vibrational-modes-plates

This project simulates Chladni figures (the vibrational modes of plates and membranes) using finite differences in Python.

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Instructions

Clone the repository:

git clone https://github.com/DC-Creatives-Club/vibrational-modes-plates.git

Create and activate a Python virtual environment called math:

python3 -m venv ~/.virtualenvs/math
source ~/.virtualenvs/math/bin/activate

Then simply run the script to produce plots with default parameters (skip the first 10 modes):

python3 chlandi.py

To run with different parameters:

python3 chladni.py --bc neumann --N 100 --modes 9 --skip 10 --save

where

• bc are boundary conditions, either neumann or dirichlet
• N is mesh size
• modes is number of modes
• skip is the number of initial modes to skip
• --save flag saves the modes separately to a folder plots

Chladni Mode Simulation

This project simulates Chladni figures (the vibrational modes of plates) using finite differences in Python.

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1. Physical background

Elastic plate (Chladni plate)

A thin elastic plate obeys the Kirchhoff–Love equation:

$$ \rho h,\partial_{t}^{2}w + D,\nabla^{4}w = 0, $$

with

• $h$ = plate thickness
• $D = \frac{E h^{3}}{12(1-\nu^{2})}$ = bending stiffness

Modes satisfy the biharmonic eigenproblem:

$$ \nabla^{4}\phi = \mu,\phi, \quad \mu = \frac{\rho h}{D},\omega^{2}. $$

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2. Discrete Laplacian

On a 1-D grid with spacing $\Delta x$:

$$ L_{1D} = \frac{1}{\Delta x^{2}} \begin{bmatrix} -2 & 1 & 0 & \cdots \\ 1 & -2 & 1 & \ddots \\ 0 & 1 & -2 & \ddots \\ \vdots & \ddots & \ddots & \ddots \end{bmatrix}. $$

On a 2-D square grid:

$$ L_{2D} = I \otimes L_{1D} + L_{1D} \otimes I. $$

This matrix approximates $\nabla^{2}\phi$. For a plate, the biharmonic operator is $L_{2D}^2$.

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3. Boundary conditions

• Dirichlet (clamped edge): enforce $\phi = 0$ on the boundary.
• Neumann (free edge): enforce $\partial_n \phi = 0$ (zero normal derivative).

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4. Eigenvalues → frequencies

• Plate: $(L_{2D})^2 v = \mu v \implies \omega = \sqrt{\frac{D}{\rho h}\mu}$.

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5. Chladni patterns

Nodal lines are given by the zero set of $\phi(x,y)$.
Numerically: plot contour lines at level = 0 to simulate where sand would accumulate.

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6. Implementation notes

• Use scipy.sparse and eigsh for efficiency.
• For realistic clamped edges, enforce Dirichlet BCs (interior points only).
• Plot zero contours (plt.contour(..., levels=[0])) to visualize Chladni patterns.
• Increase grid resolution for finer patterns.
• Skip the first few lowest modes to see higher-order “flower” and circular patterns.