Baseline. A lumped mass on a linear spring with viscous damping: x″ + 2γ x′ + ω₀² x = 0, ω₀ = √(k/m). For γ < ω₀ the motion is x(t) = e^(−γt)[x₀ cos ω_d t + ((v₀ + γx₀)/ω_d) sin ω_d t] with ω_d = √(ω₀² − γ²).

What this lab adds. A periodic force on the mass: x″ + 2γ x′ + ω₀² x = (F₀/m) cos Ωt. After the transient dies out the amplitude is A = (F₀/m) / √((ω₀² − Ω²)² + (2γΩ)²) with phase lag φ = atan2(2γΩ, ω₀² − Ω²). Resonance peaks near Ω ≈ √(ω₀² − 2γ²) and the sharpness is the quality factor Q = ω₀/(2γ). The drive-strength slider sets F₀ = f·k·|x₀|.

Honest simplifications. A single linear mode: real footbridges, buoys and suspensions have many modes, nonlinear stiffness and amplitude-dependent damping. Parameters of each case are the originals'; the drive is an addition for exploring resonance and is not part of those cases.

Independent checks. RK4 is compared with the closed-form solution (homogeneous part plus steady-state particular solution), with the free-period value, and — for a driven damped case — the late-time amplitude is measured and compared with the resonance formula. A separate Python test repeats the closed forms.

Driven Damped Spring-Mass Resonance Case Studies