equation of a flexible membrane bounded and under vacuum

[kestlerhawk]

Hello all,

sorry for the trouble, but i am completely trumped by this at the moment.

if i consider a rope hanging from two points under its own weight, i can calculate the deflection using a catenary function.

however, consider a circular cutout on the top of a box. if a sheet of my mylar is laid over that hole loosely, i can reproduce the deflection using a catenary curve, however, if i apply a vacuum in that box that sucks down on the mylar membrane, will the curve remain catanary?

i'm under the impression it will be more parabolic approaching spherical, and for our application the spherical assumption was used leading to errors i'm trying to fix right now.

i'm still under the impression the curve will be a modified catanary equation (similar towed line in water.)

i cannot find any academic material of this matter and any help would be greatly appreciated.

thank you in advance for your help.

regards,

 

[hacksaw]

Hemholtz solved this problem in the 1800's. The book is usually available from Dover.

 

[hacksaw]

an alternative is Roark's. it covers membranes in various configurations. It is not a catenary, except in an en\xtreme case, perhaps

 

[prex]

If I understand you correctly, you are setting up a circular plate with simply supported edge and a uniform load. This is a very simple problem with plates, and by the way the deflected surface would be similar (within the validity of an elastic approach) for uniform weight and uniform pressure, as both are uniform loads.
Unfortunately the deflected surface is quite different from a sphere, it is (theoretically) represented by a polynomial of the fourth order in r, whilst a sphere-like parabola would be of the second order. A nearly spherical surface can be obtained only by loading the plate with a distributed moment along the edge.

prex

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