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)
See also
https://community.ptc.com/t5/PTC-Mathcad/Malfatti-circles/m-p/687281
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.
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 ..
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!
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 ...
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.
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.
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 ...
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.
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
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:
Thanks Werner for the testing!
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.
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
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.
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!?
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
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:
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.
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.
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)
