Cutting a curved piece of metal

[benracin]

Howdy Gents! Got a question for you. I'd like to know how I can measure and draw out this piece of metal before I cut it instead of trial and error. Here's the challenge:

Lets say that I have a tube vertically coming out of a flat piece of wood. To make an easy picture lets say it's a drain vent coming out of a flat roof. I'd like to wrap the joint with a piece of metal that has a 45 degree slope to it and wraps all the way around. Like a collar with a slope away from the tube to the roof. I know that from messing around cutting pieces of paper this shape when it's flat looks like a spread out U with the upper ends of the U pointing more outwards then upwards.

It seems like the known measurements would be the distance around the tube, the vertical distance between the ground and the point on the tube where the slope would start, possibly the distance from the tube out to the larger radius the piece would make on the ground, and possibly the distance around that larger radius.

So using those measurements (I think) how can I draw this shape out on a piece of paper?

If I didn't describe this well enough PLEASE let me know. And this is a challenge of course. I bet my wife that race car guys could answer this faster then the bridge and highway engineers she know. Unless of course that bridge and highway engineer is a race car guy. :P

Oh, and no this isn't for a drain cap deal. I have a tube coming out of bulkhead type deal and I'd like to cover up that seem with a wrap around, sloping piece of metal. And ok, I'll admit it, it's for a halloween costume.

[Greg Amy]

So using those measurements (I think) how can I draw this shape out on a piece of paper?[/b]
Well, my first question is: where in the GCR/ITCS does it say you can do that...?

Anyway...

It's an easy geometric problem, just break it down into its small bits. It's a cone with the top cut off. To draw it, you need two main things: the circumference of the top of the cut off part and the circumference of the bottom.

The top is easy: it's the circumference of the "pipe" you're wrapping it around. That's pi (3.1415x etc times the diameter of the pipe.) So, start out by drawing a circle of that diameter on a piece of paper.

The next thing you need to know is the circumference of the bottom. That one is also easy, when you think of the cone sliced vertically. You know the angle is 45 degrees; you know the height from the top of the cut cone to the bottom, call this H-y, where y is the height of the part of the cone that's cut off and H is the height of the cone before it's lopped off. the radius of the outer circle is the same as H, because it's a 45-degree angle.

The long side (hypotenuse) of the cone - before lopping off - is described by Pythagorean theorem, where the square of hypotenuse is equal to the the sum of the squares of the radius. Same goes for the radius of the lopped-off cone. Ergo, the difference between the radii of the top of the lopped-off cone to the bottom is that difference between these two cones...

Then, once you have all this, you've got your numbers and you can draw it out...

OK, so now that you're totally confused, give me two hard numbers and I can figure it out for you:

- height of this lopped-off cone where it touches the pipe at its top
- diameter of the pipe

[Greg Amy]

Or...

Ok.

pipe dia = x
cone height = y

We already know what the diameter of top circle of the lopped-off cone (LOC) is: x. So, start by drawing a circle in the middle of the paper of x diameter.

Next, we need to find the height of the not-LOC, cause we need that to find the diameter of the bottom circle. That would be y plus the height of the part lopped-off. Well, that's easy: given it's a 45-degree angle, the height of the part we cut off is tha same distance as its bottom radius; the bottom radius is half its diameter, or x/2.

The radius of the bottom of the cone is the same as its height of the not-LOC; the height of the not-LOC is y plus the height of the lopped-off part, or y+x/2. So, the diameter of the bottom of the cone is 2*(y+x/2), or 2y+x

But, you can't just go drawing a circle outside of that smaller one, because it doens't take into consideration the angle that's cut out to make it a cone. So, the next thing you have to figure out is the distance between the two circumferences. To do that, slice the cone vertically again, but this time slice it at the interface between the top circle and the pipe. The height of this part - and, since it's a 45 degree angle, its base - is y. Ergo, the distance between the two circles is the hypotenuse of that triangle, or the square root of the sqaures of its base/height:

Z=sqr(2x^2).

So, draw a circle outside the smaller one with a diameter of x+z. That's where the larger circle lies.

Now, the circumference of the larger circle is pi(z); the circumference of the base of the cone is pi(2y+x). Ergo, the gap of the cone will be a segment of that circle Pi(z) minus pi(2y+x); draw that segment somewhere on the larger circle so you know where its end are, then draw two lines, one from each end of that segment, to the same point on the smaller circle and - voila! - cut away...

Hey, I'm bored today.


(P.S. Hard numbers can allow me to do the calcs and see if this rambling is correct...)

[RacerBill]

where r = radius of the pipe
h = height where you want the collar to end

c(1) = circumference of the pipe/collar at the top - 2 pi r
c(2) = circumference of the collar at the bottom - 2 pi (r+h)
h(1) = length of the collar along the surface of the collar

lay out the sheet of metal

mark the distance h(1) along one edge

at one end of line h(1) mark a curve of radius r and length c(1)

at the other end of line h(1) mark a curve of radius (r+h) and length c(2)

This should result in a shape that has two straight lines and two curved lines, one convex and one concave.

Cut the metal along the lines and bend so that the straight lines are together.

If you want the two straight edges to overlap, add the amount of overlap to the curved lengths c(1) and c(2) ie + 1/4 in.

I think that will do it. Do I get a free night in a Holiday Inn Express?


Greg: You're fast! But I have an excuse - I had to do some real work!!!!

[benracin]

Thanks for the replies guys. I haven't tried your methods yet but I thought I would supply some real numbers so we could experiment.

Dia of the top part of the cone = 10"
Dia of the bottom part of the cone = 15"
The vertical distance between the bottom and top of the cone = 2"

10"
-------------
/ | \
/ |2" \
------------------
15"

If I were to make a vertical cut on one side of the cone, then lay it out flat, I think the finished rolled out piece of paper would look sort of like:

/\ /\
/ \ / \
/ ---- ---- \
\ ---- /
\ /
------_______-----

So what I'd like to know is what it takes, and how to draw out that shape on a piece of paper so that I could make the cone.

THANKS A TON! What else would you want to do on a friday?

Well I can see my drawings came out awesome. Geesh.

[benracin]

Well lookie what I found: Clickity click click (http://mathforum.org/library/drmath/view/55375.html)

[Greg Amy]

Dia of the top part of the cone = 10"
Dia of the bottom part of the cone = 15"
The vertical distance between the bottom and top of the cone = 2"[/b]
First problem: you've specified too many conflicting parameters. What you list above will work, but if you insist on a 45-degree angle then you're gonna have to pick one of the above ones to delete...

If we were to delete the bottom cone distance requirement but keep the 45-degree slope, that would result in a bottom diameter of 14".

If we delete the 45-degree requirement but keep the 15" bottom diameter, that's an angle of inv-tan(2/2.5)=roughly 39.5 degrees from horizontal.

If we kept 45 degrees, 10 and 15 inches, but deleted the 2" height, the resulting height would be 2.5".

Sooooo...pick your poison... - GA

P.S. How's the bridge builders doin'?

[benracin]

Doh! Sorry! All of my measurements are actually off the top of my head. I'm also not smart. I was smart once, but then I forgot. I actually don't know the angle. And I'm actually batman. Scratch that, this is a Megatron costume so I guess I'm a transformer.

So we can pretend that I only know the 2 diameters and the height.

[Greg Amy]

Ok, then:

- draw a circle with a diameter of 10 inches.
- Draw another circle, same center point, with a diameter of 16.4 inches
- From the larger circle measure out as close to 4.4 inches of its circumference as you can.
- Draw two lines, one each from both ends of that 4.4 inches segment, to the same point on the smaller circle (as close to halfway across that segment/line as possible.

Cut.

I want a Reese's Cup for this. - GA

[RacerBill]

This is the second time I have been presented with a problem like this. The last time was when I was installing laser engravers at a trophy shop. They had a crystal bowl for a major basketball tournament that was wider at the top than at the bottom, and they could not figure how to design the lettering so that it was straight and level.

Their art department worked for hours on it. I said, let me try and had the layout done in less than an hour.

Never can tell what you will need from high school math!

What's your excuse, Greg?

[Greg Amy]

What's your excuse, Greg?[/b]
Mechanical engineer by education. It's a sickness...

[Racerlinn]

I can count to 10 using both hands.
But don't ask me to count to 21 if I've been drinkin...... :blink:

[benracin]

The bridge builders have come back with a spreadsheet titled "Segments of Circles". You can put in any two of the arc length, chord length, radius, height, angle, apothem and area and it will give the rest. If we could put up attachments I'd add it for ya! I'm not quite sure how it works for our situation though. To many big words.

If only I continued my Mech Engineering degree. I for some reason switch to business, then architecture, then education, and finally just turned into a web/video/graphics guy. Oh well!

[Z3_GoCar]

It's an ellipse. The minor axis is the diameter of the pipe, and the major axis is the diameter divided by the cosine of the angle. I assume this is for a bathroom vent on your roof.

James

[RacerBill]

... the cosine of the angle. ...

James
[/b]

...as in the angle of the dangle?????????

[ddewhurst]

Ben, I could send you an Eng. drawing 101 book. :o