Sum of torque

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The idea is to increase rotationnal velocity (in the lab frame reference) of disks with friction forces. Friction is energy and kinetics energy too. I try to choose good velocities, but event there is a problem, it's possible to chose one disk without mass if its rotationnal velocity decrease. For those disks that increase their rotationnal velocitiy set mass not at 0. I hope it's clear enough. Look at images, please.

Before $t=0$: I give rotationnal velocity $w_1$, $w_{2a}$, $w_{2b}$ and $w_3$, this is for launch the system, during this step I need to give energy. And after I let "live" the system like it is, no external motor. The sum of energy must be constant. I count all energies in the system, heating too. Green disk is turning around blue axis clockwise and around itself counterclockwise (green axis). Magenta disk is turning around blue axis clockwise and around itself clockwise (magenta axis) for the upper and counterclockwise around itself for the lower. If I want to guess no sliding between green and magenta disks, I need to have $w_3$ different of $w_3'$. There is only friction between magenta/magenta disks, but before $t=0$, I set $friction=0$ for launch the system. I need to set $|w_1| > |w_2|$. I can take, for example, $w_1=-10$, $w_{2a}=-7$, $w_{2b}=-5$ and $w_3=+9$ and $w_3'=-29$ (all angular velocities are labo frame reference). There is a relation between $w_{2x}$ and $w_3$ because there is no sliding between magenta and green disks.

Blue axes are fixed to the ground. Arms must turn together and gears (not drawn and without friction) force all arms to turn together at the same rotationnal velocity. If a torque is present on an arm it will be apply on others through gears. Green disks have mass. Magenta disks haven't mass.

I need to have friction between magenta disks AND in the good direction, for that I need to set $w_{2x}$ different for each green disk with $|w_1| > |w_2|$. Look the description and the image where I calculate velocities.

Note I study the sum of energy only in a transcient analysis. From $time=0$ to $time=t_x$ with $t_x$ very small.

At start:  Note there is no sliding between magenta/green disks because green disk force to turn magenta disk like gear can do, it's not a gear but consider contact magenta/green disks are like gears without friction (no heating dissipiation) and no sliding. Sure, magenta disk force green disk to turn like gear can do. But I guess no friction between magenta/green disks.

Look image N°1

Friction generate $F1$ and $F2$ forces. I noted all others forces I see. Note there is heating => energy between magenta/magenta disks. $F1$ and $F2$ create others forces. Each magenta disk receive a counterclockwise torque. They reduce their rotationnal velocity but they have no mass, so they don't decrease their kinetic energy. Each green disk receive a clockwise torque and increase their rotationnal velocity, and like they have mass, they increase kinetic energy. Image shows $Fx-1$ and $Fx+1$ forces, it comes from another basis system when I repeat them. Look image N°2

Another position: Look image N°3

To be sure trajectories are correct, I calculate linear velocities in this position: look image N°4

$P1$ and $P2$ are on each magenta disk, where there is friction. With $r'$ the radius of magenta disk. The linear velocity of $P1$ is $(d-r')(10)-r'(15)=-25r'$ and the linear velocity of $P2$ is $(d+r')(10)-r'(19)=-9r$, I counted positive the right direction. So, $P1$ move faster at left than $P2$ and forces can be like I drawn.

So, with: $|w_{2a}| > |w_{2b}| > |w_{2c}| > |w_{2d}|$ it's ok.


I need to give $Fc1$ and $Fc2$ forces for have friction between magenta disks. These forces don't work. Look Image N°5

I repeat $N$ systems like that last image

I define $H$ the energy from one magenta/magenta friction. I define $K$ the additionnal kinetic energy of one green disk. I define N the number of basic system {magenta disk + green disk}. The sum of energy increases, we have $(N-1)*H$ energy from heating . Add N*K kinetic energy from green disks. Remove $H$ for two last system: I need to give energy for give $Fx-1$ and $Fx+1$ at 2 terminal disks. The additionnal energy is $(N-2)*H+NK$, it's not 0.

Edit: with friction between magenta and green disks it could be easier to find good velocities. Maybe with no friction between magenta/magenta disks. I will calculate this. If necessary change radius of disks, magenta disks can be with different radius; The same for green disks.

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Like image shows. F5 and F6 forces (and others) increase w1. There is heating. 4 red disks decrease kinetics energy but 3 black disks increase theirs and there is heating. |w1|>|w2|

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With 'R' radius of purple disk.  With 'r' the radius of grey disk. 'F' the force from friction. 't' the time. A grey disk is rotating at w1 around red axis and at w2. Gears are turning too. I need to give kinetic energy for that. After, I let "live" the disk and gears. There is on friction between purple disk and grey disk, no between gears. No external motor. I let the device like it is and I count energy. The device works only few seconds. Purple disk is fixed to the ground. Friction generate forces F1 and F2, energy goes to heating. w2>(R+r)w1/r, with labo frame reference.

Gears are turning clockwise and counterclockwise. I guess between gear/grey_disk it's like a gear: no sliding, no heating dissipation.

Cycle:

1/ Friction is OFF. Launch grey disk and gears at w1 and w2
2/ Set friction ON
3/ Measure the sum of energy, heating too (H) !

H= | F(Rw1-r(w2-w1))t |

The sum of energy must be constant, but it decreases in this example. So with gears I can have good rotationnal velocities and forces. Like gears has mass (like disk), when grey disk decelerates, gears want to keep their rotationnal velocity due to inertia, this give forces. I can set forces like I want, for example F/2 for first gear and F/4 for second gear because forces depends of the inertia of each gear and it can be like I want. Here the delta energy is H+Ft(-rw2+Rw1+1/2rw1). I need to have w2>(R+r)w1/r for have F1 and F2 in these directions. So with the limit case w2=(R+r)w1/r the sum of energy is -1/2Frtw1+H. In the limit case H=0, the sum is not 0, it is < 0. If w2=(R+r)w1/r+x, the sum of energy is -Ftx-1/2Frtw1+ Ft ((Rw1-r(w1(R+r)/r+x)-w1)t)=-1/2Frtw1, it's not 0.

Calculations :

Interface PurpleDisk/GreyDisk: ( -1/2rw2+3/2(R+r)w1 )Ft
Interface GreyDisk/FirstGear: ( -1/4rw2-3/4(R+3r)w1) Ft
Interface FirstGear/SecondGear: ( -1/4rw2+1/4(R+5r)w1) Ft

Sum is (-rw2+Rw1+1/2rw1) Ft

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My last case, the energy increases of -rw2-1/2rw1, because I forgot one torque. Like H is FRtw2 it's not possible to have 0.

Now, grey disk rotate around itself counterclockwise. Forces are like image shows. The energy from torques is Ft ( rw2-1/2rw2-3/2(R+r)w1+1/2(R+3r)w1-1/2rw2 ) = -FRtw1

This result don't depend of w1, but for friction w2 is a parameter, H = Ft((R+r)w1-rw2), look below I verified this calculation

The additionnal energy is Ft((R+r)w1-rw2)-FRtw1= Ftr(w1-w2) and w2 < w1, the energy is positive

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I'm not sure about my forces and my sum of energy but in the first image, grey disk don't around itself. The energy from heating H=+FRtw1, and torques give the energy +Frtw1-F(R+r)rw1, the sum is 0.

In the second image, grey disk turns around itself counterclockwise. The energy from heating is Ft((R+r)w1-rw2). The energy from torque is +Frtw2-F(R+r)rw1, here the sum is 0, like that I'm sure of the calculation of the heating H for the last message where I find a sum different of 0.