---
title: "Chladni Plate Resonance Interactive Lab | Sandbox Physics"
description: "Find the quiet lines inside a resonance. Can a resonant mode stay quiet at a nodal driver?"
canonical_url: "https://sandboxphysics.com/labs/chladni-plate/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/chladni-plate/"
last_modified: "2026-10-18"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

Physics tutorial

## Bending makes a different spectrum

**Background** A thin solid plate resists curvature. Its bending stiffness, thickness and boundary conditions determine its natural vibrations.

**Why it matters** Connect the moving surface, quiet lines and a quantitative frequency account.

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 misconception** Every pale line predicts a measured sand pattern.

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

### Run the experiment

1. 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.

2. 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.

3. 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.

Physical conclusion

A square can support distinct modes at the same frequency. Changing aspect ratio splits them; driver position chooses their mixture.

Primary sources: [NASA SP-160 · Vibration of Plates](https://ntrs.nasa.gov/citations/19700009156) [Experimental simply supported plates](https://doi.org/10.1016/j.mex.2019.09.023)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Measure gravity, split motion into components, add contact forces and collisions, then carry the same laws into orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)