Push a damped mass-on-a-spring at frequency ω — how big a swing do you get, and how far behind does it lag?
▶ Launch the interactive simulationDrive a damped oscillator (ω₀=2, γ=0.5) with F=f₀·cos(ωt). At each ω wait out the transient, then LOCK IN: integrate x·cos(ωt) and x·sin(ωt) over whole drive periods — the two quadratures recover the steady amplitude A and the phase lag φ at once. Sweep ω across ω₀ and overlay the measured (A,φ) on the analytic curves.
the Lorentzian A(ω)=f₀/√[(ω₀²−ω²)²+(γω)²], peaking at the resonant frequency ω_r=√(ω₀²−γ²/2) with sharpness Q=ω₀/γ, and the phase lag φ=atan2(γω, ω₀²−ω²) that sweeps 0→π and passes through EXACTLY π/2 at ω=ω₀. Measured points land on the Lorentzian to 3–4 digits; the peak (1.97 vs 1.969) and the φ=90° crossing (2.00 vs ω₀=2.0, damping-independent) hit to ~0.05%, with A_max and Q a few % low only because the drive grid is coarse near the peak