Swing Pumping: Parametric Resonance Lab

Pump Mode

Controls

Pump amplitude ΔL/L₀ 18%

Drive ratio ω_pump / 2ω₀ 1.00

Damping c 0.15

Live Readouts

Amplitude (peak)

0.0°

Effective length L

2.50 m

Specific energy (per kg)

0.00 J

How it works

Model the swinger as a point mass on a pendulum whose length L(t) is prescribed by the rider standing up (shortening L, raising their centre of mass) or crouching down (lengthening L) — an external, rheonomic constraint, not a free coordinate. With θ measured from the vertical, the Lagrangian is:

𝓛 = ½m(L²θ̇² + L̇²) + m g L cos θ

Applying the Euler–Lagrange equation to θ alone (L is driven, not dynamical) and dividing by mL² gives the governing ODE actually integrated below, with an added linear damping term c for air drag and chain friction:

θ̈ + 2(L̇/L)θ̇ + (g/L)sin θ + c θ̇ = 0

2(L̇/L)θ̇ is the parametric-pumping term: it behaves like time-varying damping. Shortening L (L̇<0) while moving toward the bottom injects energy; the classic technique is to stand at the bottom and crouch at the extremes, i.e. pump at twice the pendulum's own frequency.
Auto mode reproduces that real technique directly: the model toggles stand/crouch every time θ crosses zero.
Fixed-frequency drive instead forces L(t) = L₀ + ΔL·cos(ω_pumpt) explicitly, with ω_pump = 2·ratio·ω₀ (ω₀ = √(g/L₀)). Resonant growth appears only in a narrow band around ratio = 1 — the hallmark 2:1 parametric-resonance condition (a Mathieu-equation instability tongue), not a Dzhanibekov-style free-rotation effect.
Growth saturates into a steady limit cycle because sin θ detunes the resonance as amplitude grows — the same nonlinearity that keeps a real swing from flipping over the bar.

Rider (stand ↔ crouch) Fixed-length chain

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