Coil wrapping surface area calculation

Questions
1

Hello,

SETUP:
My question is about calculating the length of a cable wrapped around an elongated cylinder fig. (A). I’ve found on-line calculators that will give me that data for a coil around a cylinder fig. (B). Helical Coil Calculator however I don’t know if that’s the same surface area as fig. (C).

I did a physical test with a string around a can (D). Made a mark on the can and wrapped the string around two times and cut the string at the same mark were the string started. Then squeezed the can to shape an oval fig. (E). Then rewrapped the same string around the squashed can. The string started and stopped at the same place as on the round can. I guess no surprise that the surface area didn’t change.

MY QUESTION:
But squashing the can fig. (E) and the shape (A) seem to be two different surface areas. How can I calculate the surface area of (C)?

GOAL:
My ideal would be to make a calculator that I could change the shape profile and get the wrapping calculation as in Helical Coil Calculator

RELATED QUESTIONS:
1.) Is there a geometric name for fig. A or C?

Thanks for and help in this matter.

2

In many CAD programs you can build a helix / coil. But they don’t allow to change the shape of the coil. Nor do they give you the cable length calculation.

3

Mathematica calls it a ‘stadium’ shape

The perimeter of a general stadium

StadiumShape[{{x1, y1}, {x2, y2}}, r]

can be obtained using:

ArcLength[RegionBoundary[StadiumShape[{{x1, y1}, {x2, y2}}, r]]]

giving:

2 (Pi r + Sqrt[(x1-x2)^2+(y1-y2)^2])

To calculate the length of the wire just calculate the area (perimeter times depth). And divide by the diameter of the wire.

  • For thick wire you have to compensate for ‘lost’ space.
  • For slightly better accuracy you could add the radius of the wire to the radius of the stadium shape to account for the finite thickness of the wire.
4

Sander,

Thanks for your detailed reply. Stadium as in the shape of a sports stadium, interesting. I knew PI would be involved in this.

I have a predetermined length of cable that needs to fit with in a space. Using your formula I can figure the length of one wrap and multiply it to fit the length needed.

Yes the gauge of the cable will need to be factored in as well as the pitch of the cable as it wraps, as small as that is.

Sander, again that’s for your time and knowledge with this problem.

5

Sander,
I see your from Lyon. I was close to getting a job there in Lyon but it fell through. I live in LA and was looking forward to living in France, se la vie.
BTW - Love your photos

6

Pitch does not matter in one way: namely you are covering an area (the ‘extruded stadium’) with a cable. So it is simply dividing this area by the diameter of the cable. However, you might not be able to fill up all the area! that will be much more tricky to calculate of course because you have to consider the starting positio, ending position, and the maximum curvature of the wire et cetera…

By adding an ‘r’ to the shape of the stadium you can account for the added thickness of the wire (the center of the wire is r + d/2 away, rather than just r).

The naming is (I think) introduced by the Wolfram Language. I don’t think it is a well-known geometric shape. However I find the stadium name well suited…

7

Sander, thanks for the update.
Yes the cable thickness is an issue and as you see with the link below so is the gap between the rows. But I could just add a little to the diameter of the cable to compensate.

I found this on line but is only for cylinder helix wraps. Helical Coil Calculator

Should be fun to turn this into a plugin for my CAD program. BTW I’m talking with folks here as well.