# Quick Measurement Of Small Taper Angle



## randyc

I'm re-posting a quick suggestion that I made on another forum regarding measuring taper angles.





Simple way to measure small tapers.  Place the taper between 1-2-3 blocks and measure diameter across the 1 inch surfaces with calipers, making sure that the calipers are flat against the blocks.  Repeat measurement across 2 inch surfaces.  Subtract the smallest diameter from the largest and divide the result by two.  The angle of ONE side of the taper  is the arc-tangent of this number.  (Do this on a surface plate, not the kitchen table, LOL).

Example:  The measured diameter with calipers resting on 1 inch surface is .745.  Measured diameter with calipers on 2 inch surface is .620.

(.745 - .620) / 2 = .0625 and arc-tangent(.0625) = 3.5763 degrees or 3 deg 34 min 35 sec  

For the included angle of the taper, double this result.

You can also use the 1 inch and 3 inch surfaces of the 1-2-3 blocks to gain a bit more accuracy.  If you do this, however divide the difference between the diameters by "4" instead of "2".


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## kd4gij

And in your exsample that is a 1 in 16 taper.[ 1" in 16"]  Witch is the easeyest way to set up to cut a taper on a lathe. If I don't have a dro on the lathe I use 2 dial indicators to set the compound or taper atachment.  That is the most common taper used. Besides the MT, tape  it is a marine taper and pipe taper anmost tapered pins.


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## randyc

I just thought of another easy way to measure a taper.  I made this simple tool some years ago for measuring shallow angles in a vertical milling machine vise, as shown below.





Either setting or measuring the taper involves moving the table a precise, known distance and measuring the "rise" with the travel indicator.  Use arc-tangent to either determine the measured angle or to tap the workpiece until the rise is the correct tangent (if milling the angle.)






Works good for measuring tapers, too:





NOTE:  the indicator must be exactly centered on the taper centerline.


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## randyc

If you have digital readouts on your mill it gets even easier.  Put a dti in the spindle, carefully align it with the taper centerline and zero it on the large diameter of the taper.  Crank the "X" axis until the dti is near the small end of the taper.  Carefully raise the "Z" axis until the dti is zeroed again.  Take a look at the DRO: the arc-tangent of the distance the "X" axis traveled divided by the distance the "Z" axis traveled is the included angle of the taper.


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## RJSakowski

randyc said:


> If you have digital readouts on your mill it gets even easier.  Put a dti in the spindle, carefully align it with the taper centerline and zero it on the large diameter of the taper.  Crank the "X" axis until the dti is near the small end of the taper.  Carefully raise the "Z" axis until the dti is zeroed again.  Take a look at the DRO: the arc-tangent of the distance the "X" axis traveled divided by the distance the "Z" axis traveled is the included angle of the taper.


I believe that your calculation is over simplified.  Because of the smaller diameter on the small end measurement, it sits lower in the vee block than the large end.  As a result, the bottom surface is not horizontal.  Neither is the centerline of the taper.  To illustrate, the drawing below  shows a taper of 1/8"/"resting in a vee block with measurements made 2 inches apart.  The projection to the right shows how the taper is inclined and shows a measured angle of 8.50degrees instead of the actual 7.10 degrees.  
It is possible to measure the taper with this setup but the math gets more complicated.  I find the easiest way to do this is with a CAD package.  The illustration was done in Solidworks and by making the right dimensions driven, the drawing will spit out the angle.  I expect that other CAD programs can do similar calculations.  Gotta love those CAD programs!


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## randyc

RJSakowski said:


> I believe that your calculation is over simplified.  Because of the smaller diameter on the small end measurement, it sits lower in the vee block than the large end.  As a result, the bottom surface is not horizontal.  Neither is the centerline of the taper.  To illustrate, the drawing below  shows a taper of 1/8"/"resting in a vee block with measurements made 2 inches apart.  The projection to the right shows how the taper is inclined and shows a measured angle of 8.50degrees instead of the actual 7.10 degrees.
> It is possible to measure the taper with this setup but the math gets more complicated.  I find the easiest way to do this is with a CAD package.  The illustration was done in Solidworks and by making the right dimensions driven, the drawing will spit out the angle.  I expect that other CAD programs can do similar calculations.  Gotta love those CAD programs!
> View attachment 98663



You're right.  I assumed that for small tapers the error would be negligible.  The taper must sit on a flat surface which leads to the problem of how to hold the taper in the mill vise so that its axis is aligned with the table travel (necessary for accurate measurement).

This detracts from the simplicity of the procedure, LOL, leading to such "fixes" as holding the taper in place with modeling clay and the like.  If one cared to make a simple alignment tool, however, the measurement would still be quick and simple.

Oops can't attach to an edit will have to make a new post.


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## randyc

I started sketching a simple alignment tool but it turned out NOT to be so simple.  I couldn't justify making a tool to align the taper when I might measure a taper every two years or so, LOL.  I think I'd better forget this method (unless someone can come up with a simple centering technique) and go back to the 1-2-3 block procedure  

P.S.  Thanks for pointing this out !


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## RJSakowski

randyc said:


> You're right.  I assumed that for small tapers the error would be negligible.  The taper must sit on a flat surface which leads to the problem of how to hold the taper in the mill vise so that its axis is aligned with the table travel (necessary for accurate measurement).
> 
> This detracts from the simplicity of the procedure, LOL, leading to such "fixes" as holding the taper in place with modeling clay and the like.  If one cared to make a simple alignment tool, however, the measurement would still be quick and simple.


Randy, It still is a useful technique and now I have a SolidWorks tool that makes the calculation easy.  It goes into my bag of tricks!  For those not fortunate enough to have access to SolidWorks or a similar CAD program, it could be set up as a Excel spreadsheet.  Input the separation distance between the two z axis measurements and the two z axis measurements and it will spit out the taper angle.  The beauty of your technique is the type of contact point doesn't matter since the same angle conditions exist for either measurement; just like a sine bar.   

The one issue that I have with methods like this is the relatively small difference in the two z measurements. Typical measurement errors can cause a fairly large error in the resulting taper calculation.  The Tormach DRO is capable of resolving .0001" and my digital indicator will resolve .00005" which means each measurement could deviate from actual by .00015" Since I am making two measurements, my error could be as much as .0003"  Over a two inch span, this would be an error of about .5 min. of angle.  From what I was able to glean in some earlier work, the AT2 std. for taper manufacturing is about 1/10th that tolerance.  (ISO-1947 is, I believe, the defining document).  If I were trying to make a Morse taper tool, I don't believe I would be able to determine whether it was satisfactory or not.


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## randyc

RJSakowski said:


> ....From what I was able to glean in some earlier work, the AT2 std. for taper manufacturing is about 1/10th that tolerance.  (ISO-1947 is, I believe, the defining document).  If I were trying to make a Morse taper tool, I don't believe I would be able to determine whether it was satisfactory or not.



That is ASTOUNDING, I had no idea the tolerance was that strict and I'd bet that most machinists with decades of experience would not know !!!  I've made a number of Morse taper tools - not scores by any means but probably a dozen, using either compound or taper attachment.  In both methods I (and everyone else that I know) set up for taper turning  by indicating a known good taper and adjusting the setup until there is no discernible error shown on the DTI.

After turning the taper, the normal fit check (as we all know) is to blue the part and check against a known good female taper.  If there is variance, the high spots can be carefully removed with file and sandpaper until a good fit is obtained.  I can't imagine that most small machine shops (much less the HSM) have the capability to measure angular error at the limit you mention.

I have to shake my head at that standard and consider the millions of tapers that have been turned by the hundreds of thousands of craftsmen over the decades - all were likely functional to the intended purpose.  I wonder if even one machinist in a thousand attempted to set up their machine to a taper standard.
_
(Incidentally, a few imported morse taper adapters can be useful for checking fit.  They are inexpensive and each one can serve as both a male and a female fit check.  The first photos in this post depict one of these adaptors, a MT-3 male to MT-2 female.  There is the risk, however, of dimensional variation.

We're fortunate in that we don't need to give much emphasis to interchangeability.  If we find a particular tool that well fits the lathe tailstock, as an example, then that tool becomes the standard for making any new tooling for the tailstock.)_


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## RJSakowski

randyc said:


> That is ASTOUNDING, I had no idea the tolerance was that strict and I'd bet that most machinists with decades of experience would not know !!!  I've made a number of Morse taper tools - not scores by any means but probably a dozen, using either compound or taper attachment.  In both methods I (and everyone else that I know) set up for taper turning  by indicating a known good taper and adjusting the setup until there is no discernible error shown on the DTI.
> 
> After turning the taper, the normal fit check (as we all know) is to blue the part and check against a known good female taper.  If there is variance, the high spots can be carefully removed with file and sandpaper until a good fit is obtained.  I can't imagine that most small machine shops (much less the HSM) have the capability to measure angular error at the limit you mention.
> 
> I have to shake my head at that standard and consider the millions of tapers that have been turned by the hundreds of thousands of craftsmen over the decades - all were likely functional to the intended purpose.  I wonder if even one machinist in a thousand attempted to set up their machine to a taper standard.
> _
> (Incidentally, a few imported morse taper adapters can be useful for checking fit.  They are inexpensive and each one can serve as both a male and a female fit check.  The first photos in this post depict one of these adaptors, a MT-3 male to MT-2 female.  There is the risk, however, of dimensional variation.
> 
> We're fortunate in that we don't need to give much emphasis to interchangeability.  If we find a particular tool that well fits the lathe tailstock, as an example, then that tool becomes the standard for making any new tooling for the tailstock.)_


The key is in your second paragraph.  You are essentially hand finishing to final fit.  It's actually a very sensitive technique. The amount of material removed would be quite small. Unfortunately, it requires a known good taper.
When you think about it, a proper taper has to wring to its mate in order to provide the necessary amount of friction to drive  a tool.


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## 18w

That  is a lot of interesting data regarding tolerance standards...too much for this old guy though. I again find it interesting that we with home shops think we can chase tolerances that in reality we cannot even check. A grade B cheapo surface plate or non calibrated blocks that many of us own aren't capable of giving that accurate of measurement. certainly at least in a uncontrolled temperature environment. Measuring a existing taper, turning, bluing and possibly filing or sanding has always been the homeshop and even some job shops mode of making a satisfactory taper but then we are not held to ISO standards. If a morse taper for example seats, blues well and holds tight then it is good enough in my world. Making such a taper by my method would not possibly pass a qc inspection because a file or emery cloth automatically would cause a part to fail in concentric spec if nothing else. To make a ISO quality taper would require the part to be ground. You do not see commercially made ISO quality parts finished by turning, at least I haven't. Randy C has shown a often good enough example using 123 blocks. RJ Sakowski has shown the proper way using CAD. Both with far better explanations and a better grasp of trig than I have. If I had the need for a accurate measurement of a taper using Randy's second suggestion I would put the tapered part on a sine bar and put what I thought was a correct approximation of gage blocks under the bar. At that point I would use Randy's method of traversing the taper with a dti. If it read zero (unlikely) you would know the taper by using your gage block height to do the math. If it isn't zero then you would have to do the trig for the resulting difference. If I just needed to make a part to satisfy my needs I use Randy's method. If I were to need something more accurate RJ Sakowski's CAD and Tormach programs are the cats pajamas. But if I really needed that good of part and knew it had to meet the above tolerances, Hell I would just send it out to be ground. Thank you both for great descriptions.

Darrell


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## randyc

There are many, many ways to de-frock the kitty, right ?  That's what makes this craft so fascinating.  And there are not many activities that require inventiveness, precision, fine craftsmanship, patience, mathematics and the ability to explain to your wife why you need the next $500 worth of tooling.

(An old toolmaker once told me that if I could draw it, he could figure out a way to make it.  I believed him)


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## randyc

18w said:


> ....If I had the need for a accurate measurement of a taper using Randy's second suggestion I would put the tapered part on a sine bar and put what I thought was a correct approximation of gage blocks under the bar. At that point I would use Randy's method of traversing the taper with a dti...



That sounds good to me.  The only problem might be securing the part to be measured so that its axis is aligned with the table travel.

However if one were very careful, misalignment could be accounted for by finding the "high spot" of each end of the taper using the "Y" axis movement and noting the "Y" travel.  A bit of 3-D trig would be required to sort out the actual angle but as RJSakowski noted in an earlier post, it's simple enough to make a spreadsheet if repetitive solutions are required.

In fact this entire topic almost _demands_ creation of a spreadsheet, ha-ha.  Of course the elephant in the room is that _measurement_ of tapers is not that important to the HSM - _fit_ is the important characteristic.

I made a spreadsheet for mapping the errors on a well-worn lathe bed some time ago.  Some day I might post it here, I think that it would invite interesting discussion


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## RJSakowski

The bottom line, whether a hobbyist-machinist or a professional, is that the part meets the requirements of it designed task.  I do not work to thousandths for much of what I do because that is not a requirement.  I do not, for the most part, make parts for interchangeability; certainly not for my own use.

I like to look back at how medieval locks were made.  A key was made first and the lock was made to fit.  There is an old saying" carpenters work to inches, machinsts work to thousandths, blacksmiths work to snug."  And yet when you look at some of the work done by blacksmiths (and locksmiths, silver and gold smiths) over the centuries, it is astounding in its fit and finish.  Prior to Ely Whitney, every rifle that was made was unique in that you could not take a trigger from one rifle and replace that of another, etc.  But those crude by modern standard rifles won a war and birthed a nation ( my apologies to any Brits out there; no offense intended).

When I make parts I invariably create a model and drawings and (try to) make them to print but if it is not to print I do not get bent out of shape as long as it meets my requirements.  If it doesn't, I remake the part.  You can make a lot of cool stuff without a metrology lab at your disposal.


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## 18w

Nah, I'm too dumb for all that spreadsheet 3d trig stuff. Hold the sinebar to a angleplate with a clamp or magnet and dial it in on the x axis, lay the tapered part on the sine bar against the protruding angle plate. I have seen sinebars with a groove and also ones with a side fence as well. But normally simply stand the part up and use the 123 block method. Easy peasy.

Darrell


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## RJSakowski

randyc said:


> There are many, many ways to de-frock the kitty, right ?  That's what makes this craft so fascinating.  And there are not many activities that require inventiveness, precision, fine craftsmanship, patience, mathematics and the ability to explain to your wife why you need the next $500 worth of tooling.
> 
> (An old toolmaker once told me that if I could draw it, he could figure out a way to make it.  I believed him)


Randy, I think I know that machinist. He has a shop in Cupertino.

I sent this machinist a drawing for a tapered part, about an inch long and terminating with a .020" diameter, made out of Teflon.  He sent me some parts almost by return mail.  He told me he first experimented by making .020" Teflon noodles.  He figured if he could make that, he could make our part.  He is still doing so today. (Full disclosure, they are being made on a Swiss screw machine. I don't want to discourage anyone.)  

He also made PEEK parts for us with a tolerance on one section of +.0002/-.0000.  His technique is a trade secret.  I happen to know how he did it but am sworn to secrecy, even internally in the company.


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## RJSakowski

At two minutes to midnight, I came up with a simple solution to the problem posed in post #5 above.  The difference in diameters in the taper, as outlined by Randy in post # 4 above is equal to .82843 times the measured difference in z .  Divide that result by the the x distance between measurements and take the arctan to get the taper angle.  Time for bed now.  I'll post details of the calculation later.  Suffice it to say, it works because the Vee block has an included angle of 90 degrees so the distance between the points of contact is equal to the distance between the vertex of the Vee block and the center of the circle.

THEN, I started to  think about it, the vertical section is an ellipse. So my model in post #6 above is oversimplified as well. Oh well, back to the drawing board!

Update:  This morning, I did it right and actually created models for the taper and block, created an assembly, and made a drawing of the assembly.  


A few noteworthy items: 
1. By comparing to the previous drawing, the taper angle is actually .05 degrees larger than predicted in the previous drawing. 
2. Although the distance between the measurements is 2.0000 inches, the smaller diameter is .001" less than expected.  This is caused by the fact that a taper is measured by a displacement along the taper axis but that axis is tilted by 5.06 degrees to the horizontal, making the axial distance longer by a factor of (1/cos(5.06)) degrees. 
3. The difference between horizontal and vertical measurements of "diameters" is significant due to the fact that they are actually ellipses. 
4. Finally, there is a .0020 difference between the actual difference in "diameters" and that predicted in post #5 above.  That would seriously affect the taper angle determination.

So where do we stand?  I have a sinking feeling that the determination of the vertical"diameter" will involve trig and furthermore it will require using the angle we are trying to measure as part of the measurement of that angle.  Excel has a macro called solver that does a good job of solving math like this through an iterative process but this is really getting hairy in a hurry.

So plan B? Darrell as a good idea in post # 15 above.  One problem is keeping the taper axis aligned properly, which Randy stated in post #6.  One method, demonstrated by OxTool on YouTube is to move to x position 1 and scan back and forth with the y axis to high the high spot.  Note the x, y, and z coordinates.  Move to x position 2 and repeat.  In Darrell's method he would build the gage block stack until the z axis difference is zero.  To Randy's method, I would mount the taper in the mill vice so it rests on the horizontal and vertical surfaces of the vice.  You still have to correct for the cosine effect (one two axes) but at least now you have known datums for reference.   One thing that I like about Darrell's method is that the 5" span of a sine bar increases the accuracy of the measurement.

One last thought about the Vee block method.  You can use it to compare an unkown taper to a known good taper; the difference measurements should be the same. Although it may be easier blue the taper and fit to a mating socket.

I wonder how the guys that make these tapers measure them?


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## randyc

RJSakowski said:


> ....One method, demonstrated by OxTool on YouTube is to move to x position 1 and scan back and forth with the y axis to high the high spot.  Note the x, y, and z coordinates.  Move to x position 2 and repeat...





randyc said:


> ....However if one were very careful, misalignment could be accounted for by finding the "high spot" of each end of the taper using the "Y" axis movement and noting the "Y" travel.  A bit of 3-D trig would be required to sort out the actual angle but as RJSakowski noted in an earlier post, it's simple enough to make a spreadsheet if repetitive solutions are required...



The defined points will locate a triangle that can be solved WHEN viewed perpendicular to the plane of the triangle but that angle isn't easily ascertainable from right triangle solutions because of the taper axial misalignment with table travel.  That's why the 3-D trig was suggested.

I'm still thinking that this situation arises so rarely that the simplest solution is the best for me.  (I would forget the more complex ones five minutes after reading them) 

I expect that folks who make tapers in production use go/no-go gages that are calibrated by CMM.  Of course if production is involved, many creative measuring solutions could be amortized into the cost.  But that's really not something a HSM would encounter -


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## randyc

RJSakowski said:


> ..Randy, I think I know that machinist. He has a shop in Cupertino...



If this was recent, I'd think it highly unlikely.  I worked with the guy (at Zeta Laboratories in Santa Clara) around 1979 and he would have been middle-aged by then.  He was German by the way and I'm embarrassed to say that I've forgotten his name.

There are a LOT of incredible toolmakers in the Bay Area but those that are worthy of the name using manual machinery are a slowly vanishing breed.  When I worked at Westinghouse Marine Division in Sunnyvale (1965-1968) some of the stuff those guys turned out with manual machinery was on the verge of science fiction !


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## RJSakowski

randyc said:


> If this was recent, I'd think it highly unlikely.  I worked with the guy (at Zeta Laboratories in Santa Clara) around 1979 and he would have been middle-aged by then.  He was German by the way and I'm embarrassed to say that I've forgotten his name.
> 
> There are a LOT of incredible toolmakers in the Bay Area but those that are worthy of the name using manual machinery are a slowly vanishing breed.  When I worked at Westinghouse Marine Division in Sunnyvale (1965-1968) some of the stuff those guys turned out with manual machinery was on the verge of science fiction !


I'm aware that we were talking about two different people.  The point that I was making, as you just did, is there are a good number of excellent machinists out there that can virtually work magic.


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## randyc

Ah - I took it literally when you said you thought that you knew him, LOL.


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## RJSakowski

Further taper thoughts:  As I continue to reflect on this issue, it has become obvious to me that there is no easy way to accurately measure a taper when you are not measuring normal to the taper axis.  A measurement at any other angle is measuring the major axis of an ellipse.  I know that it is possible to determine the diameter from the ellipse measurement but it will involve the taper angle that we are trying to determine  (I know because Solidworks can do the math LOL).  I suspect it will be an iterative solution.  

Plan C: OxTool (YouTube) showed an interesting way to determine tapers.  His tool was a cylinder with well defined different i.d.'s on either end.  The taper would be mounted in a lathe, indicated for zero runout.  The cylinder would carefully placed on the taper and tapped lightly until it was seated true to the lathe axis, thus assuring the the end plane was perpendicular to the taper axis.  The position of the opposite end of the cylinder was determined with a dial indicator and his DRO.  The cylinder was reversed and the process repeated.  Since the cylinder length is the same in both measurements, the difference in position is the distance between the two diameters on the taper.  There are two concerns.  The edge of the holes in the cylinder have to be sharp or at least have the exact same fillet or chamfer.  Also the diameters have to be very well known.

This is similar to the 123 block method above.

I think that I am all tapered out.  It was an interesting exercise but it is time to let it go.


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## kd4gij

randyc said:


> I started sketching a simple alignment tool but it turned out NOT to be so simple.  I couldn't justify making a tool to align the taper when I might measure a taper every two years or so, LOL.  I think I'd better forget this method (unless someone can come up with a simple centering technique) and go back to the 1-2-3 block procedure
> 
> P.S.  Thanks for pointing this out !


 

  Simple, All you need for your method is a bench center :lol: I have put a tapered collet between centers on my lathe and used a dial test indicator to adjust the compound to make my er40 collet chuck. Now I make inboard boat shafts now. last one was 3"dia x21" long


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## 18w

Having realized that I had omitted a couple of pieces that went along with my suggestion in the middle of the night. I will attempt to describe what I can not show you visually because of my computer ineptness. When I originally posted , it was with a recently completed job for a magneto shaft in mind . It was a straight shaft with a taper on the end that needed a keyway. Obviously a straight shaft is no problem to lay on a sine bar against a angle plate so you just coordinate in the x for your centerline. What everyone is describing here obviously is a cone, a morse taper for instance. I have used the x-y coodordinate move back and forth to find the high spot method and it works for my use. However there is one more way. A sinebar typically has a little fence or stop at the end. Measure the od of the cone at each end ( measuring the small end is difficult and is a approximation at best to tight tolerances.) Place the part on the sinebar with the small end toward the stop which is also the end your gage stack goes under. Now place a gage pin between the cone, the stop and the dialed in face of your angle plate protruding at least as high as your cone. This gage pin is just a approximated size at this point. A size that gives the appearance of parallelism between angle plate and cone. Place a similar sized pin on the other side touching the cone and stop. Mike across outside of pins. Add or subtract difference from large o.d. end. Select 2 gage pins of 1/2 the difference each and remeasure until outside of pin dimension and large o.d. are equal. This does 2 things. It centers your part in y and it also gives a means of measuring the taper. Tool makers use ball dimensions for finding angles to a sharp edge which is what this method does. Having a known dia. pin hard against your fixed angleplate, the stop and the cone gives you a exact known center point and a right angle between stop and angle plate. With these values you can draw a hypotenuse through the center of your pin and find known lengths and angles all the way to your sharp edge on the small diameter. This requires solving for several different  triangles. Of course this exercise eliminates the dti altogether because you already have solved the taper. A long drawn out solution for what started as a simple way I guess. Clear as mud? I will see if I can find a example of using ball dimensions on the net and post if I do. Or possibly this is something you are already aware of.

Darrell


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## RJSakowski

I think I see what you are doing.  By using two equal diameter pins in a stack with your taper the equals the larger diameter, you ensure that the taper axis is parallel to the angle plate.  Using the pins against the stop gives you a measurement point at a good diameter a known distance from the stop as opposed to trying to measure the small end of the taper.  If you measure the length of the taper and subtract half the pin diameter, you have your axial distance.  At his point, the difference between the two diameters should be twice the pin diameter.  Dividing the pin diameter by the distance between should give you the taper in inch/inch.  Take the arctan to get the half  angle.


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## 18w

Exactly. your description is far better than mine.

Darrell


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## randyc

Darrell, after two glasses of wine I'm not following that very well.

But the little bit that I THINK I understand suggests that if the large end and the small end of the taper have been accurately measured then the only thing left (to calculate the angle) is to measure the length between the large end and the small end, right ?

What am I missing, can you post a sketch or a photo ?  RJSakowski obviously understands your method (and the method obviously works since you've made acceptable parts using it) but both of your brains are clearly a lot more agile than mine


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## 18w

Just because I can describe it and was shown how many years ago doesn't mean I have a clue how to make and post a drawing with a computer Randy! LOL That is a job for Mr. Sakowski who obviously could whip us up a nice CAD drawing if he would do us the favor. Also if I recall there is a method to use two balls or pins of different dia. a given distance apart between the angle plate and the side of a cone and with the appropriate formula the taper can be resolved.

Darrell


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## RJSakowski

18w said:


> Just because I can describe it and was shown how many years ago doesn't mean I have a clue how to make and post a drawing with a computer Randy! LOL That is a job for Mr. Sakowski who obviously could whip us up a nice CAD drawing if he would do us the favor. Also if I recall there is a method to use two balls or pins a given distance apart between the angle plate and the side of a cone and with the appropriate formula the taper can be resolved.
> 
> Darrell


Darrell, a photo would be nice.  I am not sure that I understand how the sine bar comes into it from your description.  If you can provide a photo, I will be happy to draft the setup.
BTW, Darrell & Randy, since we seem to be getting downright friendly here, the name is Bob


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## randyc

Thanks Bob, my fingers were getting sore from typing out your nom de plume.

There is a short chapter in "Machinery's Handbook" regarding setting up tapers.  Apparently sine bars weren't considered, I don't know why, as the first choice.  Mostly it seems that the old guys preferred to use a couple of known and different diameter discs spaced a precise distance apart.  A pair of straight edges (vise jaw or whatever) on each side of and tangent to the discs established the angle.  There are lots of tables and math expressions to support this technique in "the book".  It makes for interesting reading even if the application won't be used ..

I suppose that the technique might be useful for devising a measuring tool, maybe along the lines of Darrell's sine bar method -


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## 18w

Bob actually the sine bar was part of the first solution that had the problem of the centerline axis alignment issue. I suggested to use the sine bar for this solution simply because it has a fixed stop for the small end of the cone and the gage pins to seat against. I am sure that having the cone on a surface plate with the small end face against the inside of one leg of a ground or granite angle and the large o.d. touching the other inside leg yields the same result with the possible exception that the gage pins would not be at right angles to the cone horizontal centerline.  By using a sine bar and approximating a gage stack the horizontal plane of the centerline of the cone would be closer to 90 degrees to the vertical standing gage pins and maybe that is not a issue. The key being a 90 degree angle on the inside corner of the angle plate and the gage pins, and cone equaling the large end o.d. Also are you familiar with the use of the two different dia pins a given distance apart to measure a angle as I mentioned in my previous post? I will see if I can find a example on the net. I truly am computer illiterate  And as I post Randy lists one example of what I describe. Thanks Randy. When in doubt go to the Handbook! I could have saved you both some typing and head scratching. lol

Darrell


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## RJSakowski

Darrell, That clears up the mystery.  The one refinement that I saw, (and maybe missed in your explanation) would be to put a third pin under the small end of the taper to raise the axis of the taper to parallel with the sine bar.  I don't think it is really necessary as any correction would be very small.

I have used pins in a similar fashion before.  I never had the occasion to measure a taper mechanically.  I designed a plug and mating socket system,used in one of our products, with a large diameter of .09" and a small diameter of .02" and a length of just over an inch.   We had to determine a method of verifying machining accuracy and we used an optical comparator to make the measurements on the plug.  The tapered end could be mounted to a cylindrical shaft so we were able to measure perpendicular to the taper axis, avoiding the problems we have been discussing.   We measured the socket dimensions using precision balls.  I determined the seating depth with SolidWorks (gotta love the CAD!) and wrote the spec.  Our tolerance requirement the much looser than that for a Morse taper so these methods worked for us.

Randy, the machinist was the gentleman  from Cupertino mentioned in post#16 and that was the part mentioned.  The plug, being made fro Teflon, posed interesting challenges for QC.

I do not have a Machinery's Handbook, unfortunately.  It has been on my wish list for a very long time but the tools and accessories budget is somewhat limited and it always seems to get relegated to the back of the list.  The last time I had access to one was about fifteen years ago and it made for an interesting read.  The fascinating thing is that most of the wisdom in that book was actually developed a hundred years ago.

Bob


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## randyc

RJSakowski said:


> ...I do not have a Machinery's Handbook, unfortunately.  It has been on my wish list for a very long time but the tools and accessories budget is somewhat limited and it always seems to get relegated to the back of the list.  The last time I had access to one was about fifteen years ago and it made for an interesting read.  The fascinating thing is that most of the wisdom in that book was actually developed a hundred years ago.Bob



Bob, you're right on target, the first edition was published in 1914.  It hasn't changed much over the years either !

I guess I'm like a kid picking at a scab but I just couldn't let go of this taper problem (which ended up being fairly simple).  Solving the taper in a milling vise would be simpler using 3-D vector analysis but the math is easier if the problem is converted to two right triangle solutions.  The first triangle is the one defined by the reference, X and Z, the second is defined by the reference, Y and Z.

The X, Y and Z dimensions are obtained as mentioned previously by touching off the high points of the taper and noting the difference between the max and min values.  The differences become X, Y and Z in the expression shown below.


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## RJSakowski

randyc said:


> Bob, you're right on target, the first edition was published in 1914.  It hasn't changed much over the years either !
> 
> I guess I'm like a kid picking at a scab but I just couldn't let go of this taper problem (which ended up being fairly simple).  Solving the taper in a milling vise would be simpler using 3-D vector analysis but the math is easier if the problem is converted to two right triangle solutions.  The first triangle is the one defined by the reference, X and Z, the second is defined by the reference, Y and Z.
> 
> The X, Y and Z dimensions are obtained as mentioned previously by touching off the high points of the taper and noting the difference between the max and min values.  The differences become X, Y and Z in the expression shown below.


I am having trouble dropping it as well.  But then I'm Polish so I have a license to be stubborn.  I think your explanation is along the same line that I was thinking.

One thing that bothered me was if you were trying to measure a 90 degree taper. you couldn't because the z distance would be infinite.  Instead of an ellipse the section would be a parabola.  I did the following drawing for my brother who is a math professor last night.



The measurement we are making is actually "l", perpendicular to the x axis.   what we need is "d" because that is how a taper is defined.  We can determine d from l but we need to know the angle a.  Now a is the same for both the large and the small diameters so we will have enough equations but my math skills are too rusty to force a solution.  Excel can do it through an iterative process, their "Solver" macro. You can pick a starting value for a based on the simple measurement  or by simply guessing. I say that based on an assumption that the process is converging.
Another factor is the x distance measurement.  We are traversing along an edge of the taper not its central axis so there is a 1/cosine factor; the distance along the taper axis is longer because it deviates in two directions (I'm assuming Darrell's first suggestion of placing the taper so the edge is coincident with the sine bar and the angle plate).  There are two right triangle formed, one in the xy plane and one in the xz plane  The included angle of both triangles is a.  The actual offset of the taper axis r = sqrt(y^2 + z^2)where y and z are side opposite in the xy and xz planes.  Since they are equal, r = sqrt(2)*y or 1.414*y.  We now have the side opposite, "r", and the side adjacent, "x" which is our distance along the x axis the we measured when we measured our difference in z.  So  the taper axis distance between our measurement points, D = x/cos(arctan(r/x)).  arctan x is the angle of the resultant vector from the y and z offsets. So we should have all we need to calculate the two diameters.

This is all very complicated and it addles my brain so I won't guarantee that I haven't made a mistake in logic or in math.  It's a good thing that I like puzzles.  I'm told it keeps me from getting senile. 

Darrell's proposal of using the gage pins is actually a simpler way to accomplish the task although I can see a potential problem with trying to balance all those components and making a measurement with a mike as well.    Another drawback is you really need a double set of gage pins , preferably by tenths, to make the measurement. However, the setup guarantees that you are measuring perpendicular to the taper axis, as required. Your original method with the 123 block accomplishes the same thing.  The only drawback is it requires a flat perpendicular to the taper axis.  OxTool's dual diameter cylinder method does the same as you can align the cylinder to be parallel to the taper axis.  The drawback with his method is you have to be able to measure the diameters accurately.  Being an inside measurement, it will be some sort of transfer measurement like gage pins.  This introduces an additional error source into the measurement.  One possibility would be to "calibrate" the cylinder using a known taper.  Based on the standards requirement that I state much earlier, that is actually a pretty good solution.  If the taper is stood on it's large end on a flat surface and a height gage is used to measure , the OxTool method is essentially the same as your 123 block method.


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## randyc

Bob, just got back to this.  I was hoping that you might check my (I'll call it a hypothesis rather than a solution) math graphically.  You are obviously skilled and FAST at CAD and I am as sluggish as molasses in the morning.  It took me most of the afternoon just to draw the taper above, LOL.



RJSakowski said:


> ...Your original method with the 123 block accomplishes the same thing.  The only drawback is it requires a flat perpendicular to the taper axis...



That *is *a problem and addressing it is only slightly less difficult than aligning the taper axis horizontally if using the mill table + DTI method.    KD4gij mentioned bench centers and that would solve ALL of the problems if the taper had concentric center features at both ends.  Maybe if the bench had female centers at each end - well that has it's own set of problems ...

Placing the small end of the taper in a hole in the measuring surface base (for the 1-2-3 block measurement) and aligning it - as much as possible - perpendicular to the measuring surface could be helpful to measure tapers that do not have one flat, axially perpendicular end.  But this would result in errors in the minimum and maximum diameter measurements due to angular misalignment.

I wonder how much error would result if the taper was carefully aligned visually to be perpendicular ?  The diameter measurements would be elliptical ... one could make another diameter measurement, 90 degrees to the first and average the two readings.  But this presumes that the taper is fixed in position - more complication !

The old-timers MUST have addressed this problem in a practical manner that doesn't include any electronic device ... I can't believe that there isn't a simple way to do this 

Using the mill table, one could block up one side of the taper, as I think you and Darrell were suggesting with the pins.  By sweeping from end to end, finding the high points, it would be possible to align the taper axis with the table travel.

Maybe with a "tenths" indicator this would be adequate.  Obviously, if the taper axis is aligned, the measurement is a piece of cake, LOL.  Somehow, however, this doesn't suggest the "elegance" of a true geometrical solution.


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## RJSakowski

randyc said:


> Bob, just got back to this.  I was hoping that you might check my (I'll call it a hypothesis rather than a solution) math graphically.  You are obviously skilled and FAST at CAD and I am as sluggish as molasses in the morning.  It took me most of the afternoon just to draw the taper above, LOL.
> 
> 
> 
> That *is *a problem and addressing it is only slightly less difficult than aligning the taper axis horizontally if using the mill table + DTI method.    KD4gij mentioned bench centers and that would solve ALL of the problems if the taper had concentric center features at both ends.  Maybe if the bench had female centers at each end - well that has it's own set of problems ...
> 
> Placing the small end of the taper in a hole in the measuring surface base (for the 1-2-3 block measurement) and aligning it - as much as possible - perpendicular to the measuring surface could be helpful to measure tapers that do not have one flat, axially perpendicular end.  But this would result in errors in the minimum and maximum diameter measurements due to angular misalignment.
> 
> I wonder how much error would result if the taper was carefully aligned visually to be perpendicular ?  The diameter measurements would be elliptical ... one could make another diameter measurement, 90 degrees to the first and average the two readings.  But this presumes that the taper is fixed in position - more complication !
> 
> The old-timers MUST have addressed this problem in a practical manner that doesn't include any electronic device ... I can't believe that there isn't a simple way to do this
> 
> Using the mill table, one could block up one side of the taper, as I think you and Darrell were suggesting with the pins.  By sweeping from end to end, finding the high points, it would be possible to align the taper axis with the table travel.
> 
> Maybe with a "tenths" indicator this would be adequate.  Obviously, if the taper axis is aligned, the measurement is a piece of cake, LOL.  Somehow, however, this doesn't suggest the "elegance" of a true geometrical solution.



I think that you are correct in the assumption that visual alignment will suffice.  The cosine error will be quite small. A 1 degree error is quite huge visually and that creates a cosine error of 150 ppm.  We cannot practically make distance measurements to that kind of accuracy.  O f course the errors can stack, but even so, measuring a diameter difference of .1 inches more or less to +/-.1 mil is 1000 ppm.. 

I really would like to find out how the" big boys" do it and it would be very interesting to see how it was done a hundred years ago.  

I did find an excerpt from "Machine Shop Practice, vol. 1" which places the taper between horizontal centers and uses a 10" sine bar to measure the taper.  They also show a method using four identical pin gages.  The the small (perpendicular) end of the taper is placed on a surface plate and the pins placed on the plate on either side of the taper.  Two identical gage blocks are placed outside of the pins and the second set of pins placed on either side of the taper.  The outside to outside distances of the two sets of pins are measured.  The vertical distance between the two sets is well know.  You now have the diameter difference and the distance difference.  The gage blocks can have a tolerance of +/- 4 ppm; the gage pins, .04 mil and the mike .05 for a maximum error  of .1 mil on each diameter measurement.  If the diameter difference is about .1", this will give you a possible error in the taper of about .4 minutes of arc.   This probably as good as we can expect without more sophisticated metrological tools.

Bob


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## 18w

Bob, thanks for referring to the Machine Shop Practice excerpt. That is a simpler version of what I was trying to describe as my example is sometimes difficult with out 3 hands or lots of clamps and magnets. But that is a picture of a similar way of doing it. I offer up as to how the big boys do it, possibly a optical comparator or  now the more popular CMM?

Darrell


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## The Liberal Arts Garage

In another time and place, I was given  an "instrument lathe" with an  unknown
taper.  Being slow in trig, I resorted to the business  card method, using sharp scissors. Chopped up cards to get a close fit, drew a sharp centerline and two 
closely measured  verticals, used calipers  to transfer diameters to stock in other
lathe, hog and file to,measurements, blue female  taper, and finish. S impler than it sounds.......BLJHB


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## The Liberal Arts Garage

As  I read the impressive math and metrologic presentation above, I see that (no
original thought from my addled brain , Pythagoris could not work in free space,
and at the very summit of Machining,we,too must visualize and file to fit. Thanks,
.........BLJHB.


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