Will a Vacuum-Formed Elastic Sheet Create a Parabolic Shape?

[CosmicVoyager]

Greetings,

I am thinking of ways to make a parabolic dish. If you apply vacuum to one side of a flat circular elastic sheet, will it make a parabolic or spherical shape?

Thanks

 

[mfb]

Based on intuition, I would expect something more complicated than those two simple shapes, but I did not calculate it.
The local curvature is given by the pressure difference (always the same) and tension in the material (depends on the stretching of this material, and that will probably depend on the position).

 

[256bits]

As always one response leads to another and what ever it wa that mfb stated that got me to realize that the problem could be simplfied by looking at any thin film sphere with a pressure difference between the inside and outside. A soap bubble for example would have a spherical shape. If you take a section of the bubble the tension of the thin film has to be tangent to the sphere.

So, by replacing one section of the sphere with a hoop and allowing the tension of the thin film to continue to be tangential to the curvature of the sphere of thin film at the intersection with the loop, the spherical shape is preserved.

Whether this theoretical model can be achieved in practice and to what degree is another question in that as the vacuum is applied there will be a torsion at the intersection which will deviate the spherical shape, and the hoop being more rigid than the thin film will not expand circumferentially and will constrain the thin film in that direction also. Centre portions will be more spherical than outer diameter portions.

 

[Baluncore]

The centre of a paraboloid is spherical, but the sides are flatter. I would expect a circularly constrained elastic membrane to take a shape very close to a parabola. Focal length would be pressure dependent.

Liquid mirror. Place a circular tank containing a fluid on a vertical axis turntable. Set it turning at a controlled rate. The surface will take up a parabolic profile with a focal length determined by RPM and your local gravity.

Liquid Mirror Constant. Focal length in mm, rotation in RevPerMin
k = 1000 * 60 * 60 * 9.80665 / ( 8 * Pi * Pi ) ' = 447,129.623
RPM = Sqr( k / focal_length)

 

[Khashishi]

I think it should be spherical, because the pressure is equal across the entire surface, so it should stretch the membrane by the same amount at every point. This will give a spherical shape since the uniform stretching gives rise to uniform curvature.

No wait, I change my mind. I think it should be a catenoid. The tension will not be totally uniform because the surface will try to form a minimal surface.

 

[arildno]

A nice way to make a theoretically perfect parabolic surface is to have a cylindrical container with a fluid swirling about the vertical axis.
The surface of the fluid will have a parabolic shape.

 

[AlephZero]

As always one response leads to another and what ever it wa that mfb stated that got me to realize that the problem could be simplfied by looking at any thin film sphere with a pressure difference between the inside and outside. A soap bubble for example would have a spherical shape. If you take a section of the bubble the tension of the thin film has to be tangent to the sphere.

The difference between a soap bubble and an elastic sheet is that the stress in the elastic sheet can have a shear component, but the stress in the soap bubble can not (because it is a fluid).

If you can arrange your experiment so the sheet is in pure tension, you will get a spherical surface not a parabolic one. If not, you probably won't get any "simple" geometrical shape. (For practical demonstrations, look at some balloon art.)

 

[Baluncore]

arildno. That is the “liquid mirror” I mentioned. http://en.wikipedia.org/wiki/Liquid_mirror_telescopes

“Swirling” is a problem. It is necessary to spin the container with the liquid since the RPM of the liquid must be constant throughout to prevent surface irregularities.

 

[arildno]

arildno. That is the “liquid mirror” I mentioned. http://en.wikipedia.org/wiki/Liquid_mirror_telescopes

“Swirling” is a problem. It is necessary to spin the container with the liquid since the RPM of the liquid must be constant throughout to prevent surface irregularities.

1. My bad, I didn't read your post. Secondly, as to "swirling": Again my bad. Apparently, I am not as good in English as I thought I was, so my word choice was inappropriate and misleading (I certainly meant that the liquid should rotate along with the container).
:sad:

 

[Baluncore]

If it was a fluid film it would be spherical. As an elastic membrane, the failure to expand around the circumference will result in a radial elongation of material near the edge. That satisfies the radius of curvature of a paraboloid at the centre and at the edge. For long focal lengths it will be close to perfect.

 

[Baluncore]

:sad:

Never mind, be happy. You are too honest.

 

[mfb]

Fluid films get spherical, I agree.

As an elastic membrane, the failure to expand around the circumference will result in a radial elongation of material near the edge. That satisfies the radius of curvature of a paraboloid at the centre and at the edge. For long focal lengths it will be close to perfect.

Did you calculate this, do you have a reference, or is that a guess?

 

[AlephZero]

That satisfies the radius of curvature of a paraboloid at the centre and at the edge.

I can't even work out what that sentence means, let alone whether it is true or not.

A paraboloid has two different principal radii of curvature, everywhere except at the centre ...

For long focal lengths it will be close to perfect.

Sure, because for long focal lengths a paraboloid is a good approximation to a sphere. But we don't know what the OP thinks is "close enough".

 

[Baluncore]

Someone, somewhere, must have performed the energy per unit area computation for a circularly bounded elastic membrane under a differential pressure. What I am saying is that it is worth further research. It is a safe prediction because for long focal lengths a spherical surface behaves like a parabola. I am pointing out that the 2'nd order effect nearer the edge will also be in the right direction.

It is based on my knowledge of paraboloid surface curvature. I have written code to optimise centre position and radius for spherical approximations to the radial zones of paraboloids. I also have experience with machining master paraboloids on a spherical grinding machine based on my computations.

I have no doubt that it all falls apart with short focal lengths. Without a membrane analysis it will not be possible to know what the exact limitations are. We still do not know the wavelength, diameter, focal length or beamwidth of the dish being considered. Is it being used as a reflector ?

 

[Baluncore]

I can't even work out what that sentence means, let alone whether it is true or not.

The concave side of a paraboloid can be approximated over a series of radial zones. Each zone is tangent to the surface of a sphere. The radius and the position of the sphere's centre along the axis changes according to the distance of the zone from the axis of the paraboloid. At no point does the tangent sphere penetrate the paraboloid's surface. The radius I refer to is the radius of the tangent sphere.

 

[AlephZero]

See here: http://www.cs.odu.edu/~mln/ltrs-pdfs/NASA-97-tp3658.pdf

The bottom line is that to get a true parabolic shape, you need a membrane with variable thickness, and also a boundary that expands radially as the pressure changes.

(And I liked the fact that the "classic" solution to this problem by Hencky has two errors, one in the math and another in the physics!)

 

[sophiecentaur]

A membrane could be vacuum formed to as near a parabola as you wanted (or could calculate) by using an appropriately profiled thickness for the sheet you start with (thick on the inside, thin and stretchy nearer the rim). Alternatively, a series of strengthening rings, spaced at varying distances from the centre, could also give a non spherical curve.
I guess you could even investigate the possibility of altering the temperature across the surface of the membrane so the modulus at the outside is less than the modulus near the centre.
Hours of fun!

 

[256bits]

The difference between a soap bubble and an elastic sheet is that the stress in the elastic sheet can have a shear component, but the stress in the soap bubble can not (because it is a fluid).

If you can arrange your experiment so the sheet is in pure tension, you will get a spherical surface not a parabolic one. If not, you probably won't get any "simple" geometrical shape. (For practical demonstrations, look at some balloon art.)

I had already explained in following paragraphs in the that a sheet is constrained and will not form an ideal perfect spherical shape. Nor for that matter any curved shape easily expained mathematically.