Sum of torque

[EOW]

No friction, no gravity, no external motor. The black arm is turning clockwise at w1. The gyroscope is turning at w3=0 like that it keeps its position in the labo frame reference. Springs give a clockwise torque on the black arm, the kinetic energy of the black is increasing. The gyroscope don't lost energy because it turns in a perpendiculary plan.

[EOW]

No gravity. No heating dissipation. The black arm is turning clockwise at w1. A motor accelerate more and more a slave disk (inertial disk). I guess w1 constant because I recover the energy from the black arm for keep constant w1. The belt don't receive a torque, all forces Fa, Fb, Fc, and Fd cancel themselves. The motor receives -F but this force is in the axis, so no torque from -F. The slave receives F, and the black arm receives a clockwise torque +FR . The energy won is +FRw1t. The energy gives by the motor is in the kinetic energy of the slave, so this energy is not lost, it can be recover later. The belt gives only forces where I drawn them (special belt), look at second image.

[EOW]

It's difficult to have forces like I drawn with a belt. With an hydraulic system it would be easier. An external hydraulic system (not drawn) gives a pressure in the hydraulic pipes. This pressure accelerates 2 inertial disks. The sum of forces on each hydraulic pipe is 0, so there is no torque. The black arm is turning at w1 clockwise. The disk1 is turning clockwise at w2, with w2=0 at start. The disk2 is turning counterclockwise at -w3, with w3=0 at start.

The black arm receives two forces F and -F, this gives a clockwise torque FR on the arm. The energy recovered is FRtw1. An external system (not drawn) recover the energy from the black arm and keep constant w1. If w1 is very high the work from FRtw1 is very high too.

The hydraulic system must give an energy E, but all this energy is in the kinetic energy of 2 disks. So this energy E can be recover later. At final, the energy gave by the system is FRtw1.

All the system is turning like the second image.

It's possible to compare with the system in the third image. The energy is conserved (I added a ring on the disk1, the mass is in te grey inner part). The only difference if the system in the first image

I think the hydraulic pump must turn at w1 like that all the device is turning at w1 except the disk1 and the disk2.

[EOW]

A big green plate has N small grey disks. Disks can turn around themselves. The plate turns at w clockwise like disks at start. I have on the plate 2*N devices (not drawn) that can ejects small black masses. These devices don't give a force or a torque on the green plate because masses are the same and I eject them with the same velocity. These masses shock grey disks and increase the rotationnal velocity of disks. Sure at final I have a mass, alone, that MUST give a torque FR (if F is the force that a mass give) during a time 't'. But I can change the radius of the mass because I have time, the energy for change the radius is 0 because the force is always perpendiculary to the trajectory, but if the radius decreases the rotationnal velocity is increasing because kinetic energy is constant=1/2mr²w². So I can shock the plate with a clockwise torque. I need the same energy for accelerate disks when w=0 or w=100, but in the case where w=100, the kinetics energy of each disk is increasing more than when w=0:

Second image, I can use springs.

Thrid image: use electromagnetic system for increase rotationnal velocity of each grey disk. Prepare a lot of masses M (black) for have a continous movement.

Don't forget, masses rotate at w so their velocity is V clockwise even the force reduce the velocity, the direction can give a clockwise torque.

The  5° image is a basic system, with only a spring for understand the main idea !

[EOW]

No gravity. No friction. Only one motor. An external device not drawn recover energy from the black arm for have w1 constant.

First image: I guess the energy is conserved in this case. It is the basic system that I replicate. At start, all the device is turning clockwise at w1. The motor need energy E1 for turn the inertial disk. There is a torque FR on the black arm but I guess this is the motor that give this energy.

Second image: I use 2 basic devices. The torque on the black arm is 2FR and the energy from the motor is the same than in the case 1/. The inertial disk has te same kinetic energy than in the case 1.

Third image: I can use N basic devices. The torque is NFR.

Fifth image: the small disk is turning at R/rw but I want the big is turning at w, so I can use magnet and like there is a sliding so the external ring can turn more slowly than the small ring