Challenge for bergerud

[dbfletcher]

Ok.. I know this has nothing to do with Carvewright, but I figured bergerud might enjoy this challenge. I have attached an STEP file that is an extruded part along a spiral. I really want to flatten the part so it can be cut out of sheet metal and then rolled. Unfortunately,the math for flattening a spiral has me stumped... if it was a cone, that is pretty trivial... but the spiral has me stumped since it is a constantly changing radius. I added the .zip extension so the file could be uploaded. Simple rename it without the .zip (dont try to unzip it) and you will have the STEP file geometry.

[dbfletcher]

here is the file in stl format as well... just in case you cant do anything with the step file. Same rules apply... just remove the .zip extension and you will have a valid stl

[bergerud]

Interesting problem. Do you have any more useful info like how the radius changes or the angle of the band? I assume the angle is constant (like 30 degrees up from the plane?) or I could just call it phi. It would be simpler if I knew the radius as a function of angle (the other angle). Is it just simple like r = h-k*theta ?

[dbfletcher]

The extrusion is 1"x .125 rectangle on a 30 degree angle with the starting point of the spiral at 10.375 (radius). The pitch of the spiral is 3" and rotations is 1.5. These numbers really are not all the important... im more interested in the algorithm for flattening such a shape. Btw.. was the step file of any value or did you have to view the stl. I assumed one of your programs would be able to open the step and you could get any measurement you needed.

[bergerud]

I opened the stl. (I did not right away find a way to open the step file.) Ok, I am on it.

[bergerud]

I have the method figured out, but, I wonder if they will be of any use to you. I will end up with some formulas that only a sophisticated math program can plot.

What I will do is to calculate the tangent vector to the center spiral curve. Then take its derivative and project that vector onto the spiral surface. Integrating the projection then gives the equation of the flattened spiral. (This is about third year level calculus.) In the best case, I will be able to actually integrate and get a nice formula for the flattened curves. Most likely, however, I will end up with an integral which has to be approximated as the curves are plotted. If you do not have a sophisticated math program like Mathematica or Maple, the method will not be of much use to you. The best I will be able to do for you is give you an STL of the flattened spiral for a given set of the parameters.

In any case, it is an interesting problem and I will report back when I get it plotted. It may be a day or so.

[dbfletcher]

I used to use mathcad several years ago... I might still have the laying around. In any case, I'm a programmer by trade so i always enjoy turning algorithms in to lines of code. I was a double major in College (Comp Sci & Mathematics) so hopefully I will be able to understand you... if not.. i'll ask! That was probably my biggest complain with they way I was taught math... I could perform mechanics, but often times I didnt fully understand how to apply the knowledge. I really wish there were way more verbal problems when I took the higher level courses... instead of mostly just drills on the mechanics.

[bergerud]

Here is a potpourri of files. The zips are, of course not zips. I believe the problem is solved here. It was harder than one would have expected. This is the kind of problem I like; figuring out how to make a shape become something else. Like the rotary jig calculations.

The form of the solution, however, requires serious computing power to plot. The pdf s are of the actual math calculations and of calculations done in the powerful Maple math program.

The png image shows the original central curve of the spiral (red) and the sides of the flat shape (black) to cut out which will twist into Fletcher's surface as it lines up with the red. The Stl and Dxf files are of the flat surface. (One could, using the STL importer, get the CW to it cut. The dxf is 3D and will not import into Designer.)

If you have a few years of calculus, you may find the method interesting as it is generally applicable to many problems.

Thanks for the cool problem Doug.

[dbfletcher]

Well.. that certainly explains why the math was giving me fits! What you did in a day, will probably take me a few weeks to digest. From a programmers perspective these are also the problems that I find most interesting... converting your algorithm to code should be kinda fun as well. There is another wrinkle to this problems, but I haven't figured out out to model it in my CAD program yet... the end of the spiral is also elevated by 4.5". The slope of the elevation should be constant along the path of the curve. Feel free to noodle that twist as well if you want. If I can figure out how to model the spiral spline in my cad program (Alibre), I'll repost the stl & step files. I know it support 3d sketching, but i dont know how to add the z component to the x/y spiral it automatically creates for me.

[bergerud]

What is this thing for?

[dbfletcher]

It is actually for work...the fabricators have always made these by trial and error. The ask if I could do it.... at first I said yes because I do it all the time when flattening conic sections.... but after i modeled it.... i realized it wasn't going to be as simple as i originally thought. The purpose of the spiral is a track extension for a vibratory bowl feeder. I've been trying to stress to the shop guys that most of the things they do by "trial and error" probably have mathematical solutions that we could apply to greatly speed up the assembly process. Everything we do is basically a one-off, and some of the bowls can take 80+ hours of tooling time to get all the features added a customer requests. Coming up with the solution for this was probably definitely beyond my current abilities, but I had a hutch you would enjoy it.

[bergerud]

That was fun. The extra 4.5 inch lift is another problem altogether. That takes the curve out of the plane and so introduces torsion. It is probably not a mathematically possible surface to make from a sheet. Some stretching would be involved. It is thin and the math would give a good approximation. The calculations really blow up and I would have to do it all by CAS. Probably take two days. Sometimes you just have to bend, weld and hammer! If you want, I will give it a try later.

[bergerud]

I played around some more today with the twisted spiral and discovered something interesting. For the geometry of this spiral (twisted or not), the angle (psi) calculated from the geodesic curvature is approximately a linear function of theta. This drastically reduces the amount of computations involved. There is hope for the twisted spiral. I will work on it some more if you think the attached plot is close. It is an approximation for twisted spiral. It is an approximation for two reasons, one because of the linear approximation and two, because the surface has to twist a little when you lift it up.

[aokweld101]

I have done the spiral before and its not easy it was trial and error. more on error. we laid it out on the concrete with soapstone on the floor and made the pattern from the layout off the floor. we didn't have the people that understood the concepts even the engineers didn't have the booksmarts to figure it out. You have to think that the people working doing the work, only had if any a ged or high school diploma. so to see berenrud throw numbers out there its impressive indeed. I can add and subtract ....so much money comes in... and ....only so much money goes out.....:rolleyes:

[dbfletcher]

I played around some more today with the twisted spiral and discovered something interesting. For the geometry of this spiral (twisted or not), the angle (psi) calculated from the geodesic curvature is approximately a linear function of theta. This drastically reduces the amount of computations involved. There is hope for the twisted spiral. I will work on it some more if you think the attached plot is close. It is an approximation for twisted spiral. It is an approximation for two reasons, one because of the linear approximation and two, because the surface has to twist a little when you lift it up.

Thanks for all you work on this. Of course my end goal is really to be able to understand what you have done. It has been quite a while since I have used such high level math, so I'm sure I'm quite rusty. Algebra, Geometry, & trig i use on almost a daily basis... but really haven't done much calculus since college. But I do intend on completely understanding at the end of the day. Thank you for you input on this problem!

[bergerud]

I am really getting into this and may find some more short cuts. This is a great problem. I would like to understand exactly what you need to do with this. Is this just a one off or will you make similar spirals of differing sizes? Do you want to be able to change the parameters? Do you want to understand the math for the sake of understanding or because you will use it for work?

I found a small error in the original calculation which I did not notice since it did not perceptibly change the result. I will post a corrected and simpler flat spiral calculation. I am still playing with the twisted spiral.

[dbfletcher]

As I mentioned, everything we do is pretty much a one off. But I'm sure there will be others that are similar in the future. I want to understand the math, because I'm always curious... I love math and like to continually learn how to apply it with real world situations. There is not clear cut line between "work"... and activities out side of work for me. Most days I take a challenge home with me and tinker and play with ideas to try and understand it better. Sometimes solutions present themselves... and other times I'm force just to put in on the back burner for a while until new inspiration hits. You can never know too much, and definitely desire to be a life long learner. If money wasn't and issue... I would have loved to have just become a profession student!

I'm sure for the "work" aspect of this, there is just a handful of sizes and parameters what could/would change. But I always like to understand the generalized approach so that it can be applied to wide variety of situations.

[bergerud]

Well here are my results so far. There are two pdfs. The regular spiral and the twisted spiral.
What is interesting about calculations like these is that the exactness of mathematics makes the calculations very messy. One can ignore all kinds of details and still get what looks like the same result. In fact, the result for the spiral and the twisted spiral are even very close.

I would understand the spiral calculations before even looking at the twisted spiral calculations. Let me know if you have any questions or if you want something plotted.

[ktjwilliams]

What ??? HUH !!!! :confused: