Parallel string walking between two threaded cylinders?

[thorq]

Hello, I have a problem that seems to have beaten me.

I want to make a cable-driven reduction system between a threaded rod (pinion) and a larger threaded cylinder. Because of string walking, the string between the two has to stay at all times straight, parallel to the ground so that triangulation is avoided.

I have the following input data:
- threaded rod M8 (7.8mm Major Diam, ~6.8mm Minor Diam, Pitch (is supposed to be) 1.25mm - but i had a macro picture next to a ruler and in a graphical program I could measure more then 1.25, about 1.33mm but can't be sure as the measurement is not exact.
- large cylinder (Diam=120mm) - this one I will 3D print with the calculated threading.

I have already done this procedure once assuming 1.25mm pitch but apparently i did something wrong during my calculations and now the threaded string on the pinion travels faster than the corresponding coil on the larger drum/cylinder.

-------------------------------------------------------------------------
Below is my (apparently wrong) calculations and logic:

The coil on the bolt moves up or down with "P-pitch" mm at every bolt rotation so the coil on the drum must travel the same vertical distance to keep the line between the two entities parallel at all times. Since only so much string can be accommodated by the bolt at a complete turn, the compensation on the drum is in the steeper angle of the "imaginary" groove.

For example, an M8 bolt typically has a pitch of 1.25mm (coarse). The threading is a helix with a radius smaller than the 4mm, about ~3.325 actually, plus the radius of the string used (ex: 0.16). I used the calculator at https://www.easycalculation.com/physics/classical-physics/helix.php and got a string helical length of ~23mm for a 3.45mm radius helix over 1.25mm of height at an angle of 60deg.

So for each rotation of the bolt the coil on the bolt travels 1.25mm on up or down and winds or unwinds 23mm from the larger drum.

To keep up with this travel, the angle of the string on the drum should travel 1.25mm in height every 23mm, which translates to about 3deg of an angle (http://www.pagetutor.com/trigcalc/trig.html).
-------------------------------------------------------------------------

 

[thorq]

Ok, I think I've found my error after a while. I haven't even wrote it in the description of the issue because I couldn't see it. So my calculations are ok except that after I get the 3deg angle for the helix on the larger cylinder/drum, I have wrongly calculated the pitch of the helix based on the diameter of the drum rather than the circumference of the drum.

So, to conclude this, the helical groove on the drum should have a 23.5mm pitch rather than my previous calculation of 12.5 because if you unroll the 120mm cylinder and draw a line at 3deg angle, the height between the starting point and the end point is this number.

But I am sure you know better math than me so this expl is most likely not needed.

 

[sophiecentaur]

Did you consider having a geared guide to lay the string on two smooth cylinders in the way you want? (like a fishing reel) It would be more complicated but it could force the string to keep parallel.
But, if the pitches are inversely proportional to the radius, it should take care of itself, I think. And your idea is much more sophisticated.
PS Can you be sure that the tension will always be enough to keep the string in the grooves?

 

[thorq]

I prefer complication in design and dumbness in mechanics. It would have been common sense to do this with a timing belt but instead I opted for Dyneema/Kevlar string. The tension on the string will be applied in such a way that it will be kept in the grooves, provided the grooves are the correct path, the path the string would take anyways if it was properly tensioned on a smooth cylinder. I want the grooves to take out any possibility of this predictable movement to go haywire.

The string will not rely only on friction on the large cylinder but while the middle of the loop will be coilde around the bolt/threaded rod, the ends will be on the larger cylinder, secured by knots.