Sum of torque

[EOW]

Look at the images.

The green disk is very small for cancel all torque on the support but it can be attached to a bigger disk with a mass. The green disk I drawn is a pulley attache to a bigger disk. If the diameter of the blue ring is 0.1m and the diamter of the green disk is 0.001m, if w=1 rpm, then the angular velocity of the green disk must be at 100 rpm. If I applie a force F=20 N (a torque of 1Nm), the support receive a torque = 0.1/2*5 - (0.1-0.001)/2*20 = 0.01 Nm and it can be very low. The work on the green disk is 100Cwt=0.001/2*20*100 = 1*t Nm. The work on the support is 0.01*w*t=0.01*t. So it's possible to compare these 2 different energies.

[EOW]

At start, all velocities are at 0. I accelerate the support clockwise, I let pulleys (all pulleys have mass) like they want to turn, all the pulleys are free to turn like they want. The green ring want to turn at -w (in the support frame reference) like the blue pulleys. But the difference of radius force the blue pulleys to follow the green ring. The support receives a clockwise torque, value = 1/10Fd. When I turn the support, I give the energy for turn only the support with all pulleys but I don't give the energy for turn around themselves pulleys. So I receive a clockwise torque and in the same time pulleys turn around themselves, I don't care about the value of this angular velocity, positive or negative, because a value is necessary greater than 0.

For find the final angular velocity of the pulley I can write:

1/2J1w1²+3*1/2J2w2²=C, C is calculated with a final value of w
and
w2=Kw1

There is a difference of 1/10 torque.

[EOW]

Pulleys can't have the same angular velocity, one must accelerate, other need to decelerate. Each pulley receives a torque FR, but the support receives the torque -F(R1-R2). And if I set the belt ON when the support is at w, 2 pulleys are turning at -w. So the work won by 2 pulleys is 2FRwt and the support lost F(R1-R2)wt.

Cycle:

1/ Turn the support with the belt OFF at w, in the labo frame reference disks don't turn, if a torque is appied on a disk its kinetic energy is increasing
2/ Set the belt ON
3/ The support receives -F(R1-R2), one disk +FR1 the other disk -FR2. FR1 increases the kinetic energy and FR2 too.  It's like -FR2 must decrease the energy of the smaller disk but not, in reality (labo frame reference) the smaller disk don't turn and the torque increase the kinetic energy of the smaller disk.

[EOW]

No motor ! it's only a transcient analysis. All velocities are in labo frame reference.

Case 1/ At left, the support don't turn, the small disk turns at -6.2rd/s, the big disk turns at -2rd/s. The big disk will drive the smaller, I need a spring for adjust angular velocities. I noted R=R2. The big disk win 6wFRt, the small disk lost 6.2wF3Rt and the spring win the rest. The sum of energy won/lost is well 0.

Case 2/ At right, it's the same case except that the support is turning clockwise at w. The sum of forces are the same. Except one thing the support receive a clockwise torque +2wFRt and like it turns it can produce an energy. Here, the sum of energy won is 2wFRt.

Forces can be like I drawn because in the support frame reference: the bigger disk turns at -w and the smaller at -5.2w, the smaller turns more quickly even the radius is not the same.

I use a spring for take the difference of angular velocities. The spring has no potential energy at start, but it could if necessary.

In the second case the device win energy.

[EOW]

The power is transmit like that:

With 2 basic devices (not drawn) :

I can reverse the angular velocity without give a torque on the support if the diameter of P2 and P3 are very small. The motor gives -FRwt to the support but receives 2FRwt from pulleys. The motor needs the energy -2Frwt, P4 gives Frwt. P4 has a mass no others pulleys. The motor accelerates only P4. Forces F1/F2 gives 2FRwt. P2 drives P3 like a gear (not drawn). The sum is well at 0. So, I need to place at least 3 basic devices in serial.

With 4 basic devices (the image):

Motor is P1
P2 is driven from P1 with the belt
P3 is driven from P2 with the gear (each pulley has a gear fixed to it)
P4 is driven from P3 with the belt
P5 is driven from P4 with the gear (each pulley has a gear fixed to it)
P6 is driven from P5 with the belt
P7 is driven from P6 with the gear (each pulley has a gear fixed to it)
P8 is driven from P7 with the belt

And for reverse the velocity of bigger disk, I can use gears with small diameter in the center of each pulley P4 and P5, like that the torque on the support is near 0. The device can win 2FRwt


The power transmit from P4 to P5 is higher if the angular velocity is high: 101 to -99 for example. If I take 3 to -1 I transmit 3FRwt to -FRwt and I think the power is reduced of 1/3. With 101/99 the ration is 99%.