Double Spring
Two masses connected by springs in series, hanging from a fixed point under gravity. When the masses and spring constants match, energy transfers cleanly between normal modes. When they don’t, energy sloshes unpredictably — coupled oscillators on the edge of chaos. Drag the masses to set initial conditions.
The physics
A fixed ceiling connects to mass m1 via spring k1. Mass m1 connects to mass m2 via spring k2. Both masses hang under gravity. The system has two degrees of freedom (the vertical displacements of each mass from equilibrium), giving rise to two normal modes — the symmetric mode where both masses move in phase, and the antisymmetric mode where they move in antiphase.
Energy transfer
When the masses and springs are equal, energy transfers cleanly back and forth between the two masses, producing the classic beat pattern. When they differ, the energy exchange becomes irregular. Damping causes the system to gradually lose energy and settle to its equilibrium position.
Phase portrait
The phase portrait plots each mass’s velocity against its displacement. For a single undamped oscillator this is an ellipse; for coupled oscillators the trajectory is a Lissajous-like curve whose shape reveals the frequency relationships between modes.