topic Re: New problem with circles in Mathcad

New problem with circles

ValeryOchkov — Wed, 23 Feb 2022 09:05:42 GMT

The belt pulls together two or three circles with D1, D2 and D3. What is the length L of the belt?

If D1=D2=D3 L=D*(3+pi)

Re: New problem with circles

ValeryOchkov — Thu, 24 Feb 2022 05:34:39 GMT

One of the solutions in attachment - Prime 6 and pdf.
But I would like to see a more elegant solution. Is it possible?
"How do you boil water in a kettle?
- You need to pour water into the kettle and put it on fire!
- And if there is already water in the kettle?
“We need to pour out the water and reduce the problem to the previous one!”

Who is the author?

Re: New problem with circles

DJF — Thu, 24 Feb 2022 13:04:37 GMT

Here's a quick solution for the two circle problem. Three circle can be solved the same way but needs a check on if the 3rd circle extends outside the belt.

Re: New problem with circles

ValeryOchkov — Thu, 24 Feb 2022 18:10:53 GMT

Thanks DFJ for the nice and simple solution.
But other simplicity is worse than theft - theft of a graphical representation of the task and the transition to three, four, etc. circles.
But from the very beginning, I focused not on two, but on three circles and on a graphical display of the problem.

Re: New problem with circles

Werner_E — Sat, 26 Feb 2022 20:53:13 GMT

I would suggest a more generic problem.

Let's not limit ourselves to touching circles, and let's not limit the number to three or four. Let's find the convex hull of any number of circles in any position and find its perimeter.

Of course we can use the very same solution to deal with touching circles as well.

Here's a nice animation with three moving circles trying to escape the rubber band:

Lets see what you can come up with ..

Re: New problem with circles

ValeryOchkov — Fri, 25 Feb 2022 05:38:20 GMT

Fine! A solution without animation is not a solution.
But a solution without the code shown is not a solution too.

Let's start with an open function L(D1, D2, D3):=...

Than L(D):=... where D is a vector.

And without crossing circles. Only touch!

I often get Werner's solutions just in time for my morning coffee.
We might think that Werner lives in Australia.

Re: New problem with circles

Werner_E — Sat, 26 Feb 2022 21:07:19 GMT

The function you are asking for is called "TouchingCircles" im my sheet - the rest is done by the already mentioned generic functions.

And no, I'm not living in Australia - no kangaroos here in Austria 😉
We call it "Senile Bettflucht" over here 😞

I'am waiting for other solutions.

My solution still fails in some special cases but I don't think I'll be interested in trying to fix that in the near (or even far) future 😉 It would require to add some if statements which usually ruins the beauty of the main solution 😞
And once the main problem is solved, my interest usually fades very quickly ...

Anyway - here is something for you to play with a bit if you like ...

BTW, a nice problem to solve would be a couple of circles in arbitrary position (not overlapping) and then wrap a rubber band around them and see how they move and come closer. But I guess that the physics involved here is beyond me and my resources.

EDIT (2022-02-26): The worksheet which was attached here calculated wrong values for the circumferences. Should be fixed in the new attached sheet.

Re: New problem with circles

Fred_Kohlhepp — Fri, 25 Feb 2022 11:52:58 GMT

"BTW, a nice problem to solve would be a couple of circles in arbitrary position (not overlapping) and then wrap a rubber band around them and see how they move and come closer. But I guess that the physics involved here is beyond me and my resources."

The physics is simple: find the minimum energy, in this case minimize the tension in the rubber band. (The rubber band simple model is Hooke's Law.) Now the math . . .

Re: New problem with circles

Werner_E — Fri, 25 Feb 2022 13:04:25 GMT

@Fred_Kohlhepp wrote:

"BTW, a nice problem to solve would be a couple of circles in arbitrary position (not overlapping) and then wrap a rubber band around them and see how they move and come closer. But I guess that the physics involved here is beyond me and my resources."

The physics is simple: find the minimum energy, in this case minimize the tension in the rubber band. (The rubber band simple model is Hooke's Law.) Now the math . . .

The rubber band alone may not be the problem even though I don't see at the moment how I would deal with a stretched rubber band alone - it won't keep the initial shape and maybe if we neglect gravity its end position is a perfect circle.
But then adding the differently sized circles (Valerys tree trunks) and their interactions with each other and with the rubber band ... Nothing I would dare to do on the side

Re: New problem with circles

ValeryOchkov — Sat, 26 Feb 2022 17:50:11 GMT

Thanks Fred and Werner for a good idea.
We take a bunch of pencils, tighten them with a rubber ring and calculate how it all falls. Friction is not taken into account, but we minimize the potential energy of pencils and a stretched rubber band.

Re: New problem with circles

Werner_E — Sat, 26 Feb 2022 19:22:08 GMT

I gave it a first quick try. Just minimizing the circumference and avoiding overlapping of the circles - no physics 😉

The result is somewhat disappointing even with a very low value for TOL. The pics show the result of the Conjugate Gradients algorithm, the Quasi-Newton is doing its job a slightly better but still not satisfying.

While playing around with this I also noticed that in my sheet the circumferences were nor calculated correctly. This is now fixed and the faulty sheet replaced by the fixed one (including the minimize attempt).

And of course the most interesting part still is missing - the movement of the circles (pencils, tree trunks) from their initial position to its end position. But I think I'll leave that to do for others ...

Re: New problem with circles

ValeryOchkov — Sat, 26 Feb 2022 21:26:21 GMT

The first step - three pencils lie on the table and are pulled together with an elastic Hoocke band. While without taking into account the force of friction. An interesting curve turned out - with a maximum and a minimum. Prime 6 in attach.

Re: New problem with circles

Werner_E — Sat, 26 Feb 2022 22:06:00 GMT

We sure need a nice animation for this!

That would mean to make the function dependent on time, not on height of the top rod.

In my opinion either neither friction,gravity and mass should be considered or all of them.

In the first case the interesting part would not be the end position but rather the intermediate steps from an arbitrary initial position to the end position. But as I had shown above, the results of the built in numeric minimize are quite disappointing.

The second case sure is more interesting but much more difficult to deal with. Friction would not only occur between two rods but also to a large amount between the rubber band and the rods. It would also mean that the tension in the rubber band is not uniform - tension would be less where the rubber band touches the rods.

Re: New problem with circles

ValeryOchkov — Sat, 26 Feb 2022 22:00:48 GMT

Animation of a new interesting pendulum. Damped if friction is taken into account. And with not linear Hooke's law!
And with 3, 4, 5... circles with not equal radius.

A good problem for one Senile Bettfluch at this nacht 🙂

I tied three pencils with an elastic band, pressed the top pencil, let it go. The pencil starts jumping for joy that I let it go.

Re: New problem with circles

Werner_E — Sat, 26 Feb 2022 22:43:43 GMT

> A good problem for one Senile Bettfluch at this nacht

Not for me, but maybe for someone who already had posted a couple of nice pendulum animations 🙂

> I tied three pencils with an elastic band, pressed the top pencil, let it go.

Ancient lead pencil at 1kg each?

> The pencil starts jumping for joy that I let it go.

That would be the animation I was looking for.

In real life I also would expect some kind of oscillation before the final position is arrived. Without friction - would we have an endless oscillation?

Re: New problem with circles

ValeryOchkov — Sun, 27 Feb 2022 11:24:04 GMT

Hi Fred & Werner.
Look at the article in the attachment. Maybe you will translate, add will become co-authors. The idea is yours. Prime 6 in attach.

Figures

Re: New problem with circles

Werner_E — Sun, 27 Feb 2022 17:39:41 GMT

Pretty interesting even without considering friction. Your last pics show that a very strong elastic is able to bring the rods in a position near(?) a perfect pyramid - all circles touching. I wonder how h could be made time-dependent?

I guess that @Fred_Kohlhepp would be able to say more about it.

EDIT: I tried m.O:=1kg and k:=5 N/m and got this:

Re: New problem with circles

ValeryOchkov — Sun, 27 Feb 2022 19:17:34 GMT

Thanks Werner for the testing!

Idiot-proof - Wikipedia

h := if(h < R, R, h) - no the local minimum.

But for the animation? h<R ?

I dream of watching the animation of this new unusual pendulum over my morning coffee tomorrow.

Re: New problem with circles

Fred_Kohlhepp — Sun, 27 Feb 2022 23:19:29 GMT

" I wonder how h could be made time-dependent?"

Well. you have gravity (or you can set it), you've defined mass and spring rate. All you need now is Newton (F = m a). Force applied to mass creates acceleration. Acceleration is the derivative of velocity with time, the second derivative of position. The rubber band exerts a force on the "pencils", which begin to move, which changes the length of he rubber band and the applied forces , , , And you have a three-body two-dimensional set of differential equations in time! NOT ME!!

And how will you treat the collisions when the pencils come together?

Re: New problem with circles

Werner_E — Mon, 28 Feb 2022 01:04:44 GMT

@Fred_Kohlhepp wrote:

NOT ME!!

So there are already two of us who won't do it 😉

Furthermore - when the three pencils are lying on the table side by side and encircled with the rubber band, in an ideal situation the won't move, right? The forces applied by the two outer pencils in the inner one cancel.

In real life situations sometimes the middle pencil will rise up and sometimes the middle one is pressed down and so the outer two will rise and then the whole thing will tumble.

Guess we should leave it for somebody who had already posted some multi body calculations and animations in the past 😈