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https://github.com/ratwolfzero/chladni_figures

Chladni Figures Simulation
https://github.com/ratwolfzero/chladni_figures

chladni chladni-figures chladni-patterns chladni-plates

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Chladni Figures Simulation

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README 

          
# Chladni Figures Simulation



This Python code simulates the nodal line patterns known as **Chladni figures** by visualizing the resonant modes of a vibrating surface. This is a computational approximation that captures the essential visual phenomenon without simulating particle dynamics.



![Chladni](Chladni_3.png)



---



## Table of Contents



- [Chladni Figures Simulation](#chladni-figures-simulation)

  - [Table of Contents](#table-of-contents)

  - [Historical Context](#historical-context)

  - [Physical Principles of Simulation](#physical-principles-of-simulation)

  - [Simulation Implementation Principles](#simulation-implementation-principles)

  - [Key Parameters](#key-parameters)

  - [Frequency Scaling Factor *k*](#frequency-scaling-factor-k)

  - [Damping Factor γ](#damping-factor-γ)

  - [Mode Superposition and γ](#mode-superposition-and-γ)

  - [Patterns for Different γ Values](#patterns-for-different-γ-values)

  - [Usage](#usage)

  - [Controls](#controls)

  - [Limitations](#limitations)

  - [References](#references)



---



## Historical Context



Ernst Chladni (1756–1827), often called the *father of acoustics*, studied how vibrating plates caused **particles like sand or powder to accumulate along nodal lines**—regions where the plate remains stationary. These patterns, now known as **Chladni figures**, visually reveal the **standing wave patterns** on the plate.



In his experiments:



- A thin metal plate is fixed at its center or edges.

- It is vibrated with a violin bow at different frequencies.

- Fine particles move away from areas of high vibration and collect along **nodal lines**, producing beautiful geometric patterns.



Chladni figures were key in understanding **vibrational modes** and laid foundations for acoustics, wave physics, and modern mechanical engineering.



---



## Physical Principles of Simulation



The displacement field of a **single vibrational mode** \$(m,n)\$ on a rectangular plate of size \$L\_x \times L\_y\$ is given by:



$$

Z_{mn}(x,y,t) = A \sin\left(\frac{m \pi x}{L_x}\right) \sin\left(\frac{n \pi y}{L_y}\right) \cos(2 \pi f_{mn} t)

$$



where:



- \$m, n \in \mathbb{N}\$ are the number of nodal lines along the \$x\$ and \$y\$ axes, respectively.

- \$A\$ is the amplitude of oscillation.

- \$f\_{mn}\$ is the eigenfrequency of the \$(m,n)\$ mode:



$$

f_{mn} = k \sqrt{\left(\frac{m}{L_x}\right)^2 + \left(\frac{n}{L_y}\right)^2}

$$



- $k$ sets the overall frequency scale. In real plates it depends on material properties, but in this simplified simulation we take $k=1$.



The **nodal lines** of this mode, defined by where \$Z\_{mn}(x,y,t) = 0\$, are where particles accumulate in real experiments to form the classic Chladni figures.



---



## Simulation Implementation Principles



Just as a real violin bow applies a nearly single-frequency drive, the simulation uses a **single driving frequency** \$f\$, summing the response of all modes weighted by a resonance term (modulating the amplitude) that includes the damping factor \$\gamma\$:



$$

Z(x,y; f) = \sum_{m=1}^{M} \sum_{n=1}^{N} \frac{\sin(m \pi x) \sin(n \pi y)}{(f - f_{mn})^2 + \gamma^2}.

$$



Here, \$\gamma\$ controls the influence of each mode: small \$\gamma\$ produces a sharp resonance, exciting primarily a single mode, while larger \$\gamma\$ broadens the response, allowing multiple nearby modes to contribute.



1. **Visualization:**

   - The absolute displacement is visualized as colormap with|Z|^0.2 to enhance contrast of nodal lines.

   Dark regions approximate nodal lines; bright regions are anti-nodes.



   - The resulting superposition (modulated by γ) roughly captures the richness of real Chladni patterns.

   While the simulation assumes a nearly single-frequency drive (like a bow), γ mimics real-world plate imperfections that broaden resonances, causing several nearby modes to be excited simultaneously.



   - The title displays the current driving frequency and the eigenfrequency of the **closest resonant mode(s)**. **Important:** the resulting pattern is a *superposition* of all significantly excited modes at frequency `f`. A small `γ` results in a pattern dominated by one mode, while a large `γ` blends several modes into a more complex, often asymmetric pattern.



2. **Approximation and Model Choice:**



   - **Particles are not explicitly simulated**.

   - The mathematical model uses an ideal flexible **membrane** (like a drumhead) under tension, with sinusoidal eigenfunctions and eigenfrequencies proportional to \$\sqrt{m^2 + n^2}\$. This simplifies the physics of rigid **plates** with bending stiffness.

   - The membrane model was chosen for **computational efficiency**, allowing real-time interactive exploration. This approach successfully captures the qualitative behavior and visual essence of modal patterns, reproducing the kinds of figures observed when a real plate is driven by a bow.



---



## Key Parameters



| Parameter    | Description                          | Typical Effect                                                                                                                                          |

| ------------ | ------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------- |

| `max_mode`   | Maximum mode numbers \$M\$ and \$N\$ | Higher values allow more complex patterns, slower computation                                                                                           |

| `gamma`      | Damping factor in mode contributions | Small γ → sharp, symmetric patterns; large γ → broad, potentially asymmetric patterns; controls resonance width, lifts degeneracy, mimics imperfections |

| `k`          | Frequency scaling factor             | Adjusts eigenfrequency scale                                                                                                                            |

| `resolution` | Grid resolution                      | Higher → smoother visual patterns, slower computation                                                                                                   |

| `init_freq`  | Initial driving frequency            | Starting frequency when simulation launches                                                                                                             |



---



## Frequency Scaling Factor *k*



In the eigenfrequency expression



$$

f_{mn} = k \sqrt{\left(\frac{m}{L_x}\right)^2 + \left(\frac{n}{L_y}\right)^2},

$$



the parameter **k** sets the **overall frequency scale** of the simulation.



1. **Physical Meaning:**



   - For a real membrane or plate, the eigenfrequency depends on **geometry** (plate dimensions) and **material properties** (tension, density, stiffness).

   - These details are collapsed into a single proportionality constant. In this simplified model, that constant is represented by **k**.



2. **Role in the Simulation:**



   - **Larger k** → shifts all resonances to **higher frequencies**.

   - **Smaller k** → shifts all resonances to **lower frequencies**.

   - Importantly, **k does not change the shape of the modal patterns**—only their placement along the frequency axis.



3. **Interpretation:**



   - Think of **k** as a **tuning knob** that lets you control where in the frequency range the resonances appear.

   - While **γ** governs how sharply modes appear and blend, **k** simply sets the “frequency scale” of the entire system.



4. **Guidelines:**



   - Adjust **k** to place resonances in a convenient range for exploration.

   - Once chosen, k can usually remain fixed, while **γ** and the driving frequency `f` are varied interactively.



---



## Damping Factor γ



In the simulation, the total displacement field is computed as:



$$

Z(x,y; f) = \sum_{m=1}^{M} \sum_{n=1}^{N} \frac{\sin(m \pi x) \sin(n \pi y)}{(f - f_{mn})^2 + \gamma^2}

$$



Here, **γ** is the **damping factor**. The following graph illustrates its primary function: controlling the resonance width and amplitude peak. A smaller γ results in a sharper, taller response, meaning only frequencies very close to the resonant frequency $f_{mn}$ will excite that mode. A larger γ creates a broader, shorter response, allowing multiple nearby modes to contribute to the pattern simultaneously.



![Chladni](Resonance_and_Damping_Factors_Graph.png)



Its role in the simulation is multi-faceted:



1. **Resonance Width:**

    - Small γ → narrow resonance: only modes very close to \$f\$ contribute.

    - Large γ → wide resonance: multiple modes contribute simultaneously.



2. **Mode Superposition & Symmetry:**

   In an ideal plate, some vibration modes are **degenerate** (same eigenfrequency), e.g., $f_{12} = f_{21}$ on a square plate. The **damping factor γ** controls how modes combine:



    - **Very small γ** excites one mode or a degenerate pair nearly equally, producing **highly symmetric, patterns**.

    - **Increasing γ** allows nearby non-degenerate modes to contribute, introducing **subtle asymmetries**.

    - **Large γ** excites many overlapping modes, yielding **asymetric, complex patterns**



3. **Mimicking Physical Imperfections:**

    - Real plates have variations in thickness, material, or boundaries.

    - Increasing γ reproduces the effect of these imperfections by broadening resonance peaks.



4. **Amplitude Control:**

    - Maximum mode contribution at resonance (\$f=f\_{mn}\$) is \$1/\gamma^2\$.

    - Smaller γ → sharper nodal lines, higher amplitude.

    - Larger γ → blended, lower-amplitude patterns.



5. **Guidelines:**

    - γ ≈ 0.01–0.03 → sharp, symmetric patterns

    - γ ≈ 0.05–0.1 → slight asymmetry

    - γ > 0.1 → diffuse, asymmetric patterns



---



## Mode Superposition and γ



| γ Regime               | Mode Contribution                                                                         | Resulting Pattern Description                                                                            |

| ---------------------- | ----------------------------------------------------------------------------------------- | -------------------------------------------------------------------------------------------------------- |

| **Small (≈0.01–0.03)** | Single dominant mode **or degenerate pair**                                               | Sharp, symmetric patterns with well-defined nodal lines.                                                 |

| **Medium (≈0.05–0.1)** | Degenerate or near-degenerate modes blend; nearby non-degenerate modes start contributing | Slight asymmetry emerges; patterns begin to merge subtly.                                                |

| **Large (>0.1)**       | Multiple overlapping modes contribute significantly                                       | Diffuse, complex, asymmetric patterns; nodal lines are less distinct|



> This illustrates how nodal lines deform as γ increases, simulating physical imperfections in the plate.



---



## Patterns for Different γ Values



| γ Range          | Description                                                                                                                               |

| :--------------- | :---------------------------------------------------------------------------------------------------------------------------------------- |

| **γ ≈ 0.01–0.03** | Sharp, near-perfectly symmetric patterns. Idealized behavior of a perfect plate.                                                          |

| **γ ≈ 0.05–0.1**  | Patterns start blending, showing slight asymmetry. Represents a plate with minor imperfections or energy loss.                            |

| **γ > 0.1**       | Strong superposition of modes results in diffuse, asymmetric nodal lines. Simulates a system with significant damping or strong imperfections. |



> **Tip:** Adjust γ in your simulation to see the transition from symmetric to complex, realistic patterns.



---



## Usage



- Use the frequency slider to explore modes.

- Experiment with damping γ to see symmetry/asymmetry effects.



---



## Controls



| Control               | Function                       |

| :-------------------- | :----------------------------- |

| Frequency Slider      | Adjust driving frequency.      |

| Next Resonance Button | Jump to next higher resonance. |

| Scan Button           | Sweep frequency automatically. |

| Stop Scan Button      | Stop automatic scanning.       |



---



## Limitations



- Field-based simulation; **does not model particle dynamics**.

- Only finite `max_mode` included.

- Damping γ is uniform; real plates have non-uniform damping.

- Geometry is idealized square plate.

- Uses a membrane model for computational efficiency, which simplifies the physics of a true plate with bending stiffness.



---



## References



- Paul Bourke, *Chladni Figures*, [http://paulbourke.net/geometry/chladni/](http://paulbourke.net/geometry/chladni/)

- Wikipedia: [Chladni Patterns](https://en.wikipedia.org/wiki/Chladni_pattern)

- Standard vibration theory for rectangular plates