3D Chladni Plate Resonance: Blender Shaker Bench, Vibrating Plate and Sand Grains

Setting up the bench...

Presets

Live readouts

Graphs

Response of the plate: rms displacement against drive frequency (log scale). Red line = current frequency.

Chladni law: resonance frequency against m2 + n2 (dots are modes of this plate)

Fraction of the sand resting where plate acceleration is below g, against time

Vibration amplitude along a line of the plate; the yellow line is the lift-off level g / omega^2

Numerical checks

running...

How to use

Pick a preset: the frequency jumps to a resonance of this plate and the sand organises itself within a second or two. Drag the red dot on the plate (or the x and y sliders) to move the drive point: for degenerate pairs this changes how the two modes mix, and at a nodal point of a mode that mode cannot be excited. Move the frequency slider slowly, or press Auto sweep, and watch graph 1: the sand only forms a sharp figure close to a peak. Try a thicker plate, a smaller side or aluminium and see every resonance shift. Raise Q for sharper lines and lower force for fatter ones. Space pauses, R scatters the sand again.

Drag to orbit, wheel or pinch to zoom, camera buttons jump to presets. The plate in the 3D page is displaced in slow motion (a visual 1.5 Hz) with the true mode shape. The shaker bench, plate and clamp were modelled in Blender with baked PBR textures; all dynamics run in JavaScript.

The science

The plate. A thin elastic plate of side L, thickness h, Young modulus E, Poisson ratio nu and density rho obeys the Kirchhoff equation with a harmonic point force F cos(omega t) at (x_d, y_d):

rho h w_tt + D (w_xxxx + 2 w_xxyy + w_yyyy) = F delta(x - x_d) delta(y - y_d) cos(omega t), D = E h^3 / (12 (1 - nu^2))

Modes. For a square plate with simply supported edges (w = 0 and no bending moment) the modes are exact products of sines, phi_mn = 2 sin(m pi x/L) sin(n pi y/L), with omega_mn = pi^2 sqrt(D / (rho h)) (m^2 + n^2) / L^2, f_11 = pi sqrt(D / (rho h)) / L^2 so the resonances follow the Chladni law f_mn = f_11 (m^2 + n^2) / 2 (graph 2). Modes (m,n) and (n,m) share the same frequency; how they mix depends on where you drive: the amplitude of each is proportional to phi_mn(x_d, y_d), which is why moving the red dot changes the figure.

Steady state. Expanding w = sum q_mn phi_mn gives for every mode a damped oscillator, q_mn = F phi_mn(x_d,y_d) / [ rho h L^2 (omega_mn^2 - omega^2 + 2 i zeta omega_mn omega) ], with zeta = 1/(2Q). The sum over 14 x 14 modes is evaluated on a 96 x 96 grid. The area-mean of |w|^2 is exactly sum |q_mn|^2 because the modes are orthonormal, which gives the response curve of graph 1 cheaply.

Sand. A grain on the plate leaves the surface when the plate acceleration exceeds gravity, omega^2 |w| > g. Such grains hop a small random distance in a random direction, longer the more the acceleration exceeds g; grains where omega^2 |w| < g stay put. A random walk whose step size depends on position piles up where the steps vanish, i.e. on the nodal lines, and the figure sharpens as the drive grows (the yellow region is where a < g). This is the lift-off picture first given by Faraday and it explains why a weak drive gives wide, fuzzy lines.

Honest simplifications. (1) Simply supported edges: real Chladni plates are held at the centre with free edges, which have no closed-form modes (the Ritz method is needed) and give different figures and ratios such as the famous (m + 2n)^p law; the principle is the same. (2) Damping is modal viscous with one Q for all modes. (3) The grains are point hoppers with no collisions, air streaming or sliding friction; the hop length is a visual scale. (4) Real plates have a mass-loaded drive point and a finite coupling; the force amplitude here is that of an ideal point shaker.

Checks run in the page. The scanned resonance peak against the closed form for f_11; the peak ratios against (m^2+n^2)/2 for five modes; the quality factor measured from the half-power width of the (1,1) peak against the set Q; orthonormality of the modes on the grid; and the sand concentration on the quiet region against its area share.