Chladni plate with Neumann conditions

In summary, the conversation discusses simulating a vibrating plate with free edges by using a matrix representing the Laplacian operator. The eigenvectors of the matrix correspond to the state of the plate, and an article suggests using a different matrix to find resonant states for a plate with free edges. The question is then raised on how to build such a matrix with Neumann conditions, and it is suggested to think about why the matrix has its current form in order to make changes.
  • #1
Smollett
2
0
Hi there,
I'm trying to simulate a vibrating plate with free edges.
If i consider a consider a plate with fixed edges, the eigenvectors of the matrix bellow (which repesents the Laplacien operator) with S as a nxn tridiagonal matrix with -4 on the diagonal and 1s on either side (making the following a n2 by n2 matrix representing a plate with n2 points).

Screen_Shot_2016_06_08_at_13_22_00.png


correspond to the state of the plate (the following example is for a plate simulated by a 4 by 4 grid), so i can generate the following images.

figure_3.png


I found an article online (in french) which indicates that it is possible, using the same method but a different matrix to find the resonant states of a plate with free edges (Neumann conditions).
My only question is how to build a matrix with such conditions?

Has anyone encountered this problem before?
 
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  • #2
Think a bit about why the matrix looks the way it does. That should let you argue for how it changes.
 
  • #3
Thanks, I know why it looks that way, I made it.
 

1. What is a Chladni plate with Neumann conditions?

A Chladni plate with Neumann conditions is a type of acoustic experiment that involves creating patterns on a metal plate by vibrating it at specific frequencies. The Neumann boundary conditions refer to the way the edges of the plate are fixed, allowing for more precise control over the vibration patterns.

2. How does a Chladni plate with Neumann conditions work?

A Chladni plate with Neumann conditions works by using a transducer to vibrate the metal plate at specific frequencies. The fixed edges of the plate allow for standing waves to form, creating intricate patterns on the surface of the plate. These patterns can be visualized by sprinkling sand or other small particles on the plate.

3. What is the purpose of a Chladni plate with Neumann conditions?

The purpose of a Chladni plate with Neumann conditions is to study the behavior of sound waves and how they interact with different materials. This experiment can help scientists better understand the physics of sound and can also be used to create visual representations of sound frequencies.

4. What are the applications of a Chladni plate with Neumann conditions?

The applications of a Chladni plate with Neumann conditions include studying the acoustic properties of different materials, creating visual representations of sound waves, and even making musical instruments. This experiment can also be used in the field of acoustical engineering to design and test new products.

5. Are there any limitations to a Chladni plate with Neumann conditions?

Yes, there are some limitations to a Chladni plate with Neumann conditions. The size and material of the plate can affect the patterns formed, and the experiment can only show the behavior of sound waves in two dimensions. Additionally, the fixed edges of the plate may not accurately represent real-world conditions, so it is important to consider the limitations when interpreting the results of this experiment.

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