[TheForge] Guesstamating

[Larry]

After trying to write this rule on my thumb,  I decided I needed a finer point pen, then I went to the very fine line pen, but found I could not read the fine print without a magnifying glass.  Though this is by far the most interesting of replies, I am afraid for my purpose, I will try the make one and see pattern till I can get the proportions I am trying to achieve.  But thanks for the
excellent information.  I plan to study it till I actually can make work in practice what you have explained here in text.  I now have a new life long goal.  Thank you Bruce, I will forever be in your debt.

Larry

Bruce Freeman wrote:

> 1) A one-way taper (to a wedge or chisel edge) draws out the (rectangular) stock to two times its length.
> E.g., tapering 6" of a 1" square bar to an edge will result in a wedge 12" long.  Conversely, if you want a 6" taper from 1" square to an edge, you need a 3" length of 1" square stock.
>
> 2) A two-way taper (to a point) draws out the (round or rectangular) stock to three times its length.
> E.g., drawing the last 2" of a 1" square bar to a point will stretch that 2" to 6".  The same is true for 1" round stock.  This is true providing you don't change shape, such as a round point on a square bar.
>
> That's the easy part.  It's more complicated if you're drawing down from one diameter to another or from one shape to another.  First you have to consider the drawing from one uniform thickness to another uniform thickness:
>
> 3) Drawing square to round draws out the metal to 4/pi, or about 1.25 times its length.  E.g., start with 12" of half-inch square stock and draw the entire length down to half-inch round stock and you'll end up with about 15" of round stock.
>
> 4) Drawing rectangular stock from one thickness to another (without changing the width) increases the length of the stock by the ratio of the thicknesses.  E.g., If we start with 1 foot of 1" square and draw the entire piece down to 1" x 1/3" strip, we'll have three feet of stock.
>
> 5) Drawing round stock from one diameter to another increases the length of the stock by the square of the ratios of the diameters.  E.g., starting with 1" diameter, 1 foot long, if we draw the entire piece to half-inch diameter, we'll have a 4-foot piece.
>
> 6) Drawing square stock from one (square) size to another increases the length by the square, just as in (5).
>
> 7) Drawing rectangular stock from one thickness and width to a different thickness AND width is best approached as a double application of rule (4).
>
> Hence to calculate the specific example you give, using these rules, you'd break the problem up into parts:  (Don't you love word problems?)
>
> a) Suppose you drew out round from 1" to a point, such that the portion of that bar from the 1" diameter to the  half-inch diameter measured 6" long.  How long would the taper be?  (This is easier than it sounds.)
> Answer:  6" x (1" / 1/2") or 12".
>
> b) How much of the 1" round stock would be consumed in (a)?
> Answer:  12" / 3  = 4"
>
> c) What length of half-inch round is represented by the section end of this taper (with diameter of 1/2" or less?).  I.e., how much half-inch round stock does it take to draw a point, 6" long?
> Answer:  6" / 3 = 2" (of half-inch round, not 1" round).
>
> d) How much 1" round has to be drawn out to make 2" of half-inch round.
> Answer:  2" / (1" / 1/2")^2 = 1/2"  [where "^2" means "squared"]
>
> e) So if it would take 4" of 1" round to draw the 12"-long point in (a) and the last 6" represents only 1/2" of the 1" stock, how much is required to draw a 1" rod to 1/2" over a length of 6"?
> Answer:  4" - 1/2" = 3.5".
>
> Next question:  Are there other ways to do this?
> Answer:  Yeah, about as many as there are folks who want to do it!
>
> My preference would be integral calculus.  Consider the final taper you want.  Slice it infinately thin.  (While you're off doing that, I'll just hypothesize about it.)  Each slice is of area pi*r^2  [where * means multiply] and of thickness dx (meaning infinitesimal, but not zero).  Now radius r varies linearly from 0.5" to 1" as the position, x, along the point runs from 0" to 6", hence
> r(x) = 0.5" + x/12
> E.g., r(0")= 0.5"; r(6")=1"
> So all we have to do is integrate from 0" to 6" the function pi*r(x)^2*dx.
> Got that?  [It's easier than it sounds.]
>
> Bruce
> NJ