Parabola! (Do doo do do do) Parabola? (do doo doo do)... Hoping for a CAD Master's Interest

[homebrewed]

It turns out your problem is an often-encountered one in the manufacturing world. The starting material for many of the things we use all the time is flat, but needs to be formed or shaped into a 3D surface. Shoes. Car bodies. Underwear. Stuff that is stretchy is easier. Stuff that's hard to stretch is, uh, harder. Google "The flattening of arbitrary surfaces by approximation with developable stripes" for some hair-raising math. And Here is an explanation of what a "developable surface" is.

But others have thought about your particular problem. See this or this. The first paper looks like it will give you a better figure (in an optical sense). The second approach looks like it will produce less waste. So pick your poison....

 

[MERLIncMan]

I very much appreciate these - I have seen one of them. I might be able to mess with Srinivasan et. al's formula. Glad to see that they had the same experimental results/suggestions as I've seen in similar literature.

I have thought often about the aforementioned 1945 RADAR dish; I like to keep in mind that Apollo was done with a slide-rule and a T square. I feel rather confident that 99.999999999999999% of engineers/mathematicians/draftsmen under the age of 30 have no idea what those things are, or how powerful a mind can be when it doesn't have binary-pixies to lean on.

How in the hell did a man conceive of the methods necessary to fabricate that RADAR dish? The first steam-turbine boat - the well-named Turbina - was designed by a man who knew nothing of ship design, and built - not by shipwrights, but by freaking Tinners! Turbina, as it turns out, has some really neat hydrodynamic properties, aside from her turbine engine, that are mirrored by pretty much every Man-O-War vessel on the sea today - and it was designed by an enthusiastic trust-fund-baby, and built by sheet-metal guys, none of whom had the least training in Marine Engineering...

All of this is to say, I'd love to CAD-CAM this thing, but it's looking more like my skills with a divider will be what gets it done.

Kinda irritating actually, but hey, you guys are awesome for helping, and if machining/fabrication were easy, everyone would do it

 

[vtcnc]

Send me your dimensions and I'll give it a whirl. The sheetmetal features in Fusion360 may be able to flatten the parabolic shapes.

 

[MERLIncMan]

Send me your dimensions and I'll give it a whirl. The sheetmetal features in Fusion360 may be able to flatten the parabolic shapes.

The shadow it would cast - being projected to a flat plane that is - would be 47" wide, 34" tall (that's about a sheet of A0 paper, or 1 square meter). I'd love to do lap-joint cuts in plate to make a grid-frame like the cardboard dividers in the boxes that liquor bottles ship in, then lay the somewhat eye shaped pieces of shiny stuff on top of that frame such that their seams line up nicely when they take the shape of the parabola.

For ease of design, say a focal point of 18" or so? That number isn't critical, but would need a proboscis cantilever member to attach the target onto which the rays are focused.

I'd love to see what you come up with!

 

[vtcnc]

The shadow it would cast - being projected to a flat plane that is - would be 47" wide, 34" tall (that's about a sheet of A0 paper, or 1 square meter). I'd love to do lap-joint cuts in plate to make a grid-frame like the cardboard dividers in the boxes that liquor bottles ship in, then lay the somewhat eye shaped pieces of shiny stuff on top of that frame such that their seams line up nicely when they take the shape of the parabola.

For ease of design, say a focal point of 18" or so? That number isn't critical, but would need a proboscis cantilever member to attach the target onto which the rays are focused.

I'd love to see what you come up with!

Modeling it won't be the problem. It will be to determine whether the software can flatten a section or not. Give me a couple of evenings to play around with it.

The issue that you are going to run head long into is that forming a flat pattern into any non-linear shape is going to involved deformation (stretching and compression) of the surface. Think of dimpling a small piece of metal with anvil and hammer. In a scaled up version of this, you have metal presses taking large blanks and forming unibody sections for passenger cars. Those blanks have been refined to stretch and form to near net finished shapes with very little trimming, if any is needed at all. My guess is that when you can come up with a final flat pattern, it will still need to be somewhat oversized in order to give some allowance for stretch and final trimming/dressing of the edges.

 

[RJSakowski]

The math needed to construct a parabolic reflector has been well known for centuries.

Figuring out the shape of the gores shouldn't be too difficult. If you look at the formula for a parabola, y = x^2, the radius of the circle located in a plane perpendicular to the central axis and a distance y from the base would be the sqrt(y) The circumference of the circle will be 2π* sqrt(y). For n gores, the width of each gore at a distance y from the base will be 2π* sqrt(y)/n. This assumes that the gore will be formed to fit the curvature. If you use flat gores, then you would need to calculate the chord of the arc at a distance. The length of a chord, w. subtending an angle A is w = r *sinA/2 where A is 2π/n and r = 2π* sqrt(y). Substituting, w = 2π* sqrt(y)*sin(2π/n).

This equation can be loaded into an Excel spreadsheet to calculate points which can be connected to generate the gores or loaded into a CAD/CAM program to generate a tool path. If you desire to have a seam, just add the seam width to the the generated widths.

 

[MERLIncMan]

Modeling it won't be the problem. It will be to determine whether the software can flatten a section or not. Give me a couple of evenings to play around with it.

The issue that you are going to run head long into is that forming a flat pattern into any non-linear shape is going to involved deformation (stretching and compression) of the surface. Think of dimpling a small piece of metal with anvil and hammer. In a scaled up version of this, you have metal presses taking large blanks and forming unibody sections for passenger cars. Those blanks have been refined to stretch and form to near net finished shapes with very little trimming, if any is needed at all. My guess is that when you can come up with a final flat pattern, it will still need to be somewhat oversized in order to give some allowance for stretch and final trimming/dressing of the edges.

Much agreed - shiny aluminum prefers to stretch rather than compress I think, but not so much as to tear and the grain of the blank sheet will likely make itself known if the stretch is too much. Bending in 2 dimensions is bad enough (hence the thin and weak steel of the carbody becoming strong by not-corrugated corrugation) - bending graceful curves in 2 dimensions....

The reflections from parabolics (known from experimenting with aluminum foil on cardboard) have a funny property of photo-interferometry such that the reflected light shows EVERYTHING. It is a curious and fascinating effect that makes dish-type mirrors irritatingly picky regarding dimensional accuracy and surface tolerance. Anything outside of 2uin shows up dramatically in the reflected light and spoils the efficiency.

It occurs to me that a proportional spin-form would serve - the mandrel moving in and out with the parabolic shape as the part rotates - but that's just insanity!

 

[MERLIncMan]

The math needed to construct a parabolic reflector has been well known for centuries.

Figuring out the shape of the gores shouldn't be too difficult. If you look at the formula for a parabola, y = x^2, the radius of the circle located in a plane perpendicular to the central axis and a distance y from the base would be the sqrt(y) The circumference of the circle will be 2π* sqrt(y). For n gores, the width of each gore at a distance y from the base will be 2π* sqrt(y)/n. This assumes that the gore will be formed to fit the curvature. If you use flat gores, then you would need to calculate the chord of the arc at a distance. The length of a chord, w. subtending an angle A is w = r *sinA/2 where A is 2π/n and r = 2π* sqrt(y). Substituting, w = 2π* sqrt(y)*sin(2π/n).

This equation can be loaded into an Excel spreadsheet to calculate points which can be connected to generate the gores or loaded into a CAD/CAM program to generate a tool path. If you desire to have a seam, just add the seam width to the the generated widths.

I'll not pretend to have understood your description - I loved geometry because it helps triangulate, and Trig because it tells me the world (ironic pun!) - but visualizing from certain equations has continued to elude me.

That being said, your process seems to be similar to how I made a cardboard globe in years past. I was able to calculate the radius from an arc, given the chord and depth of said arc. From that I derived a formula to give me the radius of a gore, with the chord of the gore being 1/2 the circumference of the globe, and the depth of that arc (there being 2 arcs in a gore for a sphere) being 1/24 the circumference of the sphere.

As the parabola is a conic, and a conic is a cousin to a sphere, with Jupiter's moon being in retrograde.... I might be able to do the same voodoo using a parabola rather than a circle.

Time for some more experiments with cardboard and a utility knife!

 

[RJSakowski]

I have a 10' parabolic reflector in my field. It was originally a dish for receiving satellite microwave signals. It is made from fiberglass with an metallic skin with paint over it. The focal length is about 42". I have never attempted to use it for a solar reflector but if I put my ear at the focal point, I could pick up a whisper a half mile away. I also can't vouch for the accuracy of the surface although it would have had to be fairly good based on its intended use.

I don't know the nature of the metallic skin. It could be a metallic laden paint similar to what was used for printed circuit repair or it could be a metallic foil glued on. If I were to use it for a solar furnace, I would probably glue aluminum foil to the surface and polish it to a specular surface. The mount for the dish is a azimuth/ elevation mount which would be capable of motorized tracking of the sun.

 

[MERLIncMan]

I have a 10' parabolic reflector in my field. It was originally a dish for receiving satellite microwave signals. It is made from fiberglass with an metallic skin with paint over it. The focal length is about 42". I have never attempted to use it for a solar reflector but if I put my ear at the focal point, I could pick up a whisper a half mile away. I also can't vouch for the accuracy of the surface although it would have had to be fairly good based on its intended use.

I don't know the nature of the metallic skin. It could be a metallic laden paint similar to what was used for printed circuit repair or it could be a metallic foil glued on. If I were to use it for a solar furnace, I would probably glue aluminum foil to the surface and polish it to a specular surface. The mount for the dish is a azimuth/ elevation mount which would be capable of motorized tracking of the sun.

Why you gotta brag man?

That's neat!

Don't put your ear in the focus when it's shiny and sunny!