Strike to hear the whole decay.
Equal masses between two walls, joined by springs that stiffen as they stretch (tension k d + k3 d³), shown in real time: to scale on top, and below the same displacements drawn sideways. Strike gives one mass, or one normal mode, the energy on the slider.
Weak strikes ring as independent normal modes. Strong ones trade energy between the modes, chaotically for three or more masses. Ghost adds a second chain started a tiny distance away: in chaotic motion the two separate exponentially. The space bar pauses.
Mass, spring constants k and k3, the rest spacing a (drawing only), and the decay time τ of the amplitude (drag on the masses; 0 for none). Apply restarts the chain at rest. The cubic term matches the linear one at the stretch d0 listed below; the motion turns nonlinear at energies around k d0².
Charts of the chain on the left, on the same clock. Click a chip to show or hide one; hover to read values, click to pin the cursor.
Mode bars show the energy in each normal mode now, Mode energies the same over time. Energy error should stay tiny. Lyapunov is the rate (1/s) at which nearby motions separate: positive for chaos, near 0 for regular motion. Separation (with the ghost on) is the ghost's distance from the chain.
A copy of the same strike, run fast enough to be heard: Pitch sets where the lowest normal mode lands, and the whole decay plays in a few seconds. While this panel is open each Strike plays a note; Replay plays the last one again with the current settings.
Save WAV saves the last note as heard (16-bit mono, normalized). Signal chooses what is heard. Chaotic motion sounds noisy and settles into steady tones as the energy decays. The charts span the whole note: spectrogram, Lyapunov exponent, energy, and λ against energy collected over notes.
A map of the motions at one energy, without damping. Two masses: each time mass 1 passes its rest position moving right, mass 2's displacement and momentum are plotted. Three masses: only motions with masses 1 and 3 moving alike, plotted each time mass 2 passes rest. Regular motions trace closed curves; chaotic ones scatter. The outline bounds what is reachable.
The chain's own crossings are dark rings. Follow tracks the chain (real time) or the playing note (sped up). Hover to pick out an orbit; click to start one there, which with Click to strike also strikes the chain.