Sine bar errors

[mklotz]

In the metalworking books, when you see a picture of a sinebar in use, the
stack used to form the angles is generally composed of gage blocks, the
accuracy of which is measured in millionths of an inch. Are gage blocks, a
moderately expensive item for the amateur, really needed? Or is it possible to
get by with a homeshop-made stack that's only accurate to a thou?

The equation for a sinebar is:

sin(A) = S/L

where:

A = desired angle
S = stack height
L = sinebar length (i.e., roller center-to-center distance)

With a little bit of differential calculus, it's possible to write the error
equation for the angle due to errors in the stack height.

dA = (1/cos(A)) * dS/L

where:

dA = the error in the angle due to an error of 'dS' in the stack height.
(For purposes of this discussion, we'll ignore the effect of any error in 'L'.)

Let's plug in some numbers...

A = 10 deg
L = 5 in
dS = 0.005 in

Then:

dA = 1.01543 * 0.005 / 5 = 1.01543E-3 rad = 0.0582 deg

or about one milliradian error. That's pretty small. Think about it this
way...If I make a one milliradian error pointing my rifle at a target 100 yards
away, I'll miss the bullseye by 3.6 in.

If I'm any kind of machinist, I should be able to machine the block I'm using
for the stack to within 0.001 in, which would reduce the error to 0.2
milliradian, or a target miss of 0.72 in at 100 yards.

The error depends on the angle for which the sinebar is set. For:

L = 5 in
ds = 0.001 in

it looks like this:

5 0.0115029
10 0.0116359
15 0.0118634
20 0.0121946
25 0.0126438
30 0.0132319
35 0.013989
40 0.0149589
45 0.0162057
50 0.0178273
55 0.0199784
60 0.0229183
65 0.0271147
70 0.0335043
75 0.0442748
80 0.0659906
85 0.131479

where the first column is the angle, A, in degrees and the second column is
the error in A, dA, in degrees.

Since a sinebar is seldom used for angles greater than 40 degrees, we can
count on an angle error of less than 0.015 deg (0.25 mrad) if we can machine
the stack block to an accuracy of one thou. Unless you're making highly
critical components, don't be afraid to machine your own blocks for setting
the sine bar.