The plate
A square plate as a grid of equal masses on vertical rails, moving up and down only, seen from above: red is up, blue down, white at rest (scaled to fit). Each mass resists bending (κ, on how far it sits off the mean of its four neighbours). The edges are simply supported (held at zero height but free to turn there), clamped (held and kept level), or free. A free plate would also slide and tilt as a whole; that motion is removed (the plate is followed in its own frame), and the readout says how much of a strike's energy it took. A bent plate also stretches in its own plane (von Kármán), and the in-plane stress stiffens it: this is the nonlinearity. It matters once the plate moves by about its thickness h, so weak strikes ring as independent normal modes, while strong ones start sharp and glide down in pitch as they decay, and the strongest spread their energy over many modes.
Strike gives the energy on the slider as a hammer blow (a short pyramid bump of velocity, one spacing of the 8 × 8 plate wide on each side) or with a mallet (mass M, contact stiffness K in Parameters) that presses into the plate over the same bump through a Hertz contact (force K δ^1.5) and lifts off. The point is in metres from the lower left corner; click the plate to move it there. The cross marks the strike point, the circle the listening point. The space bar pauses.
Parameters
Mass, bending κ, the thickness h (it sets the stretching, as the amplitude at which it matters; 0 for a linear plate), the spacing a, the decay time τ of the amplitude (drag on the masses, all modes alike; 0 for none), and τ top, the decay time a frequency-dependent loss (dashpots alongside the grid springs) gives the highest modes (at 8 × 8 masses), less for lower ones (0 for none). Apply restarts the plate at rest.
The values are for the plate with 8 masses per side. Masses per side refines the same plate: same size, strike energies and decay, with each mass scaled with the area of its cell and κ inversely (the thickness stays), so the low modes stay put and more masses add higher ones.
The stretching follows the von Kármán plate equations: the in-plane stress comes from a stress function, solved on the grid at every step. A simply supported edge may move outward but not along itself; a free edge is free in its own plane too. The nonlinear scale below is the energy of the lowest mode at which its stretching energy equals its bending energy.
With free edges the bending energy also includes the plate's twisting term (Poisson's ratio 0.3), which sets the edge conditions; the modes are computed, and their frequencies converge to plate theory (Leissa's tables) as the spacing squared (the lowest is 2.6% low at 9 × 9, 0.2% at 33 × 33). On a free plate the outer masses sit on the edge, carrying half a cell's mass (a quarter at the corners), so the free plate's side is one spacing less than the number of masses. Clamped edges hold the slope at zero too; the modes are computed (no (m, n)), and the plate is stiffer than a simply supported one, with overtones about as widely spread (plate theory: 1, 2.04, 3.01, 3.66, ...). For a simply supported plate, the normal modes are products of sines, numbered (m, n) by their half waves along x and y. Their frequencies go as m² + n² for the low ones (the continuum plate), so (1, 2) and (2, 1) share a pitch, and the ratios to the lowest, 2.5, 4, 5, 6.5, 8.5, ..., are not harmonic. The derived values list the lowest modes and how close the grid is to the continuum.
Real time
Charts of the plate on the left, on the same clock. Click a chip to show or hide one; hover to read values, click to pin the cursor.
Lyapunov is the rate (1/s) at which nearby motions separate, measured with a hidden second plate kept a tiny distance away: positive for chaos, near 0 for regular motion (a linear plate, or a weak strike). Shown over the last 10 s and since the strike; the first 5 s are left out of the y range. Energy error should stay tiny.
Sped up · listen
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.
Signal chooses what is heard, at a listening point (x, y in metres, between masses by interpolation). Save WAV saves the last note as heard (16-bit mono, normalized). The spectrogram has dashed lines at the normal-mode pitches. The Lyapunov chip shows the exponent over the last 10 s of the plate's own time through the note, and λ vs E collects it against the energy over notes, earlier notes faded (Clear starts over); both need a hidden second plate run along, which doubles the time to compute a note, so they are off at first. Notes are anti-alias filtered (motion above 20 kHz at 48 kHz is removed rather than folded back); a note says when the top modes are above that and so are not heard.