# Wrapping a Ribbon Around a Cone

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1. Nov 15, 2018

### jonny_d

Greetings everybody. This is my first post and I am looking for help with a little math/geometry/engineering problem. This has been a real brain buster for my colleague and I the past couple days so I am hoping somebody can help. I am not sure if this is the best section for it, but it applies to an engineering design I am working on so I will put it here...

I need to come up with a math model for the following problem:

A ribbon is wrapped around a cone. The cone rotates around it's axis as a machine traverses down the length of the cone (parallel to the cone axis) while applying even tension through the cross section of the ribbon. The ribbon material is of high modulus and it can be assumed to have no compliance except in the direction that it bends on to the surface of the cone. Also the ribbon must lay flat on the surface of the cone with no wrinkling or buckling of the material.

Here is a quick sketch of the problem:

Through experimentation and intuition I have determined that the wrap angle (theta) and the pitch will change as the wrap progresses. Specifically, the wrap angle will increase and the pitch will decrease as shown in the image. So for the machine to work, it will need to change the angle that it grips the material and compensate for the changing pitch as it translates along the length of the cone.

For a math model to represent the system I am thinking that it will need the following as I/O:
Inputs: cone taper angle (alpha), cone base diameter (D), initial wrap angle (theta), delta L
Outputs: pitch and wrap angle at the given delta L

I would also be interested in any advice on how to accurately model this in CAD in such a way that the ribbon is not deformed. I have access to SolidWorks and Catia.

Final note: the cone angle in my sketch is greatly exaggerated, and in actuality it will be a tapered cylinder, but I am considering it a cone here for simplicity. The actual taper is less than a degree over a length of 40 feet. This may sound insignificant but experimentation has shown that it truly matters for the application.

So does anyone want to take a crack at this? Thanks in advance!

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2. Nov 15, 2018

### Tom.G

I once worked (decades ago) at a plant that made machines to wind roving onto Helicopter rotors. So your problem isn't especially new, nor does it seem easily findable on-line. Anyhow, here is a slim lead. Hope it helps.

Filament winding processes introduce residual fiber stresses in the fiber matrix assembly that are retained through cure. Each of these phenomena can result in local residual stresses of significance in the matrix. Procedures for computing these stresses were developed under U.S. government contract and reported by Rai and Brockman (1988); thus, the code is available on request.

Found with https://www.google.com/search?&q=wrapped+helicopter+blades 3 250 000 hits, this was at the bottom of the first page of results.

Cheers,
Tom

3. Nov 16, 2018

### 256bits

Interesting problem, especially when one considers that a cone can be made from a flat sheet of paper.
And yet pasting flat ribbons onto the curved cone does present problems.

so what we can do is flatten out our cone and draw ribbons on that flat space.
One a sheet of paper, draw a large circle of radius R.
Draw another radius R such that the area between the two R's represents the area of the cone flattened out.
Draw more R's, to represent more cones.

Now drawing parallel lines of width w across the circle will represent ribbons of width w that can be pasted flatly onto the cone.
If you draw two more circles of radii R1 and R2 to represent a tapered cylinder, one can see how the ribbons can be wrapped. Say for example, starting at circle R1, the narrow end, parallel lines initially perpendicular to the radius, one can see how many wraps go around the cone to reach R2, and the changing pitch as the ribbon moves from the smaller end to the larger. Some equation should follow.

4. Nov 16, 2018

### 256bits

here is what I mean.

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5. Nov 16, 2018

### Tom.G

Consider that you are requiring the ribbon to be parallel to the cone surface at all points. This means that the ribbon path will have a component of winding direction that is parallel to the cone axis. According to the article linked below, the winding pitch is calculable based on the cone apex angle and that it will be shown later; I haven't found "later" though.

See Problem 2.2 (about 40% down the page) at: http://pi.math.cornell.edu/~dwh/books/eg99/Ch02/Ch02.html

Cheers,
Tom

p.s. This thread may get better answers in the Mathematics forums. Here is paging @mfb to perhaps move it.

6. Nov 16, 2018

### 256bits

http://www.mathcurve.com/courbes3d.gb/heliceconic/heliceconic.shtml
Or a conical spiral, but that would work only for laying a string upon the cone with a particular constant pitch.
http://www.mathcurve.com/courbes3d.gb/spiraleconic/pappus.shtml

For laying a ribbon, the geodesic has to be involved for the ribbon to lay flat.
For a general surface,
http://www.mathcurve.com/courbes3d.gb/lignes/geodesic.shtml
which does not give any intuitive sense of what the required straight lines would look like if applied for a cone.
, which is exactly what Post #3 and #4 are explaining.

Instead we go to the cone of revolution,
http://www.mathcurve.com/surfaces.gb/conederevolution/conederevolution.shtml
which gives an equation for the geodesics.

7. Nov 17, 2018

### Staff: Mentor

The mathematics part looks nicely covered already, the engineering questions are still open.
I put a redirect in the mathematics section, I think the main thread is fine here.

8. Nov 17, 2018

### Tom.G

Although you may have already addressed it, another fine point just came to mind.
Even with the traveler moving parallel to the cone axis, ideally the throat of the traveler feeding the ribbon should be parallel to the cone surface. Note that this is a rotation in two dimensions, albeit by less than one degree.

9. Nov 19, 2018

### Baluncore

I do not believe you can wind a straight tape or ribbon onto a rigid cone without distorting the ribbon. Consider a small square of ribbon glued onto the surface near the thick end of the cone. Extrapolate two opposite sides of that square as lines along the conical surface. They will diverge which means the ribbon cannot fit the conical surface.

10. Nov 19, 2018

### 256bits

That is not correct.
A sphere, a 3d surface will not accept squares lain upon its surface.
Since the cone is a 2d curved flat surface it can accept.

There can be parallel lines marked upon the surface of a cone.
Only method is using the geodesics for the ribbon to lay flat, in which case the ribbon runs out of cone quite quickly.
Or, a strip of ribbon lain longitudinally, in which case the strips will have to be tapered for no overlap.

If one uses overlap while wrapping the cone, then the conical spiral might suffice if the thickness of the warp is sufficient.

11. Nov 19, 2018

### Baluncore

But the two parallel lines will be of different lengths, which must distort or skew the tape.

12. Nov 19, 2018

### OmCheeto

You can do what I did, and actually make one, and see that 256bits is correct.
It just wont look anything like the doodle in the OP.
I didn't understand posts 3 & 4 until I doodled my own flat cone.

Quite boring, as the ribbons don't even make it around the cone even once!
But then I realized that if I doodled in extra equally spaced radial lines, I could get the ribbon to circle the cone/layers of cone quite as often as I like.

In other words, in post #4, the cones labeled 1 thru 6 aren't separate cones, they are the same cone. Layers, if you will. The radial lines represent where the cones/layers line up.

I will have to work out the mathematical solution for myself though, as the references 256bits has provided are a bit noisy.
Doesn't really look that hard. And I won't have to learn a bunch of fancy shmancy math words and phrases: directrices? "principal radii of curvature"? [from the "cone of revolution" link]

Fun problem!

13. Nov 20, 2018

### votingmachine

It might be productive to not use a circular cone but to use a pyramid shape as the basis for calculating the angles. I can wind a ribbon around a triangle. each time I flip across an edge, I fold with a fairly defined symmetry around that edge. My approaching two edges hit the triangle edge and wrap, forming a "V" shape. The "V" gets smaller with each successive fold. (I believe the change in pitch is half the internal triangle end angle ... but I'm playing around not being careful).

It seems straightforward for the triangle to calculate the change in the pitch angle, relative the the angle presented by the triangle. I have no idea if the calculation gets worse with a triangular pyramid, or a square-based pyramid, or a higher number of sides one. I doubt you need a very high number of sides to compensate for the angle.

I'm probably simplifying too much, but the problem of wrapping a ribbon around a cylinder is not much different from wrapping a ribbon around a long rectangle. And you say the angle is small between your "cone" and a cylinder. You might be able to just calculate the angular drift from the case of the triangle, and be close enough.

And of course if you use a very large number of sides for the base of the pyramid, the accuracy can be arbitrarily high.

14. Nov 21, 2018

### 256bits

It is tricky to visualize.
But take a sheet of paper and make a cone out of it - that you agree should be possible.

Undo the paper and cut out, or mark, a square.
Make a cone out of that paper with the cut out square showing.
The lines must continue to be parallel.

Maybe even better is to use some squared graphing paper.

But, you really are right in a way, ( ie I did say incorrect but I should have said with reservation because )
If one draws a two circles around the cone with the centre being at the axis, those two lines should be parallel, but curved, ( ias one circle is longer than the other. )
( Parallel curved lines ?? - equidistant along the length is probably a better description )
Take a paper coffee cup and unwrap it.
Even though the cup when on the table sits nice and sturdy, one can see that the top and bottom are actually curved when flattening it out.
So a ribbon, since it cannot follow those curved lines has to work its way up the cone ( narrow end to larger end ) to remain flat on the surface.

15. Nov 21, 2018

### 256bits

I admit, my picture is not self explanatory - thanks for the better description.
Quite boring?? And surprising too.
The ancient Greeks must have had a whole lot of time on there hands, as they just loved geometry, angles, and all this fun stuff to the max.
It really is a mind bender of sorts for cones and their spirals.

16. Nov 21, 2018

### 256bits

One engineering problem for the OP to consider.

17. Nov 21, 2018

### OmCheeto

I've gotten started on the problem, and have run into a few realizations/problems/constraints already.
This is not a trivial problem.

This makes absolutely no sense to me in three dimensions. But I think I understand what you meant, in the context of the discussion.
File it under constraints and realizations.

Constraint: The ribbon cannot intersect with the apex of the cone.
Realization: The problem is trivial if the ribbon runs parallel to the 2D cone axis. But if not, it can be rectified by rotating the 2D cone.
I'm not sure how to describe this properly, without a bunch of doodles and maths.

ps. Someone remind me in about 4 hours to take a break from this problem, and start working on cooking my turkey.

18. Nov 21, 2018

### Baluncore

I reduced the problem to a long thin cone, from the apex, infinitely extended. Included apical angle = alpha.
That conical surface can be mapped to 2D, drawn with the apex at the origin. One side of the seam lies along the x-axis, the other side of the seam is a diagonal line from the origin at an angle beta, to the x-axis. beta = Pi * alpha.

Now draw the ribbon at an angle (heading) gamma from the x-axis, to contact the diagonal seam line.
Transfer the seam contact points back to the x-axis, using arcs centred on the origin (= distance from apex).
Change plane heading of the ribbon by adding beta to gamma, then draw the next turn on the new heading.

Repeat until gamma directs the ribbon onto a heading that remains forever between the seams;
Or until the ribbon covers the apex at the origin, when the two sides of the ribbon can take quite different paths.

A double cone must then be considered, the path of the ribbon from one cone side to the other must pass through the apex point. The ribbon section must then be a tightly rolled spiral, with the sides of the spiral separated by zero in space but by the width of the ribbon in the spiralized flat surface world.

19. Nov 22, 2018

### jrmichler

AHA! So simple after it's solved. So I made a model cone 10" along the side (diagonal measurement) and 2" diameter at the base. The included angle of the flat pattern is 36 degrees. I wrapped a ribbon around it starting 2" from the tip.
ci

I then cut it open:
then c

It looks like the CAD model, with some allowance for not getting the ribbon started straight, nor wrapping it perfectly flat:

The next step is to develop the math model requested in the OP. It looks like the wrapping head angle changes by a fixed amount per revolution, which leaves the winding pitch per revolution to be calculated. Or is the ribbon wide enough that the wrapping head could be mounted on a swivel so that it tracks automatically?

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20. Nov 22, 2018

### Baluncore

Yes, each line does change direction by the same angle, my beta. With the cone surface seam along the x-axis, a simple mechanical ribbon feed will come from somewhere on the x-y plane. By drawing those ribbon centre lines on the surface to infinity we can see where they all meet, which will not be a point but more probably a caustic of some sort, if that is the right word for the silhouette of a line of tangents.

The ribbon can then be fed past a circular spool, mounted with axis vertical parallel to z-axis, centred at a position in the x-y plane), with a diameter and circumference that approximates the caustic.

There will be a 90° twist in the ribbon between the spool and the cone, a twist that may help or hinder the winding. Such a mechanical system will not handle the complexity of the spiral counter-twist near the apex, nor the situation where the ribbon is close to parallel with the seam.

As a general problem it is understood. But for a specific engineering solution we need to know the range of cone dimensions, the ribbon width and the range of the cone coverage required. Only then can the optimum linkage or mechanism be synthesised.

Last edited: Nov 22, 2018