FINDING A RADIUS
by
James R. Howe
(6/12/2001)
This article was developed to solve the problem of how to measure a
radius that is less than a full diameter. When it is necessary to inspect the
diameter of a bore we can use a dial caliper, an inside micrometer, or a
telescope gage and outside micrometer. The method is simple enough as the
measuring device can easily span the diameter. But what do you use when the
radius to be measured is less than a full diameter? We cannot use a CMM
because we don't have one. The photo readily shows the dilemma.
The radii are marked with red arrow. It is easily seen that the common
measuring tools mentioned above will not work, there is not a complete diameter.
Since it is imperative to measure these radii for wear. The following technique is
recommended.
r = (c2 + 4h2 ) / 8h *
Arc
c
h
r
Works like a charm! However, be careful with your "c" and "h" measurements.
Procedure: After tracing the radius on a suitable surface (a 3x5 index card works
well for radii under 6.00") carefully draw line "c" using a 6" metal rule. Choosing a
dimension that is easily divisible by two allows a quick midpoint setting of "c".
Using the same 6" metal rule carefully draw a perpendicular line "h" from the
midpoint of "c" to the "arc". Using a dial caliper carefully measure "h" from the
midpoint of "c" to the very outside of the "arc". Now that the dimensions are
known use a calculator to determine "r". Multiply "r" by two for diameter.
To check your calculation set the dial caliper to the calculated "r" value. Using the
6" metal rule extend line "h". Place the dial caliper on "h" with the very tip of the
caliper jaw on the outside of the "arc". Rotate the caliper and observe how
closely it follows the "arc" from the one intersection with "c" to the other
intersection of "c". It should follow exactly. If it does not then your calculations
and/or measurements are probably in error.
The measured values can change dramatically so be careful with your technique.
(see below discussion on ERRORS)
J.R. Howe
QA-Akron
*The Equation is from: The MACHINERY'S HANDBOOK 23
ERRORS IN THE PROCEDURE FOR FINDING A RADIUS
r=(c2+4h2)/8h
Beginning with a real world radius of 1.7745 or diameter =
3.549 (optimum of 3.547/3.551) for a SK330-6 slide-loc. We
can set c at 2.5 and hold it while we vary h. we can then
determine how sensitive the equation is when h varies. To
do this we must know what h is under optimum conditions.
If r=1.7745, and c=2.5, then we can solve the equation for h:
r=(c2+4h2)/8h
1.7745=(6.25+4h2 )/8h
14.196=(6.25+4h2)/h
14.196h=6.25+4h2
0=6.25+4h2-14.196h
0=4h2-14.196h+6.25
or 0=h2-3.549h+1.5625
This is a quadratic equation solvable by using
x=-b +/- (b2-4ac)1/2 / 2a
x={3.549 +/- [(3.5492)-(4x1x1.5625)]1/2 } / 2x1
x={3.549 +/- (6.3454)1/2 } / 2
x=(3.549 +/- 2.519) /2
x= 1.03/2 and/or x= 6.06/2
x= .515
and/or x= 3.034
Since the 3.034 does not fit in our radius world we take the
.515 as optimum h. We can now plot r for each value
assigned to h +/- .001 increments from the optimum .515.
R=(C*C+4H*H)/8H
R
1.77449
C
2.5
D = 3.54898
H
0.51
0.511
0.512
0.513
0.514
0.515
0.516
0.517
0.518
0.519
0.52
D
3.574
3.569 FOR EVERY +.001 CHANGE IN H
3.564 THE DIAMETER DECREASES -.005
3.559
3.554
3.574
3.549
-3.525
3.544
0.049
3.539
3.534
3.53
3.525
C
2.495
2.496
2.497
2.498
2.499
2.5
2.501
2.502
2.503
2.504
2.505
D
3.537
3.539 FOR EVERY +.001 CHANGE IN C
3.542 THE DIAMETER INCREASES +.002 TO +.003
3.544
3.547
3.561
3.549
-3.537
3.551
0.024
3.554
3.556
3.559
3.561
THE RADIUS AND HENCE THE DIAMETER CHANGES
TWICE AS FAST WITH A CHANGE IN h AS IT DOES
WITH A CHANGE IN c
H
0.515