A thin rectangular plate, simply supported at its edges, seen from above: red is up, blue down, white at rest (scaled to the recent largest displacement). It is a grid of equal masses with bending stiffness and von Kármán stretching: a bent plate stretches in its own plane, which stiffens it once it moves by about its thickness h. There is no damping.
A force in the shape of one normal mode (the driven mode) pushes the plate. In a linear plate such a force moves that mode and nothing else, so any other mode that moves got its energy through the stretching. The force is always in phase with the driven mode's own velocity, so it adds energy whatever the mode's frequency (which rises as the plate stiffens), and a controller sets its strength so that the plate's total energy follows the target.
The target is given against Enl, the energy of the driven mode at which its stretching energy equals its bending energy: well below 1 the plate is nearly linear. Ramp raises the target steadily, the same number of periods for each factor of ten, up to the top of the slider; untick it to hold the energy where it is. Reset starts again in the driven mode at 10−3 Enl. Time is counted in periods of the driven mode (linear), and Speed sets how many pass per second; at high energies the plate needs many more steps per period, and the actual speed (shown) can fall below the setting. The space bar pauses.
Plate and drive
Masses across is the number of masses across the short side; the long side gets as many as keep the cells nearly square. Aspect is the ratio of the sides. A square plate has pairs of modes at the same pitch ((1, 2) and (2, 1)), which trade energy easily; a non-square plate splits them. The default 1.3 keeps the lowest modes at least 1% apart in pitch.
The modes are products of sines, numbered (m, n) by their half waves along x and y. The list gives each one's pitch relative to the lowest. Perturbation starts the plate with a tiny random motion in every other mode, a millionth of the starting energy in all; without it the plate starts in the driven mode exactly, and by symmetry only modes with the driven mode's symmetry can then grow (for (1, 1): odd m and odd n), at first; round-off seeds the rest. Any change here starts again.
Loss drains energy from the modes, fastest at the top of the band, as a sink for the energy the cascade brings there (without it the energy has nowhere to go and spreads over every mode up to the grid's highest). Each mode's amplitude decays on its own, the top mode in τ top driven periods and a mode of pitch ω in τ top (ωtop / ω)k, k from the rate's power of ω: with ω (k = 1, the dashpot loss of the other apps) the driven mode decays in about 40 τ top on the default plate, with ω² in about 1700 τ top, with ω⁴ practically never. The drive makes up what the loss takes. The loss can be switched and changed during a run.
Units are scaled away: for the von Kármán plate only the aspect ratio and Poisson's ratio (0.3) matter once energy is given against Enl, displacement against the thickness h and time in periods.
Energy
Against time in driven periods, on a log scale in units of Enl: the total energy (black), the target (dashed), the driven mode's energy (blue) and the sum over all other modes (orange). The mode energies are the linear ones, (velocity² + ω² displacement²) / 2 per mode, averaged over the time steps of the mode map below; once stretching matters they no longer add up exactly to the total.
Modes
Every normal mode of the plate, one row each, ordered by pitch with the lowest at the bottom; the label gives (m, n) and the pitch relative to the driven mode (marked ▸). Time runs across, the whole run as in the energy chart. Colour is the mode's linear energy, (velocity² + ω² displacement²) / 2, averaged over each time step of the map (sampled 16 times per driven period), on a log scale reaching 12 factors of ten below the largest so far: dark is next to nothing, pale yellow the most. Averaging matters because a mode pushed off its own pitch by the others swings in energy within each cycle.
Hover over the map to read a mode and its energy at that time. The time steps start at half a period and double whenever the map fills, so a long run still fits.