Sum of torque

[EOW]

To simplify the study of this device. Think with an external theoretical device that add the energy FRtw1, like that all rotationnal velocitiy w1is  constant. I give FRtw1 and I receive Ft((R+r)w1-rw2), the difference is: Frt(w1-w2) with w1>w2, the energy is positive. I can't keep w2 constant, because I need the force F4 (and F5), if w2 constant => w3 constant and F4=0. So, w2 must change with time. Maybe a little sliding between grey disk / brown disk can be adjust the kinetic energy.

Heating gives H=Ft((R+r)w1-rw2)
Torques need Ft ( rw2-1/2rw2-3/2(R+r)w1+1/2(R+3r)w1-1/2rw2 ) = -FRtw1

Here F4=1/2F but it's possible to have another value it depends of the inertia.

[EOW]

I'm not sure when I drawn direction of rotation, so I give an example:

w1 = 10 rd/s
w2 = 8rd/s
w3 = 12rd/s

w1 = 10 rd/s, the arm is turning clockwise
w2 = 8rd/s, the grey disk is turning clockwise at w1 around red axis (labo reference) and grey disk is turnin around itself counterclockwise at 2 rd/s (arm reference)
w3 = 12rd/s, the brown disk is turning clockwise at w1 around red axis (labo reference) and brown disk is turning around itself clockwise at 2 rd/s (arm reference)

I guess |w2| < |w1| < |w3| . There is a relation between w2 and w3 at start because there is no slinding between grey disk and brown disk. In the arm frame reference w2 = -w3 at start.

[EOW]

In the case with the arm fixed and the purple disk is turning, I have a work from torque at -FRtw1+Frtw2+kFrtw2-kFrtw3, heating don't change, the sum is at kFrtw2-kFrtw3, it is 0 too. 'k' is kF force from brown disk. If there is a little sliding between the grey disk and the brown disk, the sum is always kFrtw2-kFrtw3. So there is more heating but +kFrtw2-kFrtw3 seems to be higher because w2 seems to increases more than w3.

[EOW]

First image:

Another idea. A motor is fixed to the arm accelerates the red disk, more and more. The slave disk (grey) is accelerated in the same time, no sliding between the red disk and the grey disk. All the system is turning at w1 around the magenta axis. I drawn all forces. The motor receive a negative  torque, it must give energy E1. The slave disk and the red disk receive its energy E1 and increase their rotationnal velocity. But the grey disk has the force F6 that incresase w1. In the contrary, F5 don't work. The sum of work seems to be positive.Like the grey disk is turning in the other direction that w1, choose the rotationnal velocity of the grey disk > w1 at start.

Second image: no motor after t=0

w3 > w1, and w2=w3 (or not exactly). All rotationnal velocities are counterclockwise. Before start, launch the device with w1, w2, w3 and set friction between disks at 0. At t=0, set friction ON, there are forces like I drawn. F6 can give energy to the arm via w1. Disks slow down but it is compensated by heating from friction.

[EOW]

Like this there is a torque on arm.