How to draw a frustrum?
 

I want to have a 3" steel tube pass through sheet metal at a shallow angle, maybe 10-20 degrees. I'm going to weld the tube in place around the whole tube so as not to loose much of the sheet metal's structural value. To do this I need to make a template of the cut I will have to make in the sheet metal so it will but up against the tube before I start welding.

Attached is my guess at the formula for this. I cut this figure out of a piece of paper, and it looked like it worked for a 45 degree angle as shown. Though the ends look a little pointy and weren't quite as good a fit as the rest seemed to be. I suppose I could do this with cardboard and get a better idea of whether I am on the right track.

And news flash - that would be the cotangent (or 1 over the tangent), not the tangent, if I am on the right track: two otherwise ordinary intersecting arcs one diameter apart in the middle.


http://forums.pelicanparts.com/uploa...1174636195.jpg

Or does a diagonal slice through a cylinder create an ellipse or some kind of parabola or paraboloid? If so, any tips on how to draw the template?

This is going to make my brake ducting abrasion proof, or so I hope.

Walt Fricke

Not quite related but close. You can get a tube mitre software program here http://www.tonyfoale.com/ . Click on the freeware link and go to "Tube porfile cutting guide" .

I quote from the instructions: "You can input sizes of two round tubes that you wish to join together by notching one of them, the programme will then allow you to print a paper template which can be wrapped around the tube as a guide to cut the right shape. It will handle tube sizes of up to about 80 mm. diam. This limit is set by the size of the paper in your printer. This programme has no use unless you are actually constructing a tubular structure."

Regards

Johan

Here's what I'm getting on the Mac, by starting with your circle, skewing a copy of it 45 degrees, centering everything, then creating an elipse that picks up all the outer points. We do stuff like this in the shop (sign fabrication) and that's the method I use. No guarantee, but it might be worth cutting a template and seeing if it fits. The "fabricator's trick" above should produce the same shape. BTW, this shape should work for any tubing diameter @ 45 degrees, by enlarging/reducing so that the vertical dimension = tubing diameter.

http://forums.pelicanparts.com/uploa...1174657651.jpg

For the record, the shape is an ellipse w/ a minor axis = the diameter of the tube. The length of the major axis is determined by the angle between the plane surface and the tube axis.

If the tube mitre program doesn't work you can always use a flashlight, the shape of the pattern on the plane surface is a circle when the angle of incidence is 90&deg and transforms into the ellipse that you want as this angle is changed. The size of course should be determined by the length of the minor axis.

Bill,

I suspect the flashlight trick works if the beam is focused and not diverging, otherwise you would get a pattern for a cone.

I like the 'cut a piece of tubing at that angle and use it as a template' method.

Les

Old trick is to place 2 pins in a piece of paper 1/3 the diameter of the tube in question apart.
Then cut a piece of string and tie the ends together so that it is the diameter of the tube.
Place the string on the paper (flat) and put your pencil in the circle of the string.
Start drawing the line by pulling the string tight against the pins and continue in a circle.
You will end up with an elipse.
If the dimentions are correct, the one direction will be the diameter of the pipe, and the other will be the extended elipse.
Old style way of fabrication.
(just realised that I am showing my age...lol)
Bob

LOL, Bob - Are you a former sign painter? That's the way we used to draw elipses. Now we use the Mac for everything.

That would seem to work at a fixed angle, but as the angle changes, so does the ellipse, no? The thin slice method seems to be the quickest and easiest. I'll bet Bill V. has used sophisticated methods to cut a hole where a mast passes thru a deck.

I have used the 2-points spread apart method for creating elliptical arches in construction many times. I've never had a formula for the distance between the two points. It's always been a matter of trial and error. I'll try to understand what Bill said. :confused:

Here's another way to draw an elipse, similar to the "string" method, but creating points instead. Pulled this out of an old sign painter's book, which I haven't looked at in at least 15 years!

But really, all this method does is create the same thing that you can make on your Mac or PC. Even the most simple graphics software will have an elipse tool. Just draw a box that represents the height and width of the elipse, then draw an elipse to fit within it.

In the drawing above, I just drew a circle, then "skewed" a clone of it 45 degrees, thereby establishing the height and width for the elipse. Try it - it's extremely simple.

If you were going through a plane on a 30 degree angle, you'd just skew the circle 30 degrees to get the width, etc. etc.

Then - that pattern will work for ANY diameter tubing, at that angle, by enlarging or reducing to get the height of the elipse to equal the tubing diameter.

Forgot the attachment:

http://forums.pelicanparts.com/uploa...1174665189.jpg

I may be wrong but it seems to me that most of the responses are missing the main point.

Assuming 2.5" tube, at an angle of 20&deg to the plane surface. He needs to be able to describe the unique ellipse that has a minor diameter of 2.5". The hard part is determining the ellipse that is defined by the angle.

the standard 2 foci method w/ a string doesn't work because he doesn't know where the foci belong other than somewhere on the major axis equidistant from the intersection of the major and minor axis

yes, the flashlight needs to have a nice focused beam but again assuming a minor axis < ~3" the parallax error will be small

Easily done on a computer, no? IE:

http://forums.pelicanparts.com/uploa...1174666493.jpg

Why not make a carboard template through trial and error? How many iterations could it take?

Seems like if you can only narrow your angle down to 10 to 20 degrees, it may be an iterative process anyways?

Good luck,

Doug

I did this a couple of months ago but you could try plugging the lower equation into XL.

0 < theta <2 pi
z = 0 (since you want the x-y plane
p = radius of the pipe
phi = angle the pipe makes with horizontal (? not sure of Greek alphabet)

vary theata from 0 to 2*pi

http://forums.pelicanparts.com/uploa...1174667249.jpg

here is what was in the XL cells but unfortunately I named everything. sin_phi and cos_phi are the value stored in a cell

x =p_rad*COS(theta2)
y =p_rad*cos_phi*SIN(theta2)+(Zhgt+p_rad*sin_phi*SIN (theta2))/cos_phi*sin_phi

below is 60 deg

http://forums.pelicanparts.com/uploa...1174667579.jpg

Wow- what a great mix of old-timey and new, high-tech techniques.

Here's another http://vatsaas.org/rtv/tools/computationtools.aspx#transition

But I'd try the flashlight first. Strong halogen beam held as far as possible - definitely not an LED type.

I can draw an ellipse in 2 seconds in autocad, and print it so that a template can be made. I'd be happy to send a PDF of the drawing if that would be of help.
-Scott

While calculating to get the shape of the ellipse might be fun and a challenge. If the panel is slightly irregularly shaped, or the tube passes through the panel skewed in more than one axis things could get interesting.

These tools are really handy to have around. they make fit up of tubing a breeze. Unfortunately the 3" pipemaster is $105... tough nut to swallow for a infrequently used tool.
We have 1-1/2",1-1/4", and 1" at the shop.

http://www.pipemastertools.com/

http://forums.pelicanparts.com/uploa...1174683233.jpg

http://forums.pelicanparts.com/uploa...1174683248.jpg

Thanks, guys. I was correct in assuming that a lot of us love a challenge.

RS9 - My first thought, too, was just to cut a pipe. However, I doubt my ability to cut a 3" pipe at a fairly acute angle accurately, so I want to hold that one in reserve. For those more adept or with better tools that certainly would make it easy.

Flattop - I'm saving that site for when I fabricate a new roll cage. I haven't tried it using a nearly infinite radius for the second tube.

Wavey and others with computer skills (mine being quite primitive) - I don't think I have a program that will do this, but may see if I do. Especially one that says it will put an ellipse inside a box, as I think I now know how to figure the box.

And thanks for the trammel method - that's cool, and may be easier than the string method for my purposes. The cut with my hand jig saw in the sheet metal is probably going to be more irregular than a quick trammel pattern.

I confess to being in the generation of all who mentioned the string method for an ellipse. Remembered it from high school, though not the details. This plus Google refreshed me.

Bill V: By knowing the figure is an ellipse, I think I have it. I know the minor diameter. As you say, how do I correlate the major diameter with the angle, given the minor?

How about this: Major diameter = minor diameter/sin phi (or however you want to denominate the angle). Derivation later in the post. Rick - you have your Greek right, but do I have my math right?

I like the flashlight notion. Maybe a laser pointer rotated around a tube fixed in place? Another one I had thought of was to use a rod held parallel to the tube axis on its OD as a scribe.

Plus I may want to punch through lower down where I am going through the somewhat complex sheet metal box tubing of the unibody just in front of where the front sway bar drop link is. The relatively flat sheet metal is a bit high up. Using light or a rod ought to allow me to scribe on irregular surfaces as well as part of the simpler vertical. Thinking of doing this is what made me get some fairly thick wall muffler tubing, thinking I could minimize the weakening of the unibody structure in this area.

Googling around led me to www.cleavebooks.co.uk/scol/callipse.htm. That is an ellipse calculator. Enter the major and minor diameters and it gives you the foci and string length also.

FYall's I, if I have things right, an angle of 5 degrees gives a major diameter of 38 some inches. Scratch that one. 15 is 12.85, and 30 is 6.6 inches, so I now have something to play around with for a reality check before I start cutting. The good news is that if I start at a larger angle, and that causes problems, every point for a smaller angle is outside the first cut (except right at the minor axis), as Wavey's diagrams show.

DW - this may be my iteration: The practical problem is that this part of the front fenderwell area is a tight fit. I am using 3" hose, which doesn't help. In the past I ran it flush with the fenderwell, and where my slicks rubbed on it I armored it so they wouldn't rub through, at least not quickly. The reduction in turning radius was inconsequential for racing. But I don't know the max angle that will clear everything. I think I may start with 30 degrees and get smaller. I may have to cut some sections of a tube anyway unless I want to cut first and measure later, so to speak - but wait: I can just cut some scrap sheet or cardboard rectangles 3" across at various angles and use those. Two dimensions should work fine for this clearance checking.

Scott - I may want to take you up on your offer of autocad and a PDF. Let's see how I do with cruder methods.

Rick - I had forgotten that Excel offers charting/diagramming. But I think it would take me longer to learn to do that than the string or trammel methods would take. Excel frustrates me because I use it infrequently, so I forget many of the conventions/commands in between. For instance, I take Club Race registration lists (done alphabetically by driver, but including car number, color, type, and class as well), and sort by car number (what a scrutineer needs when told to find a car by number). Always have trouble getting all the columns to sort together, rather than just the car number one.

OK, for the piece de resistance (hope pride doesn't go before the fall again):

Here is how I derived the new formula for major diameter:

Draw two parallel lines representing the diameter D of the tube/cylinder/unique case of a cone (as the mathematicians like to see a cylinder). Draw a diagonal line representing the angle to the plane through these two lines. From either of the two intersections so created, erect a perpendicular to the other line. You now have one side of a right triangle D, and the opposite angle phi, which is your intersecting angle. The major axis should be the hypotenuse. Pulling out my cheat sheet on trig, opposite over hypotenuse = sine of angle. Maybe this is embedded into Rick's more complete formula, which I bet includes the formula to find foci or otherwise describe an ellipse knowing major and minor axes?

Thinking about this, it may only work if the plane (cut) is parallel to the minor diameter? I hope that is good enough for my purposes, as I hope to be able to angle the tube to the rear only, and not down as well (starting inside the body, in the direction of the air flowing). But maybe if I need a down angle as well, perhaps all I need to do is figure out how to angle the template down at one end when applying it? This thinking in three dimensions is tricky.

Again, thanks, guys. I hope you had some fun shaking the cobwebs on a Friday morning. I sure did.

Walt

PS:

Tim - I had seen those nifty variations on the single plane body shape tool. And the price! I might buy the one for inch and a half if I ever get to making my own cage (plan - bend and cut and fishmouth everything, tack weld, and take to real welder to finish).

And the URL I gave doesn't work, but if you chop off some of the end you will get to the site that lists their various calculators.

Walt

it's just a circular profile (aka contour) gage... neat idea but way overpriced.