Calculating the Length of String on a Reel
Date: 12/07/2001 at 10:25:54 From: Larry Schworer Subject: Calculate string on a reel What is the formula for calculating the length of wire that can go on a reel (spindel) where the reels may have different diameters, hub diameters, and widths, and the wires may have different diameters? Once a reel and wire combination are selected they do not change for that reel combination. I wish to write a program to make this calculation. Thanks, La
Date: 12/07/2001 at 12:06:15 From: Doctor Peterson Subject: Re: Calculate string on a reel Hi, Larry. The answer will be similar to that for a similar question we have answered many times, about the amount of material (such as carpet) on a roll: How Much Carpet is Left? http://mathforum.org/dr.math/problems/gillette.03.27.99.html We find the length by equating the volume of material on the roll to its volume laid out flat. There is always a caveat in this calculation, however: we have to assume that the material retains the same thickness as it bends around the roll, and is not either compressed or wound loosely. The calculation is very sensitive to the actual thickness of the material on the roll. The same is even more true in your case. The wire may be wound in different ways that will incorporate different amounts of air space between turns, so it is hard to be sure how much of the volume on the reel consists of wire. The best thing to do would be to measure an actual reel of known length in order to determine this ratio and adjust the formula, rather than blindly trust theory. Having said that, let's look at the problem. For simplicity, I'm going to pretend that the wire has a square cross section, since then there would be no need for space between turns. Your measurements will determine how closely this corresponds to reality. Suppose the reel has hub diameter Di, outside diameter Do, and width W. Then the volume of wire on it is V = pi(Do^2 - Di^2)W/4 Suppose you have a wire with length L and square cross-section with side Dw (the diameter of the wire). Its volume is V = LDw^2 Setting these equal, we get LDw^2 = pi(Do^2 - Di^2)W/4 and we can solve to find L = pi(Do^2 - Di^2)W/(4Dw^2) Now suppose you measure an actual reel, and the actual length is X times this calculated length (let's hope not too far from 1). Then you can take X as the typical correction factor, and multiply any calculated length by X to estimate the actual length when you don't know it ahead of time. This correction factor may well be more or less constant for all reels wound by the same method. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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