Chladni figures

The plate uses Kirchhoff–Love theory and the Rayleigh–Ritz method with free outer edges. At each frequency, 80 modes are combined with their amplitudes and phases. A moving centre prescribes vertical displacement; in force mode the centre has fixed displacement and slopes. These are two different fixture conditions.

Sand uses a qualitative stochastic model: drift is zero, while random jumps (noise multiplier 20) are stronger in high-vibration regions. Movement slows in quiet areas, so sand accumulates there. Collisions and bouncing are not simulated. Amplitude is normalized. Sand rearrangement time is not physically calibrated. This is not a hertz-accurate prediction of a real experiment.

Frequency control: 10–5,000 Hz. For the square and circle, orders 15 and 17 were compared only below 1500 Hz: natural frequencies changed by up to 1.5% with a moving centre and 5.3% with a fixed centre. Triangle convergence is not established: higher orders are sensitive to ill-conditioned matrices. Accuracy above 1500 Hz and agreement with a real experiment have not been verified; the current set contains 80 modes.