# Measuring the diameter of small holes



## mklotz (Jan 31, 2008)

Accurate measurement of the diameter of small holes can be difficult for the home shop machinist. Digital calipers aren't the right tool because most of them have minute flats on the inside jaws which make the caliper read low. Bore gages are great tools but they have limited range and are very expensive for the amateur. An inside mike will produce accurate readings but they seldom can measure smaller than 0.2". Split ball gages can go lower than 0.2" but they require practice to develop the right "feel" to obtain an accurate measurement.

The cost-effective solution for the HSM is a set of plug gages. These are two inch long precision ground steel cylinders that come in graduated sets that cover a range of diameters in small increments. Probably the most useful set for the HSM is the one that covers 0.062" to 0.25" in steps of 0.001".

[Aside: These sets come in plus or minus tolerance. As the names indicate, the plus can be slightly larger but not less than the indicated size while the minus can be slightly smaller but not larger than the indicated size. For checking hole sizes, you'll want the minus type.]

Ok, now for the tip that is the subject of this post. Few people realize that they can use their plug gages to measure holes larger than the largest pin in their set. This is done by inserting three pins into the hole rather than just one.

If you have three circles (the pins) all mutually tangent to each other, it's always possible to draw a circle around these three circles that is tangent to each of the smaller circles. [People with a math background will recognize this as the Outer Soddy circle.] By adjusting the three pins chosen we can make this larger circle any diameter we wish up to some maximum size.

Putting some numbers to it, with the set mentioned above we can measure any size hole up to a maximum of 0.5365". As an example, to gage a 0.5" hole, we would use the 0.215, 0.230 and 0.249 pins and the resulting Soddy circle would have a diameter of 0.5" with an error of only 0.0000003" - close enough for government work.

Now, deciding which three pins to use is a bit of mathematical misery. I've written a (free) program, PLUG, available on my page to do the calculations for you.


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## S_J_H (Jan 31, 2008)

Marv, What an Excellent post! And much thanks for the program! I would have never thought to use 3 pins!

Steve


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## zeusrekning (Jan 31, 2008)

Marv, You are the man!! For some reason at my shop I have never ordered pins over .500" . I don't know why because we use them often. But now I won't need to.
Again , Very impressed and disappointed I never thought of it ???
Tim


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## mklotz (Feb 1, 2008)

Thanks for the kind words, guys.

I've long contended that there are many instances where mathematics applied to machining can produce useful, and sometimes innovative, results. Machinists seldom have the math background to do this for themselves and most mathematicians assume that, since machining is such a long-established practice, there remains little to improve via mathematical techniques.

Since many hobbyists come from technical vocations, they represent a rich resource for improving and innovating the way we work because they can apply their specialized knowledge to manual machining tasks that are largely ignored by the currently computerized commercial developers. Hobbyists are largely unconstrained by considerations of time and schedule so they can afford to 'waste' time developing tools and techniques that will save other hobbyists time and effort (and, sometimes, money).

The most difficult part of doing this is to 'get your mind outside the box' and begin to view the things you do from the viewpoint of your own area of expertise. Next time you have an especially difficult or onerous machining job, try to see if there is anything in your vocational toolbox that could be applied to the task.

As an example of what I'm talking about, consider the task of cutting a metric thread on a lathe with an Imperial leadscrew. Conventional wisdom says that the threading dial cannot be used (that's true) and consequently, the half-nuts must stay engaged until the thread is finished. This requires that the motor must be reversed to drive the carriage back to the right to prepare for the next pass on the thread being cut.

But, here's how a mathematician would think about the problem. When I move the carriage, I have to move it an integral multiple of the pitch of the thread being cut in order to guarantee that the tool will still track the thread. However, that distance is not a multiple of the pitch of the leadscrew so the half-nuts won't reengage at that distance. For the half-nuts to reengage, I have to move the carriage an integer multiple of the pitch of the leadscrew.

Those two pitches are related by a rational number, 25.4 (mm/inch). So, there will always be distance at which I've moved both an integral multiple of the metric pitch AND an integral multiple of the leadscrew pitch.

I can disengage the half-nuts, move that distance and, not only will the half-nuts reengage but the cutting tool will also be tracking properly to cut the metric thread.

Now, that distance may not be practical in some cases. For instance, if that distance is longer than the lathe bed, this insight is useless. Nevertheless, there will be many cases where the distance is practical and, with lathe DROs more common, precisely moving that distance is a lot simpler than it used to be.

If you want to explore this technique, take a look at the STICK program on my page.


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## Brass_Machine (Feb 1, 2008)

My son thinks I have some strange heroes. While he is into spiderman and batman, my heroes are Marv, Birk, Mcgyver, Wes, Rick, Tin, Rob, Mike, John, Steve* ... well you can see where this going?!?

Eric

*these names are in no particular order, nor is the list complete!


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## mklotz (Feb 1, 2008)

Before this gushing goes too far, I want to make it very clear that...

I DON'T WEAR MY SHORTS ON THE OUTSIDE OF MY CLOTHING.


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## Brass_Machine (Feb 1, 2008)

mklotz  said:
			
		

> Before this gushing goes too far, I want to make it very clear that...
> 
> I DON'T WEAR MY SHORTS ON THE OUTSIDE OF MY CLOTHING.



What about blue tights?????

 ;D


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## Don Huseman (Mar 1, 2008)

In regards to measuring the diameter of hole with three pins. My father had a Babbitt bearing shop and I spent a lot of time there during the summers and then later on i had a company called Keelco right next door to his company. I later found out about "The Home shop Machine magazine. I showed this magazine to my father, who was a mechanical engineer. He told me to get that kitty **** magazine of his desk and went back to doing his work. I took my magazine out of his office, hurt to the quick.
well Later on I became a salemen for him and was out in the field when I came accross a babbitt bearing that was only 120 degrees of a circle. I had never seen a bearing like that but there it was. with out 180 degrees I didn't know how to measure the Id of the bearing. Well I got back to my fathers bearing company and asked my father how to measure the Id of this 120 degree bearing. He said that they always measured it on a horizontal boring mill. I said no this bearing is out in the field how do you measure it. He said" he will have to get back to me with the answer. 
Well many month went buy and no answer was given to me. I was looking in the "home Machine shop magazine when I came accross the solution. It used three cylinders that you would stack up and measure the width with a mic. Well the very next day I took the magazine and layed it out on his desk and said here is the solution to the problem of how to measure the 120 degree bearing out in the Field and I want $10,000 for the solution. He told me to go """" in my hat and told me to get out. Latter on when we had lunch his curiosity got the better of him and I have a big mouth so I showed him.
Don


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## mklotz (Mar 1, 2008)

Don Huseman  said:
			
		

> It used three cylinders that you would stack up and measure the width with a mic.



Don,

That sounds like something that could be useful to some of the members here. How about showing us a diagram and the associated math that you used to do that?


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## Don Huseman (Mar 4, 2008)

I tried to find the article in the home machine shop magazine but have not been albe to find it yet. It was published back in the 80ies can any body find it.


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## mklotz (Mar 4, 2008)

I don't see why one needs to use any pins at all. Basically, to find the radius of a partial arc, all one needs is a depth mike.

As shown in the attached drawing, drop the mike into the curved form such that it forms a chord and then measure the sagitta (s) with it.

If

C = chord length
S = sagitta length

then the radius of the arc, R, is given by:

R = (4*S^2 + C^2)/(8*S)

If you don't have a depth mike or the base on the one you have is the wrong size, simply drop any piece of rectangular stock into the curve. Measure the sagitta by using the depth rod on your calipers from the top of the stock and subtracting the height of the stock. 

View attachment scan0001.tif


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## Don Huseman (Mar 8, 2008)

Does any one know what Sagitta length is. Other than Marv


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## cfellows (Mar 9, 2008)

See Picture:


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## jpaul (Mar 10, 2008)

Sagitta? Is that the same as the Mid Ordinate? If so, the Mid Ordinate is the perpendicular bisector of the chord. As such it extends to the Radius point of the arc. The Mid Ordinate, m = R(1-cos Delta) where Delta is the Central Angle.

Marv mentioned that solutions to machining problems may often be found in disciplines outside of machining. As a surveyor, I have used the Mid Ordinate to One, solve for the radius of a curve by measuring the chord and the mid ordinate and, 2. to layout an arc by setting the beginning and end points of the arc, and a mid point on the arc measured from the mid point of the chord a distance equal to the calculated Mid Ordinate.


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## mklotz (Mar 10, 2008)

Yes, Paul, that's it.

If you draw the arc and its associated chord in isolation (i.e., without the rest of the circle) you'll note that it looks like an outline drawing of an archer's strung bow. When you add the sagitta, it looks like an arrow nocked into the bow.

The Greeks and Romans saw this same symbolism when they looked at it. The Latin word for "arrow" is "sagitta", hence the term that is used in mathematics. We also encounter this bit of Latin in astronomy. The name for the constellation known as the "archer" is "Sagittarius" which, in Latin, translates into "shooter of arrows".


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## cfellows (Mar 10, 2008)

I'm familiar with the term sagitta from my telescope making experience. Sagitta references the depth of curvature on parabolic mirror in a reflecting style telescope.

Chuck


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## mklotz (Mar 10, 2008)

For completeness if someone stumbles across this post in the future, let me expand on this business of measuring radii.

For the example I've shown above it's assumed that the radius is concave so that it's possible to drop a depth gauge into the concavity and make the necessary measurements. But suppose you've got a solid chunk of material with a radius on one of the edges and you want to know that radius.

What you do is sit the radiused edge on two rollers of known diameter spaced a known distance apart and measure the (minimum) height of the radius above the surface plate.

The associated mathematics are a little messy as you can see by examining the attached sketch and derivation. The program MRADIUS from my page sorts it all out for you.


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