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

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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	});
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	17	<body>
	18
	19	<h1>Rolling wire-strip calculator: equations</h1>
	20
	21	<p>The calculations performed by the <a href="rolling.html">rolling
	22	wire-strip calculator</a> were derived by examining experimental data.
	23	We might not have considered all of the necessary variables. Anyway,
	24	here’s how it currently works.
	25
	26	<p>Let’s suppose we start with square wire, with side $S$,
	27	and we roll it to thickness $t$. Then we find that the
	28	wire’s width is
	29	\[ w = \sqrt{\frac{S^3}{t}} \]
	30	Rearranging, we find that
	31	\[ S = \sqrt[3]{w^2 t} \]
	32	For round wire, we assume that the cross-section area is the important
	33	bit, so a round wire with diameter $D$ ought to work as well as
	34	square wire with side $S$ if $S^2 = \pi D^2/4$, i.e.,
	35	\[ D = \sqrt{\frac{4 S^2}{\pi}} = \frac{2 S}{\sqrt\pi} \]
	36	Volume is conserved, so if the original and final wire lengths
	37	are $L$ and $l$ respectively, then
	38	\[ L S^2 = l w t \]
	39	and hence
	40	\[ L = \frac{l w t}{S^2} \]
	41	Finally, determining the required initial stock length $L_0$ given
	42	its side $S_0$ (for square stock) or diameter $D_0$ (for
	43	round) again makes use of conservation of volume:
	44	\[ L_0 = \frac{S^2 L}{S_0^2} = \frac{4 S^2 L}{\pi D_0^2} \]
	45
8a0fc4a2	46	<p>[This page uses <a href="https://www.mathjax.org/">MathJax</a> for
ef43c701 MW	47	rendering equations. It probably doesn't work if you don't enable
	48	JavaScript.]
	49
	50	</body>
	51	</html>