When a damped oscillator is driven by $F = F_0\cos(\omega_d t)$, it eventually oscillates at $\omega_d$ (not $\omega_0$). Steady-state amplitude $A = F_0/\sqrt{m^2(\omega_0^2 - \omega_d^2)^2 + b^2\omega_d^2}$. Maximum amplitude occurs at $\omega_d = \sqrt{\omega_0^2 - 2\gamma^2} \approx \omega_0$ for light damping. At resonance, the amplitude is limited only by damping: $A_{\max} = F_0/(b\omega_0)$. Real-world examples: soldiers breaking step on bridges, tuning radio circuits, Tacoma Narrows Bridge collapse.