Forced Oscillations

Forced oscillation: a driving force F = F₀·cos(ω_d·t) drives a damped oscillator at frequency ω_d (different from natural ω₀).

Steady-state response: x(t) = A_d·cos(ω_d·t − φ), where A_d depends on ω_d.

Amplitude A_d depends on (i) driving force F₀, (ii) ω_d vs ω₀, (iii) damping b.

Maximum amplitude occurs at RESONANCE ω_d ≈ ω₀ — sharp peak for small damping.

Phase φ between drive and response: 0 at low ω_d, π/2 at resonance, π at high ω_d.

Energy is continuously supplied by drive and dissipated by damping at steady state.

Transient response (initial conditions) dies out exponentially — only steady-state remains.

Examples: child on swing, AC circuit driven by source, building swaying in wind, tuning circuits.