Damped, Driven Oscillator and Resonance

A mass on a spring with friction, pushed by a steady sinusoidal force: $x'' + 2\gamma x' + \omega_0^2 x = F_0 \cos(\omega t)$. Push at just the right frequency and the response spikes; push off-frequency and nothing much happens. This is resonance, and it controls radio tuning, glass shattering, and why bridges have forbidden marching frequencies.

integrated by RK4 with $\omega_0 = F_0 = 1$. The top panel shows the live oscillator response $x(t)$ (cyan) against the drive (faint orange). The bottom panel is the steady-state amplitude curve

with a cursor on the current drive frequency. The resonance peak sits at $\omega_r = \omega_0\,(1 - 2(\gamma/\omega_0)^2)^{1/2}$, just below $\omega_0$ for non-zero damping. For small damping the peak amplitude is approximately $Q = \omega_0 / (2 \gamma)$ times $F_0 / \omega_0^2$. Slide $\omega$ across the peak and watch the response grow and contract; raise $\gamma$ to broaden the peak.