Sizing a Connector

Date: 06/01/2001 at 15:09:15
From: Tom Banghart
Subject: Overall diameter on any given number of diameters bunched 
together

I have to come up with a specific diameter for anywhere from 2 to 91 
diameters that are bunched together.

The application is round wires. I am given the number and the diameter 
of wires that I have to fit into a round connector. The connector is 
compressed after the wires are bunched together (trying to get them as 
round as possible). I need the overall diameter so I can size the 
connector before it is pressed.

I know the sum of the diameters is not accurate.

Date: 06/01/2001 at 23:10:01
From: Doctor Peterson
Subject: Re: Overall diameter on any given number of diameters bunched 
together

Hi, Tom.

For small numbers of wires, the answer will depend very much on the 
number and arrangement of the wires. For only two wires, of course, 
the bunch will have twice the diameter of one wire - but not much of 
the connector will be filled. For three wires, they would form an 
equilateral triangle, and we could figure out the diameter of the 
largest circle that fits around them. (I get about 2.15 times the 
diameter of a wire.) Each larger case requires a new calculation, 
because the best arrangement will be different.

For large numbers, we can approximate the diameter, assuming they are 
fitted together as tightly as possible. If I just imagine an infinite 
number of circles filling a plane, packed together in a triangular 
arrangement, I can calculate that the circles will take up 
pi/(2 sqrt(3)) of the total area, which is about 91%. If I now draw a 
larger circle (with radius R) that contains N of the small circles 
(with radius r), the total area of those N small circles should be 
about 91% of the area of the large circle. This gives me the equation

    N * pi r^2       pi
    ---------- = ---------
        pi R^2   2 sqrt(3)

Solving for the ratio R/r, I get

    R/r = sqrt[N * 2 sqrt(3)/pi] = 1.05 sqrt(N)

So I would estimate that the radius of a circle containing N tightly 
packed circles will be somewhat larger than the square root of N times 
the radius of each of the small circles. For small numbers, this will 
be much too small; but for large N it might serve as a good estimate.

Let me know whether the large-number approximation is suitable, or you 
need specifics for a certain range of small N. I may be able to help 
if you don't need too many cases.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/