Wilberforce Pendulum

A Wilberforce pendulum couples vertical motion z and twist θ about the vertical axis. This linear model measures z relative to static equilibrium and starts with zero velocities. The side view shows vertical motion; the top view shows twist. Drawing dimensions are illustrative.

Potential energy V=½kz²+½κθ²+k_czθ, with k_c=g√(kκ), 0≤g≤0.95. Here g is a dimensionless coupling, not gravity. The total includes the signed coupling energy; vertical and torsional energies alone need not add to it. Damping removes energy at rate γ(mv²+Iω²).

Match frequencies sets κ=Ik/m. The two coupled normal modes still have different frequencies. The displayed modal beat period is 2π/(Ω_high−Ω_low), including the damping shift; it is absent without coupling or an oscillatory mode pair. It characterizes interference, not a guarantee of full energy transfer for arbitrary initial conditions or strong damping.

Motion is evaluated analytically in mass-weighted normal coordinates, including overdamped modes. Step advances0.1s. Pause stops simulation time; parameter changes reset the result. Extra mode/energy values are expandable. Linear ideal springs, uniform velocity damping, no lateral swinging or nonlinear material behaviour. Large twists may exceed the validity of the linear model. A hidden page pauses until Resume is pressed. Playback advances at most 0.05 s per frame. After one simulated hour, Replay restarts with the same settings.

Inputs immediately show the initial position and predicted energy exchange. The dashed vertical marker indicates the current time in a precomputed window, including future values. The window moves in blocks, up to 20 s, shortened for fast modes. Edit Time to inspect any instant from 0 to 3600 s; it pauses playback. Reset restores the current initial inputs and keeps all views. Only exact input strings are stored, never motion, time or plots.