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
It's stronger than vodka.
I used this in 1968-69 to blow up heaps of coal that died in the cold when I worked as the head of a boiler house in Kazakhstan.
Werner, Fred!
And what, my formula for determining the length of an elastic band is not correct? At first I calculated it through the betta angle, but Alan suggested a simpler formula.
Val,
Your equations aren't wrong! My first posting was wrong (which Werner pointed out.)
But your first equation
is only valid when the center cylinder is in contact with the ends (b < 60 deg) since it's Pythagoras' theorem and you need the distance between the centers to be 2r.
I think to solve the motions we need to look at the forces that develop. I'm working on it, but it may take a few days . . .
Ignoring friction (while it will yield unrealistic results) will reveal the characteristics of the oscillator Val envisions, if it's there. The basic solutions for pendulums and weights on springs start by ignoring friction. Which equation below have you seen more often?
One thing that interests me is the effects of the mass of the cylinders and the strength of the elastic band, and their effects on the motion. If (for example) the elastic is very strong then the condition of minimum length will be the resting state. As the band gets softer and the cylinders heavier we would reach a point where the three cylinders all lie on the table surface. Is there a weight/strength relationship that would settle out at some intermediate angle?
Attaching my "in work" file (clearly not done!) for comment/correction.
You sure are right that omitting the friction is mandatory, at least in the first attempts. And it sure would be interesting to see if you can achieve the oscillation Valery expects.
But I wondered about how the tension in the whole rubber band would affect a single cylinder. Especially I am absolutely not clear about the effect the part of the rubber band around the circle would have.
In your sheet you make the down force on the upper circle dependent on the change of length of the whole rubber band and I wonder if its not just the part of the rubber band I sketched in red in the following pic (as you can see I am absolutely unsure about the arc part of it). The ratio of this part of the band to the full circumfence varies as the circles start moving.
Just some vague thoughts of someone with very limited knowledge in applied physics.
EDIT: Just read that a band of elastics would be different from a normal spring as its something like an entropic spring which stores its energy as thermal energy which is also the reason a rubber band gets hotter when its stretched ... ?? Does this mean that Hooke can not be applied? (https://www.quora.com/Which-force-is-used-for-stretching-a-rubber-band)
Hmmm, I knew it would be beyond me ...
All good questions:
"In your sheet you make the down force on the upper circle dependent on the change of length of the whole rubber band and I wonder if its not just the part of the rubber band I sketched in red in the following pic (as you can see I am absolutely unsure about the arc part of it). The ratio of this part of the band to the full circumfence varies as the circles start moving."
If we are ignoring friction (and I think we've agreed at least in first efforts,) then the elastic band can slide freely over the cylinders, and the tension in the band will be uniform along the band and related to the change in the total length.
"Just read that a band of elastics would be different from a normal spring as its something like an entropic spring which stores its energy as thermal energy which is also the reason a rubber band gets hotter when its stretched ... ?? Does this mean that Hooke can not be applied?"
Once again, you are correct. A rubber band is a poor spring. If you do an experiment you will discover that it has hysteresis, the loading curve is different from the unloading curve:
So we could find a different expression for the force/extension curve and even account for whether the band was expanding or contracting. But the assumption that it obeys Hooke's Law, like ignoring friction, is a good first approximation.
Thanks for clarifying.
I am already looking forward to your further results in this matter (which we both said that we would not deal with further 😉 )
Actually I would have thought that the downward force resulting from the elastic on the top circle would be larger in the first picture below than in the second one (the total length of the rubber band is the same in both cases)
" I would have thought that the downward force resulting from the elastic on the top circle would be larger in the first picture below than in the second one (the total length of the rubber band is the same in both cases)"
Once again you're right! The band exerts a tension force on each side of the central cylinder that's aligned with the band. So the vector sum of these two forces sums to zero in the horizontal direction and to a smaller value as b decreases. When the three cylinders align in a horizontal row the vertical force from the band is zero. (This is part of the reason why I'm not convinced that this arrangement will ever oscillate.)
(which we both said that we would not deal with further 
Yeah! Well, it's nice to have a discussion!
@Fred_Kohlhepp wrote:
When I was a student, MPEI had the most modern electrical engineering classroom with TVs at that time. There was a note in the main newspaper of the USSR "Pravda". It was written there that the lecturer presses a button and a Hysteresis loop (with a large letter!) appears on the TV screen as if alive.
The lecturer became famous throughout the country, but many approached him and asked: "And how is old Hysteresis doing there?"
New new problem.
Three cylinders of radius r are connected together in the form of a pyramid (see the Photo) and fit tightly into a pipe with radius R. The radius of the pipe begins to increase. How will the position of the cylinders in the pipe change? A variant of the problem: the cylinders are connected by an elastic band of different elasticity.
We can get an oscillator without a rubber band!
