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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

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- Thread starter CosmicVoyager
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- #1

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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

- #2

mfb

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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).

- #3

256bits

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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.

- #4

Baluncore

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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)

- #5

Khashishi

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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.

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.

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arildno

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The surface of the fluid will have a parabolic shape.

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AlephZero

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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.)

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Baluncore

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“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.

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arildno

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“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:

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Baluncore

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Baluncore

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Never mind, be happy. You are too honest.arildno said::sad:

- #12

mfb

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

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AlephZero

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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".

- #14

Baluncore

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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 ?

- #15

Baluncore

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

- #16

AlephZero

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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!)

- #17

sophiecentaur

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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!

- #18

256bits

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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.

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