How do two-dimensional surfaces vibrate under forced oscillations?

In summary, the conversation discusses the subject of "Chladni plates" and the search for information on how a two-dimensional surface vibrates under forced oscillations. The recommended resources for understanding this topic include mathematics books on partial differential equations, "Fundamentals of Acoustics" by Kinsler, Coppens, Frey, and Sanders, and online resources such as MIT's OpenCourseWare lectures.
  • #1
Chen
977
1
Hi,

I'm looking into the subject of "Chladni plates":
http://www.physics.montana.edu/demonstrations/video/3_oscillationandwaves/demos/chladniplates.html
For a lecture I'm supposed to prepare, and I'm looking for information on how exactly a two-dimensional surface vibrates under forced oscillations. It's no secret that the motion of a 1D string is governed by the simple wave equation. So which equation governs, for example, a thin square plate?

Thanks,
Chen
 
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  • #2
Chen said:
Hi,

I'm looking into the subject of "Chladni plates":
http://www.physics.montana.edu/demonstrations/video/3_oscillationandwaves/demos/chladniplates.html
For a lecture I'm supposed to prepare, and I'm looking for information on how exactly a two-dimensional surface vibrates under forced oscillations. It's no secret that the motion of a 1D string is governed by the simple wave equation. So which equation governs, for example, a thin square plate?

Thanks,
Chen

This is more of a mathematics problem, so I recommend you look in mathematics books dealing with partial differential equation. In Mary Boas's text "Mathematical Methods in the Physical Science (2nd Ed)", she has a treatment on 2D vibrating circular membrane in Chapter 13 on PDE, giving you all those Bessel function solutions.

Zz.
 
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  • #3
I have the 1982 version of "Fundamentals of Acoustics" by Kinsler, Coppens, Frey, and Sanders. The index reads:

CHAPTER 4
The Two Dimensional Wave Equation: Vibrations of Membranes and Plates

4.1 Vibrations of a Plane Surface

4.2 The Wave Equation for a Stretched Membrane

4.3 Free Vibrations of a Rectangular Membrane with a Fixed Rim

.
.
.

4.7 Forced Vibrations of a Membrane

.
.
.
 
  • #4
Thank you both, I'll try looking at those books when I get the chance.
If anyone knows of an online resource for this information, even if it's a bit simplified at first...

Thanks :)
 
  • #5
That would be the case for membranes, but not for thin plates (much like the difference between the vibration of strings and that of solid bars).
 
  • #6
Chen, I just deleted my post after rereading the OP. Hadn't read your last post before I did that - sorry.
 
  • #7
These may help a bit. I also picked up a copy of this (used) which is, IMO, a good book for PDEs:

https://www.amazon.com/gp/product/0139586202/?tag=pfamazon01-20

http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-306Spring2004/7CE34382-EFED-4C3E-9AAA-E3EB930C2AFC/0/hristinas_lec14.pdf

http://ocw.mit.edu/NR/rdonlyres/Mathematics/18-306Spring2004/F4CA6117-8F21-486A-809A-5BC3998AA38E/0/hristinas_lec15.pdf
 
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1. What is the concept of "vibration" in relation to 2D surfaces?

Vibration refers to the periodic motion of a 2D surface around a stable equilibrium position. This motion can be caused by external forces or internal energy within the surface itself.

2. How does the geometry of a 2D surface affect its vibration?

The geometry of a 2D surface can greatly impact its vibration behavior. For example, a flat surface will have different vibrational modes compared to a curved surface. Additionally, the size and shape of the surface can also affect its natural frequency of vibration.

3. What factors influence the frequency of vibration in a 2D surface?

The frequency of vibration in a 2D surface is influenced by various factors such as the material properties, geometry, and boundary conditions of the surface. The stiffness and mass of the surface also play a role in determining the frequency.

4. How can we measure and analyze the vibration of 2D surfaces?

There are various techniques used to measure and analyze the vibration of 2D surfaces, such as laser vibrometry, accelerometers, and strain gauges. These tools can capture data on the displacement, velocity, and acceleration of the surface, which can then be analyzed to determine its vibrational characteristics.

5. What are some real-world applications of studying the vibration of 2D surfaces?

The study of vibration in 2D surfaces has many practical applications. For example, it is crucial in the design and analysis of structures such as bridges, buildings, and aircraft. It is also used in fields such as acoustics, seismology, and electronics to understand and control the behavior of materials and systems.

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