Sum of torque

[EOW]

I think the integrals are good, I had a problem of coef in my program and it is ok now. Like that I have the sum of forces at 0 and without the green center I have the sum of torques at 0 too.

But with the green center I can have a difference in the sum of energy.

I have the definite integral so it can't be a problem of accuracy

[EOW]

I drawn an image to show how is the black axis, without the balls and the red wall to watch details. Spoke are not in the same plan like that they don't interact with balls. The semicircle can only turn around the black axis and the black axis is fixed to the black arm. The semicircle can only rotate around itself but there is no torque on it (because it is a part of a circle and the forces of pressure are perpendicular to the surface). The semicircle is in an unstable position but in theory there is no torque on it and in practice it's possible to correct with an external device the position.

I don't wrote but I show an image with a rotation of the device, but note:

The red wall is free to rotate around the fixed red axis, a motor force the red wall to rotate clockwise and follow the half disk
The black arm (telescopic) is free to rotate around the fixed green axis, it has the force F2 at end (black dot). The black arm rotates around the green center and increases its length
the semicircle is free to rotate around the black dot but it don't rotate around it because there is no torque
The trajectory of the black dot is a circle with a radius of 1.5 and the center is the red axis.
I used balls inside the half circle but it's possible to use another shape like ellipse.

The black axis is not connected with the red arm.

[dieter]

I see you're very creative. As usual this is way over my head ^^ nevertheless: keep up the good work!

[EOW]

Thanks. Maybe someone could understand what I'm trying to explain

[LibreEnergia]

Thanks. Maybe someone could understand what I'm trying to explain

Noether's Theorem tells us that for every differential symmetry there is a corresponding conservation law.  In my opinion if you are finding anything other than zero change in energy while integrating over a path that starts and ends in the same physical positions then your analysis is wrong.

In this case 1, The motion is continuously differentiable, ie it has no inflection points, and it does represent a mathematical  'symmetry' , in this case a rotation,  so it meets the conditions for which Noether's theorem applies.

In my opinion you will find the error in analyzing the way you are calculating the integrals numerically, in that small errors introduced when summing numbers containing large differences in numerator and denominator give rise to an error in the totals.