It's only 3 objects in free rotation. Not easy to build in practise the free rotation, but like Algodoo don't compute friction, if someone can test with another simulator it can be confirm (or not) these results, the test is very short to do. Nobody has another simulator ? With an axis for Object 1 it seems the system increase energy too. P = 18.5 rd/s * 0.026 m /2 * 88 * cos(60) = 10.5 W

edit: cvf2

In some cases, F1 don't work around its trajectory, F2 too. But F1/F2 give a torque, no ?

I understood too in the system where there is not direct torque from F(cg1)/F(cg2). In the system with trajectory like first image showing, forces are like second image shows. Center of gravity of Object2 moves like red circle. F1 and F2 apply a torque, this torque is exactly the same for Object1. For Object2, F1 is separated by d from F2, there a torque but F1 works more than F2. The difference of power is F*w*d.

If I change angular velocity to 6 rd/s, forces are divided by 8 and angular velocity by 3, the power is divided by 24, it's linear. I change frequency, this don't change the result.

I compute all torques, it's only the difference of trajectories that give energy. It's logical. When I find 22*3 = 66 W Algodoo find less: 15 W. It's possible to move red and orange objects and look at energy after one object touch green object, if forces are like I show before, the energy increase. Red object give 66 W and orange object give 171 W with calculation (without friction). Algodoo give 15 W for red and 35 W for orange object. Maybe the difference is somewhere else than friction. Note that orange and red object must be like image shows for have best result (if red and orange objects touch green object in the same side the energy is lower). Maybe the difference come from my method of compute torque. I think Algodoo compute with mesh 2D and I compute with center of gravity maybe the result is not the same because each object has its own trajectory. Like Algodoo don't give force of a part of object I can't compute torque like that.

I added the scene with unstable forces, energy increase only when forces are in a special position.

bad scene

In this last scene I can recover 1.3 % at frequency 12000 Hz. If it's an error of Algodoo it's 0.25 J of 18 J at this frequency and with stable forces I don't think it's possible. Someone else can test with another software ?

If blue object turn at constant rotational velocity around axis "axis" (external system not drawn limit its rotational velocity). Orange object is forced to be like image because there is Fc force. If I apply forces F1 and F2 on orange object, it will turn and I increase potential energy. But in the same time blue object has a torque from -F1/-F2 because distances d1 and d2 are not equal. The energy from blue object can be recover by external system and limit rotational velocity of blue object. With a rotational velocity clockwise, orange object can receive a standard torque. F1/F2 on orange object will slow down orange object and decrease potential energy but this energy can be recover in the same time.

I tested with Algodoo. The motor give 0.15 Nm * 9 rd/s = 1.35 J each second (in fact less because I take the last rotational velocity not the mean). Algodoo give 2 J, look at the scene please.

Another test:

rotational velocity at start of big object = -6.28 rd/s

rotational velocity at start of small object = 6.28 rd/s
rotational velocity at end of small object = 7.77 rd/s

energy at start for all system = 114.56 J
energy at end for all system = 116.408 J

Motor torque  = 0.15 Nm

Loss without motor for all system at 6.28 rd/s = 0.1 W

Power from motor = (7.77+6.28)/2*0.15 = 1.05 W

Power from Algodoo = (116.408-114.56)/1 + 0.1 = 1.95 W

Near the double because when torque is giving to small object the big receive the same free. Why not exactly the double ? because small object increase its rotational velocity at 7.77 rd/s but the big increase its rotational velocity to 6.34 rs/s  only. The energy is 0.946 W from the torque 0.15 Nm. So the result find by Algodoo is 1.95W and I must find 2 W !!! very close.

If I take stem and object at end like image shows. I give rotational velocity to stem around axis "x", object rotates around "x" not around "y". The system has less energy than if object turns around "y" too. Stem turns at w around "x", object turns around "x" at w, but don't turn around "y". Object and stem don't have same rotational velocity (w compare to 0). Now I can brake from stem to object, object add a torque (and I can recover energy from brake) to stem and add its rotational velocity, the system add energy.

I apply F3 to object, stem receive F1, axis "y" receive F2 and F4 ? so, all these torque cancel themselves but the system has more energy at end.

The system in this case seems to lost energy, but it's possible to give rotational velocity to object around axis "y" and accelerate the object around "y". At start all turn at w around "x" and object turns at w around "y". After, I accelerate rotational velocity of object around "y".

I think it's works only if moment of inertia are different for object and stem (around their axis), and it is I think. Energy giving by a torque is torque*angle, but like moment of inertia are different, angle will be different even torque are the same in value.