Vibrating rectangular membrane

[d11420]

Does anyone know what the boundary condition for a free edge of a vibrating rectangular membrane would be?
The fixed condition is easy, but what about a free edge?

 

[FeX32]

The bending moment and shear force vanish at the free end


Fe

 

[d11420]

Wouldn't that be for a thin plate? I'm thinking of a very thin membrane, so there wouldn't be any moment or shear to begin with.

 

[FeX32]

I don't think you can analyze stresses with membrane theory alone.
I am not 100% sure though.


Fe

 

[TERIO]

Maybe, membrane force in the direction normal to the free edge = 0

 

[d11420]

I don't think it would be a force boundary condition, I was thinking along the lines of maybe a gradient??

For the fixed edge displacement is 0 for sure. But if it's free...

 

[TERIO]

Displacement gradient (strain) = 0 -> force (stress) = 0

 

[electricpete]

The terminology that I have heard is that membrane is the 2-D equivalent of a string (the PE results from tension), while a thin plate is the 2-D equivalent of a beam (the PE results from bending).

Are you interested in a beam or a membrane?

The term "fixed" does not compute in my world for membrane, since it implies a moment applied to enforce zero slope... you can't do that for a membrane/string.

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[electricpete]

Correction in bold:
Are you interested in a plate or a membrane?

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[d11420]

I mean membrane. Maybe I should have said clamped instead of fixed. So one of the edges could be held down, making displacement 0. What I don't know is what the condition would be for a free edge (not clamped)

 

[electricpete]

Here's what I know about strings. There are two types of boundary conditions:
pinned: displacement = 0
free: applied axial force =0

At first glance, it seems like we are missing something... don't we need two equations per side like we do on a beam? No, only one equation per side for string since it is a lower order differential equation than a beam.

I am guessing a membrane is a relatively straightforward extension of a string... but I'm not sure.



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[electricpete]

The free would probably be pretty uninteresting for a string... but at any rate the condition is that there is no applied tension or force at the boundary.

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[GregLocock]

How long is piece of string?

To use your string analogy, the bc at a free end is that there is no force in the tensile direction.

So with a membrane the bc is that there is no stress normal to the free edge.

Is that enough?

Cheers

Greg Locock


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