[TheForge] Guesstamating

[Bruce Freeman]

1) A one-way taper (to a wedge or chisel edge) draws out the (rectangular) =
stock to two times its length. =20
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?)=20

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". =20

b) How much of the 1" round stock would be consumed in (a)? =20
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. =
=20
Answer:  2" / (1" / 1/2")^2 =3D 1/2"  [where "^2" means "squared"]=20

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? =20
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=20
r(x) =3D 0.5" + x/12 =20
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. =
=20
Got that?  [It's easier than it sounds.]

Bruce
NJ




GHS wrote:Larry wrote:

> >
> > Is there a rule of thumb to use when drawing steel as to how much to =
start with?  For
> > instance, say I wish at draw out a taper on a piece of 1 inch round =
bar to 1/2 inch and wish
> > the taper to be 6 inches long.  How much stock should I start with?   =
This is assuming I wish
> > to make more than one, or have two ends of the same stock tapered =
equally.
> >
> > Larry