Length of a helix vs composition of forces

[activ25]

I corrected the sum of work

[activ25]

the calculations

[activ25]

I updated the calculations. No external gravity. No friction (to simplify the calculations). No external motor. The bolt doesn't turn around itself before I start to count the sum of energy.

[activ25]

with a torus helix composed of multiple segments join by cardan shaft. The nut must pivot around 90°, the pivot can be small if the number of segments is big.

[activ25]

To prevent to move the center of mass of the bolt I can move the nut down and more closer to the axis A1. The bolt rotates only relatively to the nut (no translation). Like that it is easier to the calculation AND for the real device.

The cycle:

The device has reach its angular velocity and it is like I drew. I can set OFF/ON the helix of the nut.

1/ The nut is set ON and just after the nut moves down and more closer to the axis A1, in the same time I set ON the motor between the nut and the bolt (the motor M in previous message) I need an energy for that, in the same time I recover an energy from the axis A1 and from the movement of the nut
2/ I set OFF the helix of the nut, just after I recover the energy from the rotation of the bolt around itself (the motor rotates the bolt around itself not only relatively to the nut)
3/ I move the nut up and farther to A1 (it must be like I drew at the image at start)

LOOP STEP 1/

2 images: first: at start, second: at final