Simple Formula for Calculating Tapers

[bluwolf]

For those with a metal lathe AND are mathematically challenged like me, there's a simple formula for calculating tapers. I may be posting the obvious but a couple people have asked me how, so I figured I'd put it here for anyone that might be interested.

1. Large Diameter minus Small Diameter = X

2. X times 28

3. Answer of #2 divided by Length of Taper = Angle to set Compound Slide

Example:

Large Diameter .750
Small Diameter .500
Length of Taper 2.000

1. .750 - .500 = .250

2. .250 x 28 = 7

3. 7 divided by 2.00 = 3.5 degrees

Maybe this will help a few people.

Mike

 

[Penultimate]

Elegantly simple, now I don't have to slog thru trig tables.
Thanks

 

[Dalecamino]

Thanks for posting this Mike. I just might use it sometime :biggrin:

 

[Rich L]

For those inclined, the actual half angle equation is

angle = arctan (∆ø/2/length)

in this case = 3.58º If it matters, that's a little over 2% in half angle off the rough calculation.

Mike's approximation is linear.

I would guess for most in this forum that difference won't make a hoot because you're probably not trying to match a standard (Morse, Jacobs, Jarno, B&S, etc). But if you are, it will.

Attached chart shows the percent error in the approximation across a ∆ø (difference in diameter) from .300 to .010 with a constant taper length of 2 inches. At the big end of the big taper of .300 the approximation produces an error of a little over .003 in radius which is very significant in matching taper work. At the other end of the taper spectrum the error is about .0001 - insignificant. At the mid range where a lot of the standard tapers reside the error is roughly .0015 which can still be problematic in taper matching.

BUT, like I said, not much of a hoot for most things done here. Just fun to play around with the math.


Cheers,
Rich

 

[BKelley]

MikeThanks for the formula. This is the kind of information that is helpful. Wish I had it this morning when I turned that Civil War pen.

Ben

 

[Dalecamino]

Rich, thanks for the edumacation. My head started hurting on the second line. :redface: I like my etch-a-sketch just fine for now.:biggrin:

 

[EBorraga]

Pi has to be in there somewhere

 

[randyrls]

The IAP Wiki has a page showing the standard sizes and tapers for the standard Morse Tapers from MT0 (yep that's a zero) to MT7
http://www.penturners.org/wiki/Morse_Taper_Dimensions

 

[Rich L]

Pi has to be in there somewhere

arctan is "pi-less" -- just produces an angle in radians. The calculations for the graph have the conversion to degrees built in (180/π) to compare to Mike's approximation.

Cheers,
Rich

 

[Penultimate]

Rich
For those of us with Sieg mini lathes the compound slide is graduated in 2 degree increments. I think the error using bluwolf's formula is inconsequential compared to the interpolation error of setting the angle. I try to set up my tapers using whole number angles by adjusting the taper length. Usually my tapers are subtle so if the taper stops before the end it's ok.

 

[azamiryou]

Rich
For those of us with Sieg mini lathes the compound slide is graduated in 2 degree increments. I think the error using bluwolf's formula is inconsequential compared to the interpolation error of setting the angle.

I wouldn't dream of using those markings to align the compound. I use a protractor for different angles and a machinist's square when I want it straight. My protractor is marked in half-degree increments, and I can set it between the markings if I need to.

 

[NewLondon88]

Who wants some pi?

3.1415926535897932384626433832795028841971693993751
058209749445923078164062862089986280348253421170679 821480865132823066470938446095505822317253594081284
811174502841027019385211055596446229489549303819644
288109756659334461284756482337867831652712019091456
485669234603486104543266482133936072602491412737245
870066063155881748815209209628292540917153643678925
90360011330530548820466521384146951941511609...

 

[Rich L]

Rich
For those of us with Sieg mini lathes the compound slide is graduated in 2 degree increments. I think the error using bluwolf's formula is inconsequential compared to the interpolation error of setting the angle.

I wouldn't dream of using those markings to align the compound. I use a protractor for different angles and a machinist's square when I want it straight. My protractor is marked in half-degree increments, and I can set it between the markings if I need to.

I certainly agree and I'd never use the compound to cut any taper I intended to replicate after tearing down the setup with an exception I'll mention below.

I'd like to say that there's nothing wrong with Mike's approximation if that's the error you can deal with. There's error in everything but if ~2% is OK with you then go for it. My post was info.

On the other hand, if you want repeatability and accuracy from setup to setup most serious folks use taper attachments and plentiful use of dial indicators and, if you have the wherewithal, more expensive machinery.

In the past, I've temporarily fastened (superglue) some stops to my compound protractor so that the setup was repeatable to my goals.

I think if you want better repeatability using something akin to Mike's approach that would be to setup your taper to an angle to start with - right on a mark. You'll probably be well within a 2% margin in angle measurement that way instead of calculating an angle with almost a guaranteed 2% error and then trying to find it without a vernier on your protractor.

BUT, like I said originally, maybe that error means not a hoot to you.

Cheers,
Rich

 

[cnirenberg]

Mike,
I still have the sheet you wrote the formula on when I bought your lathe....... I use it all the time.

 

[cnirenberg]

Who wants some pi?

3.1415926535897932384626433832795028841971693993751
058209749445923078164062862089986280348253421170679 821480865132823066470938446095505822317253594081284
811174502841027019385211055596446229489549303819644
288109756659334461284756482337867831652712019091456
485669234603486104543266482133936072602491412737245
870066063155881748815209209628292540917153643678925
90360011330530548820466521384146951941511609...

Charlie,
My OCD or CDO (it just has to be in order...) can't handle it.

 

[NewLondon88]

Charlie,
My OCD or CDO (it just has to be in order...) can't handle it.

Sorry.. did you need the rest of it? :tongue:

 

[azamiryou]

Pi is wrong.

I'm not kidding. It's not that the value is wrong, it's that the concept is wrong. Pi is the ratio of a circle's circumference to its diameter, but the "circle constant" should be the ratio of the circumference to the radius, which happens to be equal to 2 Pi. Proponents call this new constant "tau".

And it's absolutely true. You know all those confusing formulas with pi in them? They all get much simpler and more intuitive if you use tau instead of pi.

If you are the least bit interested in math, please check out the 5-minute video at the bottom of this page. If you are more interested in math, read the manifesto (click on the big yin-yang at the top of the page).

Tau Day | No, really, pi is wrong: The Tau Manifesto by Michael Hartl

 

[GoatRider]

Who wants some pi?

3.1415926535897932384626433832795028841971693993751
058209749445923078164062862089986280348253421170679 821480865132823066470938446095505822317253594081284
811174502841027019385211055596446229489549303819644
288109756659334461284756482337867831652712019091456
485669234603486104543266482133936072602491412737245
870066063155881748815209209628292540917153643678925
90360011330530548820466521384146951941511609...

OR in music: Mathematical Pi (Full Song) - YouTube

 

[Rich L]

Pi is wrong.

I'm not kidding. It's not that the value is wrong, it's that the concept is wrong. Pi is the ratio of a circle's circumference to its diameter, but the "circle constant" should be the ratio of the circumference to the radius, which happens to be equal to 2 Pi. Proponents call this new constant "tau".

...

You had to expect a response on this, right?

The value of pi is known as irrational and transcendental (so is "tau") and the concept that defines it is correct. The video argues that the notation in using pi instead of tau is clumsy and ugly. Don't necessarily disagree but many beers are worth that discussion. The fundamental mathematics in both notational approaches is still unaltered.

Don't you know that tau = taper ?? :biggrin:

By the way, the video is 50 minutes, not 5!!!

Cheers,
Rich

 

[bluwolf]

Wow... This would be fascinating if I knew what you guys were talking about. I've heard all the names and terms, but I've never put them together in a coherent sentence.

I'm not making fun, as I said, I'm mathematically challenged. This is exactly the reason I'll stick to my original formula. I'll just know to stay away from building space shuttle parts:biggrin:

Mike

 

[azamiryou]

By the way, the video is 50 minutes, not 5!!!

please check out the 5-minute video at the bottom of this page.

Scroll down. The first video is 50 minutes... I haven't even watched that! But the last video is only 5 minutes and gives a good overview of the advantages of tau over pi.

 

[Rich L]

Wow... This would be fascinating if I knew what you guys were talking about. I've heard all the names and terms, but I've never put them together in a coherent sentence.

I'm not making fun, as I said, I'm mathematically challenged. This is exactly the reason I'll stick to my original formula. I'll just know to stay away from building space shuttle parts:biggrin:

Mike

Little did you know you would have opened up this can of pi filling! :biggrin:

Scroll down. The first video is 50 minutes... I haven't even watched that! But the last video is only 5 minutes and gives a good overview of the advantages of tau over pi.

Ah! OK. Elegantly illustrative baking lesson.

Cheers,
Rich