M066 · A driven bending plate

Chladni Plate Resonance

Does driving a plate at one of its resonant frequencies always make that pattern vibrate? No. Resonance needs the right frequency and a driver placed where that pattern moves. On one of the pattern's quiet lines, the driving force does no work on that pattern, so it stays still even at its own frequency; other patterns, off resonance, still move a little. On a square plate two patterns can share one frequency, and the driver's position decides how they mix.

Move a point driver over a thin plate in three dimensions. Tune to a resonance, stretch the plate to split two modes that share a frequency, and quiet a mode by driving it on one of its nodes.

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

Edge support

b_{\rm clamp}

0.00\,

Horizontal mode · auto-tune

m

2.00\,

Vertical mode · auto-tune

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\,

The plate is assumed uniform, with aluminum-like properties, isotropic elasticity and no membrane tension. Simply supported edges have zero displacement and bending moment; clamped edges have zero displacement and slope. The bronze border marks the rest plane, and its vertical thickness is schematic.

The plate uses twenty-five exact sine-product modes for simple supports, or twenty-five Rayleigh–Ritz modes for clamped edges, all with the same damping ratio, driven by a prescribed sinusoidal point force. Only the steady response is shown, without the transient after switch-on. Change the selected indices, then use Tune to selected mode.

Physical coordinates lie in the rest surface, and positive displacement points up. The height envelope sets one magnification for the whole playback, shown in the readouts. The map shows the long-time RMS motion, and the trace uses physical units. Slow playback counts time in fundamental periods, not in real seconds.

Illustrative grains use a seeded random walk whose step variance rises with local calculated RMS displacement. The initial pattern has 850 preparation steps. Diffusion steps, grain size and vibration amplitude are rescaled independently; this model does not predict real sand lift-off, friction, impacts or air-driven inverse patterns. Clamped-mode indices label dominant trial functions and need not count exact nodal lines. High Ritz modes converge more slowly.

The model covers only small transverse motion and linear elasticity. Sand impacts, friction, sound synthesis, air loading, real material calibration, free plate edges and contact forces are not included. Keeping a finite number of modes limits the spatial detail.

The isotropic Kirchhoff–Love plate uses 25 exact modes for simple supports or 25 approximate Rayleigh–Ritz modes for clamped edges, with specified modal damping. Illustrative grains follow a rescaled, amplitude-dependent random walk.

Start with the essentials

Focus question: Can a resonant mode stay quiet when its driver sits on a node?
One-sentence intuition: A mode resonates only when the drive matches its frequency and pushes where the mode moves; a driver on a node cannot couple to that mode at all.

Core mathematical model

Specified parameters

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

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

Bending equation

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

This is Kirchhoff–Love theory for a plate of uniform thickness made of an isotropic, linear elastic material. The prescribed modal damping is added after projecting onto the modes.

Modes of a simply supported rectangle

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

The indices are positive integers starting at one. At the edges the displacement and the normal bending moment are zero, but the slope is not clamped.

Complex response of each mode

\begin{gathered}Z_k=\omega_k^2-\Omega^2+2i\zeta\omega_k\Omega\\Q_k=\frac{F_0\phi_k(x_d,y_d)}{M_k Z_k}\end{gathered}

The displacement is the real part of the modal sum times the complex driving phase. Modal mass is one quarter of total plate mass for sine products, and total mass for the area-normalized Ritz modes.

Input power equals dissipated power

\begin{gathered}W_d=\sum_k\phi_k(x_d,y_d)Q_k\\\overline P_{\rm in}=-\frac{F_0\Omega}{2}\operatorname{Im}W_d\\\overline P_{\rm loss}=\sum_k\zeta M_k\omega_k\Omega^2|Q_k|^2\end{gathered}

In steady state, the average power fed in equals the average power lost to viscous damping. This does not mean the energy is constant at every instant.

Clamp a plate

w=0,\qquad \partial_n w=0

Clamping fixes slope and displacement. Twenty-five admissible polynomial products give Rayleigh–Ritz approximations; indices name dominant trial functions. The square fundamental frequency parameter is about 35.99, versus 19.74 for simple supports.

Common difficulties

What the gold grains represent

Typical misconceptionThe simulated grains predict a real sand experiment's migration speed and exact pattern.

Better mental modelPlate grains illustrate a random walk with lower motility where vibration is weak. They use a separate dimensionless clock. Real lift-off, friction, impacts, air flow and support flexibility need additional models. The drum shows displacement directly.

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 goes quiet, but other modes that are off resonance 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: Two modes with swapped indices share one frequency on the square but separate on a rectangle.
03
Separate physics from display
Double thickness, then vary only the height display gain.
What to observe: Doubling the thickness doubles the natural frequency. The display gain leaves the physical trace and the power unchanged.

Primary sources: NASA SP-160 · Vibration of Plates Grain diffusion · Abramian et al. (2025)Experimental simply supported plates

Chladni Plate Resonance

Bending makes a different spectrum

Core mathematical model

Specified parameters

Bending equation

Modes of a simply supported rectangle

Complex response of each mode

Input power equals dissipated power

Clamp a plate

Common difficulties

What the gold grains represent

Run the experiment

Drive on a node

Split equal frequencies

Separate physics from display

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