Sum of torque

[EOW]

A force F is needed for generate friction, so energy is needed for have friction. Force F is apply from an angle 'a1' to another angle 'a2'. It's not possible to have friction from 0 to 360° but only a part of 360°, it depend of the configuration.This force give energy to disks because F increase w1 of each disk. Friction don't decrease F, so all energy from F is giving to w1. Green arrows are forces from friction. Magenta arrows are forces that I need to give energy. 

Edit: it's possible to add forces F at bottom and at top, like I'm sure I don't lost energy from forces F.

[rc4]

Works of forces

F is the value of green or magenta force

wdisks=+N1/2mr²((w1−w2i)²−(w1−w2f)²)

with w2f<w2i, w2f=w2 at final, w2i=w2 initial

Wfriction=2(N−1)Fdw3t with w3 the mean of w2

WF1=2dF−2dF=0
WF2=2dF−2dF=0

Wmagentaforce=−2Fdw3t

Sum=+N1/2mr²((w1−w2i)²−(w1−w2f)²)+2(N−1)Fdw3t−2Fdw3t

Sum=+N1/2mr²((w1−w2i)²−(w1−w2f)²)+2(N−2)Fdw3t

Sum of energy:

Before t=0, the system (N disks) has the energy N(1/2md²w1²+1/4mr²(w1−w2)²)

At final, the system has the energy:

N(1/2md²w1²+1/4mr²(w1−w2)²)+N1/2mr²((w1−w2i)²−(w1−w2f)²)+2(N−2)Fdw3t

[rc4]

I made a mistake in my calculations:

Works of forces

F is the value of green or magenta force

wdisks=+N1/2mr²((w1−w2i)²−(w1−w2f)²)

with w2f<w2i, w2f=w2 at final, w2i=w2 initial

Wfriction=2(N−1)Frw3t with w3 the mean of w2 **************************** I noted 'd' but it is 'r'

WF1=2dF−2dF=0
WF2=2dF−2dF=0

Wmagentaforces=−2Frw3t

Sum=+N1/2mr²((w1−w2i)²−(w1−w2f)²)+2(N−1)Frw3t−2Frw3t

Sum=+N1/2mr²((w1−w2i)²−(w1−w2f)²)+2(N−2)Frw3t

Sum of energy:

Before t=0, the system (N disks) has the energy N(1/2md²w1²+1/4mr²(w1−w2)²)

At final, the system has the energy:

N(1/2md²w1²+1/4mr²(w1−w2)²)+N1/2mr²((w1−w2i)²−(w1−w2f)²)+2(N−2)Frw3t

[rc4]

I posted here :

http://physics.stackexchange.com/questions/143377/one-disk-ring-in-double-rotation-and-sum-of-energy

for have a reply, maybe if you up (+1) the question, more people will be interested about it

[EOW]

The problem with my last idea are the trajectories. I can't have green forces in this direction because points don't move like I thought. For that, I need to have a rotationnal velocity higher than w1. I use for this gears, that will increase rotationnal velocity. These addionnal gears has no mass (in theory) like that they don't lost energy.

For the cycle: give rotationnal velocity w1 and w2 (and kw2), this is for launch the system. And after let "live" the system like it is, no external motor. There is only friction between magenta/magenta disks. Forces are like that because kw2 > w1. Here I can take w1=10 clockwise, w2=-7 counterclockwise and kw2=21 clockwise.

First image: At start: all disks (or rings) are turning around blue axis at closckwise w1. All bigger green disks are turning at w2 counterclockwise around green axis. All smaller magenta disks are turning at kw2 clockwise with k =3 (in this example but k can be higher). Green disks have mass. Magenta disks haven't mass. Note there is no friction between magenta/green disks because it's gears (and in theroy I consider no friction here).

Second image: Look at forces. Friction generate F1 and F2 forces. Note there is heating => energy. F1 and F2 create F3, F4, F5, F6, F7, F8, F9, F10 like image shows and 2 additionnal -F3 -F4 not drawn. Each magenta disk receive a counterclockwise torque. They reduce their rotationnal velocity but note they have no mass. Each green disk receive a clockwise torque and increase their rotationnal velocity, and like they have mass, they increase kinetic energy. In the lab frame reference, the kinetic energy is 1/2md²w1²+1/4r²w2² with m the mass of the disk, d the lenght of the arm and r the radius of the disk. Fx-1 and Fx+1 show the forces come from another basis system when I repeat them.

Third image: repeat N systems

I define H the energy from one magenta/magenta friction. I define K the additionnal kinetic energy of one green disk. The sum of energy increases, we have (N-1)*H energy from heating . Add N*K kinetic energy from each green disk. Remove H for two last system, I need to give energy for give Fx-1 and Fx+1 at 2 terminal system. The sum of energy is (N-2)*H+NK.

Edit: like before, blue axes are fixed to the ground. Think with k very high (radius of magenta disks are very small), it's easy to see forces can be like I drawn but not all around the circle just like I drawn at start with arms horizontal.