phi\)
phase lag of displacement
\(\mathrm{rad}\)
Method
Derivation 1: Add a periodic drive
The drive supplies energy at a chosen angular frequency. Damping removes energy, so the long-time motion reaches a steady response.
Damped oscillator
\[m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=0\]
Driving force
\[F_d(t)=F_0\cos\omega t\]
Driven model
\[m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=F_0\cos\omega t\]
Derivation 2: Describe the steady state
After transients die away, the oscillator moves at the driving frequency, not at its own natural frequency.
Steady response
\[x=A\cos(\omega t-\phi)\]
Natural frequency
\[\omega_0=\sqrt{\frac{k}{m}}\]
Amplitude response
\[A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(b\omega/m)^2}}\]
The denominator compares spring-inertia mismatch with damping.
Derivation 3: Interpret resonance
With light damping, the amplitude is largest near the natural frequency. More damping lowers and broadens the peak.
Light damping
\[b\ \text{small}\]
Near peak
\[\omega_{\mathrm{res}}\approx\omega_0\]
Finite amplitude
\[b>0\Rightarrow A_{\mathrm{peak}}\ \text{finite}\]
The graph shows steady-state amplitude against drive frequency. It is not the displacement-time graph.
Rules
These are the compact results from the derivations above.
Driven oscillator
\[m\frac{d^2x}{dt^2}+b\frac{dx}{dt}+kx=F_0\cos\omega t\]
Steady response
\[x=A\cos(\omega t-\phi)\]
Amplitude response
\[A=\frac{F_0/m}{\sqrt{(\omega_0^2-\omega^2)^2+(b\omega/m)^2}}\]
Resonance estimate
\[\omega_{\mathrm{res}}\approx\omega_0\quad\text{for light damping}\]
Checks
- Resonance is a steady-state effect.
- Damping limits the peak amplitude.
- Phase lag changes with drive frequency.
- Driving at high frequency gives small response.