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### Measuring a Cylinder for Out of Round Dimensions

Date: 03/23/2005 at 21:17:05
From: Bill
Subject: How to measure true out of round?

I have been asked to referee a discussion with regard to whether a
cylinder meets a out of round requirement of +/-0.001".  If I measure
the cylinder with a standard OD micrometer I find a 0.008" difference
at 0 degrees and 90 degrees.  If I use a 3 point ("V") micrometer I
find a 0.001" difference at the same two positions.

My answer was yes based on the V micrometer.  The response was to
prove the use of the OD micrometer is in error.  This I cannot do.  In
my early years I was taught to use a V micrometer, and I never
questioned it.  Now 50 years later I must!

I realize that the standard OD micrometer is measuring diameter and
the 3 point micrometer is measuring radius but cannot prove it or how
this proof would relate to out of round measurements.

The correct irregular shape can have a constant diameter but be
visibly out of round.

Date: 03/24/2005 at 10:56:35
From: Doctor Douglas
Subject: Re: How to measure true out of round?

Hi, Bill.

It's certainly possible that the cylinder is out of round.  Here's
an example:

A             OD micrometer measures AC and BD = 8 mil

x           y
V-micrometer measures Cxy and Byz = 9 mil

B             D

w           z
C

The shape is bulged-out at {x,y,z,w}, leading to a larger measurement
from the V-micrometer.  Even if both micrometers are perfectly
calibrated, you can have this effect--the OD micrometer measures the
diameter of the object straight across, and the V-micrometer gives you
the diameter of the circle that goes through the 3 points (e.g.
{C,x,y}).  Because these two quantities are not necessarily the same
(except for a true circle), this discrepancy can occur.

It so happens that I have a colleague who is a machinist, and while he
doesn't work with ultra-high precision measurements, I think his
input along the way, and in doing so I think I can directly address
guarantee roundness (although if a part doesn't satisfy the OD
measurement it is certainly out-of-round)].  A "V micrometer" is
better, but ultimately can suffer from similar limitations.

If the object is truly round, an OD micrometer will indeed tell you
the diameter.  If you rotate the object by a few degrees, then the
you can be sure that, at a minimum, the diameter varies by the
micrometer variation, and that the part is at least out-of-round by
that amount.  To make this check properly, you can make this OD
measurement at many different rotation angles.  So you can use an OD
micrometer to confirm that an out-of-round part is *NOT* within the
roundness specification.  To say that a part *DOES* conform to the
roundess specification is a different question, however.

An improved measurement can be done by placing the cylindrical object
in a precision V-groove (usually 90 degrees), and then using a
sensitive gauge to detect small displacements in the top surface of
the cylindrical object as it is rotated.  This is better for two
reasons:  (1) the measurement can be done easily as the object is
continuously rotated and (2) at any given instant the measurement
involves three simultaneous points on the circumference of the object.
You can probably see intuitively that using three points is better
than two; it certainly is closer to the in-round formness
specification.  The gauge can be outfitted with a stylus in the shape
of a very small ball, so that its angular orientation with respect to
the V-groove apex doesn't move, and so that it can be easily
accommodated by any recesses in the object surface.  Thus if the
stylus moves, it will reflect out-of-roundness measurements more
accurately.

Clearly for best results one should make measurements at many
different rotation angles.  The main problem, however, with these
methods is that even if you measure at all rotations of the object, it
is possible for the contact points where the object touches the jaws
of the micrometer or the V-grooves to move on those (planar) surfaces.
As a result, the spin-axis of the object is not confined to a single
point (which we would call the center).  Thus each individual 2-point
or 3-point measurement need not reflect the radius from a center of a
best-fit circle to the surface of the object.

So to sum up, using a V-groove device is better than a standard OD
micrometer.  If you must use either of these devices, you should make
measurements at as many rotation angles as you can afford.  And then
you must realize that even if the object meets the spec as measured by
these devices, it may still not be truly in-round.

An example of a shape that illustrates this phenomenon is the Reuleaux
Triangle:

Reuleaux Triangle - Eric Weisstein's World of Mathematics
http://mathworld.wolfram.com/ReuleauxTriangle.html

This shape is a curve of constant width (or constant diameter):

Shapes of constant Width
http://www.cut-the-knot.org/do_you_know/cwidth.shtml

and as such, will always give you the same diameter when measured with
an OD micrometer.  You can rotate this object continuously in the jaws
of an OD micrometer and the OD reading won't change.  But you can
clearly see that it is out-of-round.

In theory, a way to avoid some of these measurement problems is to
somehow locate the rotation axis of the object, and then measure its
radius (using a sensitive gauge) as it is rotated about that axis.
It's not clear how one would do this in practice, though.

- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/

Date: 03/27/2005 at 22:02:48
From: Bill
Subject: Thank you (How to measure true out of round?)

Doctor Douglas -

Thank you for your time.  I believe you have answered my question.
The philosophy of testing is usually very interesting.

Wish you well.

Bill
Associated Topics:
High School Geometry
High School Practical Geometry

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