Here is a problem that I cannot prove.

Picture a tubular steel bushing or sleeve. Perfectly round. ID= 2.00000" (theoretically exact), OD=2.05", Length 1".
This is turned and bored on the end of a piece of solid shaft, then parted off.

Through handling or whatever, the part becomes slightly elliptical. Lets say X and Y axis differs by .005"-.010".

Now the customer measures the ID with a tenth reading three legged bore micrometer, recently calibrated and certified. Of course he gets varying readings at different radial positions around the part. But EVERY reading is OVER 2.001 " so he insists that the part is oversize.

I made a plug gauge that was 1.99975" on one end and 2.00025 on the other. The plug will easily true the sleeve to perfectly round and the big end will not GO.

I tell him that he cannot measure an ellipse or oval accurately with that mic....but because EVERY position is over 2.001", he still insists that it is oversize even if it were round.

Does anyone know of a mathematical or geometrical way to prove this ?

I can only prove it in this physical manner. But not convincingly enough it seems.

Basically: Would EVERY possible position that you measure an ellipse with a three-legged bore mic read OVER the actual diameter if the ellipse was originally perfectly round ?

 

Thanks,
RC

Here is the type of mic I a referring to.

I believe the mathematical solution for finding the distance between any three equi-spaced points on the circumference of an ellipse is extremely complex and would therefore be difficult to demonstrate to anybody.

Having said that someone is bound to prove me wrong and solve your problem.

In consolation, when I was employed in engineering I remember that diameter and roundness tests were two entirely different things and the plug gauge was the master reference for hole sizes. The attached picture shows two seven sided coins, neither of which is round however measuring the diameter of either with a standard micrometer will show it to be the same diameter at any point of measurement.


Tweakie.

I assumed that it would not be easy.
I trust the plug over the mics but he is hard to convince.
Neat coin trick too.
Thanks Tweakie,
RC

After some more serious thought, an idea comes to mind - If the OD of your piece parts are accurate then a jig /gauge (which is essentially a block with an accurately bored hole or collet) into which your part could be inserted to maintain it's roundness whilst it is measured with the tri-leg micrometer may possibly allow the ID to be measured correctly ?.

Tweakie.


(no trick - the coins are real)

Tweakie,
I realize what you meant about the coins, just a neat arrangement of arcs. Should have said "Neat geometry".

I'm sure what you suggest would work fine....but the mic is with the customer in NY and I am in VA. I could send him a ring for the OD or I could buy a mic like his but hope I don't have too. $1500+. I may go visit again soon and will be prepared.

But...the plug gauge doesnt lie. So I'm sure the parts are within tolerance.
We'll see.
Thanks Tweakie,
RC

RC
A couple of things first for what it's worth.
The problem with doing rings are that the material, in your case, is solid and even if say it was cored, there are internal stresses in the material. Now as you machine .... turn outside then some inside boring check ...... over and over ...
then on the last finish passes  all is in to spec. Then you part off and it eggs because of some residual stress still exits
but is held back because it's still attached to the bar. If the bar or cored bar were normalized you " may " not get as much springing after cutting off. Normalizing the part is nothing more than heating and cooling the material in a controlled manner and it does cost money.  I know this is different but i had 3" round s/s x 6 foot lengths machined down to 2.750 +- .001 and gun drilled for a 1.0000+-.001 +- .003 concentric over any 3' length at $1000 / ft cost. I had go / no go  3' foot long gages made  hardened and ground for checking by the inspector.
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I use and have a set of B&S Intramikes,  they will show out of round easily but about one circumferential location only and you would measure in four quadrants or more. But frankly speaking, a go no go gage would be better, as i actually like to use pin gages.  You can't relate the mike measurement to eccentricity along the bore and certainly not if it's egged. The pin gage can do this for you and mounted between test centers you can relate the outside along it's length. When using these mikes they click when enough turning force has been to seat the three faces. The force is quite small (10  in oz? ) and if it's egg shaped, and unless it's a really a thin wall, i don't think you would duplicate the go no go test as it is not
forcing the ring wall out.

The easiest way i could think of, based on measurements, would be to draw the piece in cad and then you can scale / stretch that piece to a around, to prove that even though measured wrong, it is round. Don't know how this would come out in reality or how convincing it would be.
But then that's what a go no go gage can do except that your adding some force to get it to shape. Certainly would be easier than trying to prove it mathematically. 

What is important is the end use of the piece and how it was specified.  So if the guy wants a piece that's a true cylinder / ring with no outside
force required to have it in that shape and to dimension that's one thing. Not having any understanding of end use or concerns is hopless if you
look come at it from a practical point of view.

RC, post your thoughts and I'll think more about it if you wish,
RICH

Hey RICH,
 I understand about the stress and such totally. The example I gave was simply to imply a theoretically perfectly round part before parting off.
Sorry if that was in any way missleading.
The ovality is not the issue. It is acceptable. The mating part is an interference fit and is solid which "trues" the sleeve.
They are just trying to say the ID is oversize by measuring the "egg" with the bore mics. They take multiple readings and average them then take that as the actual diameter. I say they are NUTS.

That said, let give another example: Machine a perfect ring. 2.000000000" ID, 3" OD, 1" long. (or pick your own dims.)
Stable, normalized and stress relieved.
Now, put it in a vise, Kurt precision for example, and apply just enough pressure to "egg" it .005" to .010".(or more to illustrate)
NOW measure it with the 3 point bore mics. Would it read over, under or both depending on the radial position of the mics?
If I had a mic of this type, of any size, I would try this and it would verify what I'm guessing....I think.
Thanks,
RC

RC,
Let me try it out. and will get back shortly.
RICH

RC,
The ring is 1.875"OD + - .OOO5  and  1.3338 ID+ - .0002    so a little thick.
I compressed it 0.006" about 90-270 deg , so 0-180 deg on ring is parrell with vise jaws.
With one of the mikes prongs at 90 deg ID= 1.3330
    "      the same          "          180   "  ID= 1.3368

I think they are nuts also for taking an average. In fact shouldn't they be summing the squares and taking the square root?
I can do a ring with a smaller wall around 2.5"od x .050 wall if you wish tomorrow.
MODIFIIED:
OOPS: only go to 2" with the mikes so it would have to be samller but what your quessing would be the same.
RICH

That's great RICH, thanks for testing it for me.
Up to .003" over with .006" compression.
The only thing they did different was random positions...not knowing exactly where the X and Y axis's were. And by chance then, all dims. were over the nominal dia.
I am much more confident that I am correct now......and he is more NUTS.
Thanks again RICH.
No further testing necessary,
RC