M066 · A driven bending plate

Chladni Plate Resonance

Move a point driver over a three-dimensional thin plate. Tune a resonance, split a degenerate pair by stretching the rectangle, and suppress a mode by driving on its node.

Selected natural frequency $0\,\mathrm{Hz}$

Selected / fundamental frequency $0$

Gold-point displacement $0\,\mathrm{\mu m}$

Gold-point RMS displacement $0\,\mathrm{\mu m}$

Actual height magnification $0$

Displayed mechanical energy $0\,\mathrm{\mu J}$

Mean input power $0\,\mathrm{\mu W}$

Excitation & geometry

Make a resonance visible

Selected horizontal mode

m

2.00\,

Selected vertical mode

n

2.00\,

Drive / fundamental frequency

f_d/f_{11}

4.00\,

Height / width

b/a

1.00\,

Plate thickness

h

2.00\,\mathrm{mm}

Driver horizontal fraction

x_d/a

0.32\,

Driver vertical fraction

y_d/b

0.36\,

Modal damping ratio

\zeta

0.01\,

Height envelope / shorter side

H_{\rm env}/L_{\min}

0.32\,

Uniform aluminum-like parameters are assumed: isotropic elasticity, no membrane tension. All four edges enforce zero displacement and bending moment, while slope may be nonzero. The bronze perimeter marks the rest plane; vertical thickness is schematic.

Twenty-five sine-product modes, driven by a prescribed sinusoidal point force with equal modal damping ratios. Only the steady response is shown, with no switch-on transient. Change the selected indices, then use Tune to selected mode.

Physical coordinates lie in the rest surface; positive displacement points upward. The height envelope sets one fixed-in-time magnification, reported in the readouts. The map shows long-time RMS motion and the trace uses physical units. Slow playback uses fundamental periods, not wall-clock physical time.

Small transverse motion and linear elasticity only. No sand or friction dynamics, sound synthesis, air loading, real material calibration, free-edge plates, clamped plates or contact forces. Finite modal truncation limits spatial detail.

Simply supported isotropic Kirchhoff–Love plate, 25 bending modes and specified modal damping. The amplitude map is calculated displacement, not transported sand; free and clamped plates require different modes.

Start with the essentials

Focus question: Can a resonant mode stay quiet when its driver sits on a node?
One-sentence intuition: Resonance needs both frequency matching and spatial coupling. A nodal driver has zero coupling to that mode.

Core mathematical model

Specified parameters

E=70\,\mathrm{GPa},\quad\nu=0.33,\quad\rho=2700\,\mathrm{kg/m^3},\quad F_0=0.2\,\mathrm{mN}

Uniform material parameters are teaching assumptions. The small force keeps this linear plate in its small-deflection range.

Bending equation

D\nabla^4w+\rho h\,\ddot w=p,\qquad D=\frac{Eh^3}{12(1-\nu^2)}

Kirchhoff–Love theory, uniform thickness and isotropic linear elasticity. The prescribed modal damping is added after projection.

Simply supported rectangle

\phi_{mn}=\sin\frac{m\pi x}{a}\sin\frac{n\pi y}{b},\quad \omega_{mn}=\pi^2\sqrt{\frac{D}{\rho h}}\left(\frac{m^2}{a^2}+\frac{n^2}{b^2}\right)

Positive integer indices start at one. At the edges displacement and normal bending moment vanish; the slope is not clamped.

Complex modal response

Q_{mn}=\frac{F_0\phi_{mn}(x_d,y_d)}{M_{mn}(\omega_{mn}^2-\Omega^2+2i\zeta\omega_{mn}\Omega)},\quad M_{mn}=\frac{\rho hab}{4}

The displayed displacement is the real part of the modal sum times the complex driving phase. All retained modes participate.

An independent power account

\overline P_{\rm in}=-\frac{F_0\Omega}{2}\sum_{mn}\phi_{mn}(x_d,y_d)\operatorname{Im}Q_{mn}=\sum_{mn}\zeta M_{mn}\omega_{mn}\Omega^2|Q_{mn}|^2

In steady state, mean input power equals mean viscous dissipation. This does not assert constant instantaneous energy.

Common difficulties

An amplitude map is not sand

Typical misconceptionEvery pale line predicts a measured sand pattern.

Better mental modelThis map is computed RMS displacement in a linear model. Granular motion, friction, air and real support compliance are outside its scope.

Run the experiment

01
Drive on a node
Compare Four lobes with Drive on a node. Keep the selected frequency unchanged.
What to observe: The selected mode is suppressed, although other off-resonant modes can still move.
02
Split equal frequencies
Try Degenerate pair, then Break the symmetry. Use Tune to selected mode after changing the aspect ratio.
What to observe: The two swapped mode indices share a square frequency but separate on a rectangle.
03
Separate physics from display
Double thickness, then vary only the height display gain.
What to observe: Thickness doubles natural frequency. Display gain leaves the physical trace and power unchanged.

Primary sources: NASA SP-160 · Vibration of Plates Experimental simply supported plates