[dep-ui] / rolling-eqn.html

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  <title>Rolling wire-strip calculator: equations</title>
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<h1>Rolling wire-strip calculator: equations</h1>

<p>The calculations performed by the <a href="rolling.html">rolling
wire-strip calculator</a> were derived by examining experimental data.
We might not have considered all of the necessary variables.  Anyway,
here&rsquo;s how it currently works.

<p>Let&rsquo;s suppose we start with square wire, with side&nbsp;$S$,
and we roll it to thickness&nbsp;$t$.  Then we find that the
wire&rsquo;s width is
\[ w = \sqrt{\frac{S^3}{t}} \]
Rearranging, we find that
\[ S = \sqrt[3]{w^2 t} \]
For round wire, we assume that the cross-section area is the important
bit, so a round wire with diameter&nbsp;$D$ ought to work as well as
square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \]
Volume is conserved, so if the original and final wire lengths
are&nbsp;$L$ and&nbsp;$l$ respectively, then
\[ L S^2 = l w t \]
and hence
\[ L = \frac{l w t}{S^2} \]
Finally, determining the required initial stock length&nbsp;$L_0$ given
its side&nbsp;$S_0$ (for square stock) or diameter&nbsp;$D_0$ (for
round) again makes use of conservation of volume:
\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \]

<p>[This page uses <a href="http://www.mathjax.org/">MathJax</a> for
rendering equations.  It probably doesn't work if you don't enable
JavaScript.]

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Commit	Line	Data
ef43c701 MW	1	<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN"
	2	"http://www.w3c.org/TR/html4/strict.dtd">
	3	<html>
	4	<head>
	5	<title>Rolling wire-strip calculator: equations</title>
	6	<script type="text/x-mathjax-config">
	7	MathJax.Hub.Config({
	8	tex2jax: { inlineMath: [['$', '$'], ['\\(', '\\)']] }
	9	});
	10	</script>
	11	<script type="text/javascript"
	12	src="http://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS_HTML">
	13	</script>
	14	<link rel=stylesheet type="text/css" href="rolling.css">
	15	<head>
	16	<body>
	17
	18	<h1>Rolling wire-strip calculator: equations</h1>
	19
	20	<p>The calculations performed by the <a href="rolling.html">rolling
	21	wire-strip calculator</a> were derived by examining experimental data.
	22	We might not have considered all of the necessary variables. Anyway,
	23	here’s how it currently works.
	24
	25	<p>Let’s suppose we start with square wire, with side $S$,
	26	and we roll it to thickness $t$. Then we find that the
	27	wire’s width is
	28	\[ w = \sqrt{\frac{S^3}{t}} \]
	29	Rearranging, we find that
	30	\[ S = \sqrt[3]{w^2 t} \]
	31	For round wire, we assume that the cross-section area is the important
	32	bit, so a round wire with diameter $D$ ought to work as well as
	33	square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
	34	\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \]
	35	Volume is conserved, so if the original and final wire lengths
	36	are $L$ and $l$ respectively, then
	37	\[ L S^2 = l w t \]
	38	and hence
	39	\[ L = \frac{l w t}{S^2} \]
	40	Finally, determining the required initial stock length $L_0$ given
	41	its side $S_0$ (for square stock) or diameter $D_0$ (for
	42	round) again makes use of conservation of volume:
	43	\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \]
	44
	45	<p>[This page uses <a href="http://www.mathjax.org/">MathJax</a> for
	46	rendering equations. It probably doesn't work if you don't enable
	47	JavaScript.]
	48
	49	</body>
	50	</html>