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5 | <title>Rolling wire-strip calculator: equations</title> | |
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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 |
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47 | rendering equations. It probably doesn't work if you don't enable |
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