A thin rectangular plate, simply supported at its edges. The card shows one of three views. Modes (the default) has one cell per normal mode, m half waves along x and n along y, coloured by the mode's energy (log scale, smoothed over a few periods), the driven mode outlined; hover to read a cell. Vibration shows how much each point of the plate moves, the RMS displacement over the last few periods, white to dark red. Plate shows the plate from above as it moves, red up, blue down, white at rest; it changes many times a second and flickers. In the last two the circle is the listening point for the spectrogram and the sound; click to move it. The plate starts paused: press Run. 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 (set by the Ramp slider, 5 to 2000), up to the top of the slider (103 Enl); untick it to hold the energy where it is. With very fast ramps the controller (time constant 2 periods) falls behind the target. 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.
Listen
The run is recorded at the listening point (the circle on the plate) as it is computed, and can be played back from any time while the computation goes on. Time is sped up so the driven mode sounds at the chosen pitch: one driven period of the plate lasts 1/60 s of sound at 60 Hz, so playback goes through 60 periods each second, usually faster than they are computed. If it reaches the end of what has been computed it waits there and goes on (with a short fade) once about 1.5 s of sound is ready, so it is heard in stretches; to hear it unbroken, let the run compute ahead first (or raise Speed) and then play.
Click the energy chart, the spectrogram or the mode map to choose where playback starts (dashed line); the solid line is what is being heard. Velocity is heard by default (closer to the sound a plate radiates, brighter than displacement); the spectrogram shows the same signal. Steady level divides by a slowly released peak follower, so the sound stays at one level while the energy rises a hundred thousand times and what you hear change is the timbre; as computed uses one scale for the whole run (set by the loudest part recorded when Play is pressed), so the start is faint.
The recording keeps 512 samples per driven period, so frequencies up to 256 times the driven pitch are heard (15 kHz at 60 Hz).
Spectrogram
The signal chosen under Listen (velocity by default) at the listening point (the circle on the plate; click the plate to move it), as a pickup there would see it: what one hears, rather than the energy of each mode. A mode with a node at the point is missing here even when it holds energy; the mode map below sees every mode. Frequency is in units of the driven mode's linear pitch, on a log scale; the ticks on the right mark every mode's linear pitch (the driven mode's in blue). Time runs across, the whole run, as in the charts above.
Each column is the spectrum of the last 32 driven periods (so lines 1/32 of the driven pitch apart are just told apart, and changes show with a lag of about 16 periods), one every 4 periods. Per time colours each column relative to its own loudest frequency, so the shape of the spectrum shows at every energy; off, colours are relative to the loudest moment of the run, so the quiet start is dark. 80 dB from black to pale yellow. A line that moves up as the energy rises is a mode stiffened by the stretching; lines at sums and multiples of the driven pitch are its harmonics, bound to it; a broad smear is the plate's motion no longer made of a few clean tones.
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.