I was trying to figure out some kind of formula for cutting a hole in a flat piece of metal that would later become a cyl. The hole will accomadate something like a round tube or pipe. I know how to calculate the height and width but not the arc between the 4 points?
Can you explain a little further what you are doing? What do you mean “would later become a cyl”? “Cyl” means “cylinder”? You’re going to bend the metal into a cylinder? With a hole in the side of the cylinder? And what “4 points” are you referring to in your last sentence?
What you need are pipefitter’s templates. They can be found as actual plastic templates, complex drawing instruments, or software. Do a search and browse through the many that show up.
Basically, you draw an oval on the flat sheet, then roll it into a tube whereby it becomes a perfect fit for a round tube butting the main tube. Most people just cut the hole directly in the tube using a variety of methods. A hole saw in a jig is the simplest.
If you have some of the tubing already rolled to diameter, just hold the side pipe in place and carefully trace onto paper wrapped around the main pipe, then unroll the paper and use it to mark the sheet.
Dennis
A flat pattern that would be used to lay on a flat piece of steel that would be bent into a drum shape or cyl.The piece to be inserted in the whole will be another cyl shape or pipe.
The 4 points are top and bottom and width. The height is the same as a circle and I have a formula that will give me the width.
The pattern would be used to send to a water jet cutter to cut sheet metal.
I don’t think there’s going to be a simple formula since the shape will vary so widely depending on the sizes involved. If the cylinder you’re making is much larger than the pipe you’re going to jam in its side, then the hole you’ll make in the flat piece of material will be almost a perfect circle. OTOH, if the cylinder you’re making is the same size as the pipe you’re going to jam in its side (like this), then the hole you’re making in the flat piece of material will look like a pair of lips, i.e. rounded at top and bottom and coming to a sharp 90-degree point at left and right.
I was taught this sort of projection in my freshman-year drafting class ~30 years ago, and I was able to do it manually in 2-D CAD software a few years ago, but I doubt I can communicate the procedure verbally; it involves marking maybe 10 points around the perimeter of the pipe, and using a three-orthogonal-view drawing to project how those points will wrap around the main cylinder. A machine shop that’s been tasked with this will probably just end up using 3-D solid-modeling software to model the cutout in the main tube and then just “unwrap” the tube to make the waterjet model.
Google around for fishmouth/saddle cuts, and maybe you’ll find a good process.
I will check that out, I couldn’t figure out how to search it.
It presents an interesting mathematical challenge, though. What is the general planar equation of the closed curve, C, such that, when the plane is then wrapped into a cylinder of radius r₁, C forms a circle of radius r₂? (ETA – below)
As a point of curiosity, is C always an ellipse, regardless of r₁ and r₂?
If my math-fu were better, I’d have a go at this. But, it’s interesting. Or fascinating, as Mr. Spock might say.
ETA – The question really should be this, I believe: What is the general planar equation of the closed curve, C, such that, when the plane is then wrapped into a cylinder of radius r₁, C forms a circle *such that a pipe *of radius r₂ would fit perfectly in the circle?
See my previous post: for the special case of r1 = r2, it’s definitely not an ellipse.
And for any case where r1 < r2, I’m pretty sure it’s not an ellipse either.
For r1 << r2, the shape approaches a perfect circle (which is an ellipse with a=b).
I don’t have any advanced math skills but like you I found the problem kind of interesting. I often do little home projects requiring lay outs and as a rule on things like this I end using a file and fit method. If you figure out an actual formula let me know. Thanks for clarifying the problem.
Well, Mathworld says the parametric equations of the curve of intersection of two cylinders of radii a and b are
x = b cos t
y = b sin t
z = ± sqrt(a[sup]2[/sup] - b[sup]2[/sup] sin[sup]2[/sup] t)
That’s not quite what we want – we want the equation of the curve when the larger cylinder is unrolled to a plane. I don’t have time to investigate this further right now, but it’s an interesting problem, and I’m sure the solution is on the Web somewhere.
Anyone with a waterjet, probably has a CAM package designed for sheet metal that could do this easily. Any pattern or drawing you come up with will have to be converted to a .DXF file anyway.
Well okay but to be a little nit-picky I believe you have the Rs backwards. r1 is the radius of the cyl, and r2 is of the pipe. r1 has to be greater than r2. When r1 = r2, you essentially have a T joint of equally sized pipes.
And yes, when r1 >> r2, then C approaches a circle of radius r2. (I’m not sure C is an ellipse, so I’m not using a and b).
I would imagine this is a routine sheet metal operation. I wonder how they did it before computers?
It was done by layout methods. As a machinist I rarely even do trig (just draw it in CAD and dimension) anymore and its been forever since I’ve laid out anything out at the day job. In my hobby home shop, I will occasionaly break out the dykem and dividers.
That’s the stuff I was taught in drafting class - no computers, just pencils, straightedges, French curves.
Looks like there’s a free DXF generator here that should solve your problem.
This page explains the old-school drafting method I was talking about. In step 6 he shows that there is, in fact, a formula for generating the correct shape for two pipes of arbitrary radii.
Before modern CAD systems sheet metal flat patterns were generated for formed shapes by using templates, rules of thumb for bend offset and distortion, and often some amount of trial and error, especially in the case of complicated forms like conical shapes or multiple axis bends and forms.
Most production-grade modern 3D parametric CAD modelers have what is called “sheet metal” modules that will generate a flat pattern from a formed product, compensating for bend radius offsets, highlight the need for relief features, and some will even compensate for bend distortions to some degree of reliability.
There is a simple way to do this; though. Take a large piece of heavy vellum or poster board, roll it in the same shape as your actual form, and then use a can around a pole mounted light to generate the silhouette of the hole you want. You can then take a picture of the hole (as inline as possible) with a camera, or premark a grid pattern on the vellum and measure where the silhouette falls to get enough points to generate the flat pattern. However, be aware that when you remove material the edges of the hole are going to distort some, and if the hole is big enough the entire piece might warp when you try to roll it, requiring some stiffening rings, doubler places or relief cuts to allow it to maintain shape. If you are just doing a one off it might well be easier to just tack on a second heavier section and then cut out the opening with a cutting torch.
Stranger
The last one shown is what I was trying to visualize. The generators is what I will end up using but it was nice to see a drafting method that I could use.
The light never occured me, great idea. I use it on occasion for other odd shapes, I should have thought of it.
That’s not an ellipse. And as r1 → ∞ and for r1 >> r2, does that curve approach a circle? It should, right? Only if the pipe intersects the cylinder perpendicularly.
Also JA, or the joint angle, when I described above it was the trivial case for when the r2 pipe intersects the cylinder at 90°. But the joint angle θ can be 0 < θ ≤ 90°. And the intersecting pipe, if θ < 90°, does not have to have its axis lying in the same plane as the axis of the cylinder.
This gets more and more fascinating!
So I believe these are the key variables:
r1 = cylinder radius
r2 = intersecting pipe radius
θ = the ∠ between axes of cylinder and pipe, if they were coplanar
ψ = the ∠ of rotation of the pipe, perpendicular to the plane defined by sweeping through ∠θ, if the pipe’s axis is not coplanar with the cylinder’s axis
There are probably more elegant descriptions for those, but I believe those are the variables to describe the general problem, which I’ll state like this:
What is the planar equation of the closed curve, C, such that when the plane is wrapped into a cylinder of radius r1, a pipe of radius r2 and intersecting the cylinder at angles θ and ψ perfectly fits the hole described by C?