[TheForge] Guesstamating

Mon Apr 1 10:20:03 2002

Rules of "thumb" are meant to be written on the brain!  Look again, you'll =
see the rules are really very simple.  It's in the application they become =
complex. =20

That's the downfall of "simple" rules, and that's the reason the "hard =
way" (integral calculus) turns out to be eaiser.

Nonetheless, you CAN get the answer using the rules of thumb.

Bruce
>>> Larry <[email protected]> 03/28/02 10:36PM >>>
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  =3D 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 =3D 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 =3D 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" =3D 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) =3D 0.5" + x/12
> E.g., r(0")=3D 0.5"; r(6")=3D1"
> 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

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