Damped, driven oscillator and the resonance curve

A single mass on a spring with linear damping is driven by a sinusoidal force, $x'' + 2\gamma x' + \omega_0^2 x = F_0 \cos(\omega t)$. After a transient the oscillator forgets its start and settles into a steady oscillation at the drive frequency, with an amplitude that peaks when the drive is tuned near the natural frequency $\omega_0$. The top panel shows the live response $x(t)$ against the drive; the bottom panel is the steady-state amplitude-versus-frequency curve with a cursor at the current drive frequency, so you watch the response grow as you approach the resonance peak. Light damping (high quality factor $Q$) makes that peak tall and narrow; heavy damping flattens it. The readout reports $\omega$, $\gamma$, $Q$ and the resonant frequency. This is the physics behind tuning a radio and why marching troops break step on a bridge.

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.