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New problem with circles

ValeryOchkov
24-Ruby III

New problem with circles

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)

D1D2D3L.pngD1D2L.png

 

See also

https://community.ptc.com/t5/PTC-Mathcad/Malfatti-circles/m-p/687281

 

49 REPLIES 49

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?

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.  

DJF_0-1645707735213.png

 

 

ValeryOchkov
24-Ruby III
(To:DJF)

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.

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.

K2.gif

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

T2.gif

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

MC2.gif

Lets see what you can come up with ..

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!

Бревна.png

PS

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

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 ...

Werner_E_0-1645768361888.pngWerner_E_1-1645768369826.png

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.

 

 

"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 . . .


@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

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.

pens.png

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.

Werner_E_0-1645902932238.png

Werner_E_1-1645902940059.png

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 ...

 

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.

D-D-D-L.png

 

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.

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.

 

> 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?

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

1-oOo.png2-ooo.png3-o-o-o.png4-PE.png5-PE-Plot.png6-PE-Plot.png

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:

Werner_E_0-1645983560088.png

 

Thanks Werner for the testing!

Idiot-proof - Wikipedia

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

ValeryOchkov_0-1645988909983.png

But for the animation? h<R ?

 

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

 

" 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?  


@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 😈

 

And what about this?

five balls covered with an elastic transparent film. What is the area of the film? Will there be a pendulum here

8-5-Spheres.png

My try to solve this ODE problem - see please Mathcad 15 (with an error) and Prime 6 in attach.

I am not sure is it correct.

Check please!

Not sure if its correct. What about the force the elastic has on the outer two rods which in turn force the center one to go up?

 

IN MC15 you forgot to define h.max and used the wrong syntax for odesolve.

Werner_E_3-1646503311902.png

Furthermore the solve block fails if the initial height is just 2 or lower (actually a minimial height of r=1 should be allowed).
We have to lower t.end to make the solve block work again. And the height definitely should not go beyond the value shown by the horizontal marker in the plot!?

Werner_E_4-1646504036256.png

 

 

Thanks!

See please the attach

Not sure if its realistic. Do we have a perpetuum mobile just because friction is ignored? I would rather have expected a slowing down of the oscillation because of gravity.

And then - shouldn't the model also work with an initial value for h smaller than 2.1?

I started into this, making several simplifying assumptions: the center cylinder can only move vertically was one.  

 

My derivation for length of the elastic band presents a different solution than Val's

Fred_Kohlhepp_0-1646572484178.png

and I would like to see my error.  (I solved with respect to the angle b, so some thought to compare is needed.)

 

Beyond that difference:

     A pendulum works and oscillates as energy moves from potential energy to kinetic energy and back.  As the center cylinder decends its speed needs to increase and then decrease, I'm not sure how that works in this situation.  Clearly if the center cylinder is at its top (hmax) the potential energy is maximum, but it's potential energy zero point would be level with the other two (h = R).

 

Attached Prime 4 file.

 

I think it should be that way:

Werner_E_1-1646579671503.png

No matter how you move around the circles, the three arcs where the rubber band touches, always add up to a full circle.

 

>  A pendulum works and oscillates as energy moves from potential energy to kinetic energy and back.  

Yes, you sure are right and my question about the perpetuum mobile and that I would have expected a slowing down wasn't very clever 😉 Of course energy moves back and forth endlessly if it is not converted into heat by friction.

 

Calculations are calculations, but an experiment is needed.
I bought three cans of an Energy drink - Wikipedia. On the can it is written that absolute energy is stored there. In one - kinetic, in the second - the potential of a raised can, and in the third - the potential of a stretched elastic band.
I took an elastic band, which binds banknotes.
I did an experiment. First in static - see photo. Tomorrow I will buy good oil, lubricate everything with it and try to get the cans to vibrate - a pendulum.

OOO-Energy-Drink-1.JPGOOO-Energy-Drink-3.jpgOOO-Energy-Drink-2.JPG

 

PS

В России часто говорят так - эта проблема настолько сложная, что ее без пол-литры (бутылка водки 0.5 л) не решишь.

Можно было вместо трех банок с энергетиком взять три бутылки водки. Для эксперимента. 

In Russia, they often say this - this problem is so complicated that it cannot be solved without half a liter (a bottle of vodka 0.5 l).

It was possible to take three bottles of vodka instead of three cans of energy drinks. For an experiment.

The math model isn't exactly right, but the animation is very believable.

OOO-ODE.gif

PS

Der berühmte Mathematiker Gauß sagte einmal, als er über ein komplexes mathematisches Problem nachdachte: - Ich sehe die Lösung bereits klar, aber ich weiß immer noch nicht, wie ich sie erreichen werde.

Maybe its even like this (of course the animation is a fake for demonstration w/o any realistic model behind)

pencils.gif

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