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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 
experience will be helpful here.  I also add my own (mathematical) 
input along the way, and in doing so I think I can directly address 
your task of showing that an OD micrometer is insufficient to 
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 
micrometer reading will stay the same.  If the reading varies, then 
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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