Math for machinist tapers...

[NeoTech]

Well i got my filthy hand on this tool holder that states BT40.. after some googling i found that the taper is of the type "7/24".. well that is some ancient way of telling degrees or angle i guess.. 7 inch per foot.. right.. How do i convert that to well.. modern measurements.. like degree of angle....

[AdeV]

According to my digital angle gauge, it's 7.8 degrees.

I did try a bunch of maths, but in this case, the universe disagreed  (this might be because my desk is not totally level, I am not sure, trig suggested an answer of 8.297 degrees = tan^-1((7/2) / 24), assuming the ratio really is 7 in 24.

[NeoTech]

After way much googling the closes i found was this Angle = atan(7/24). but in the case with the BT40 holder i cant seem to figure out if its included angle or not.. if its included that would be *2..

[AdeV]

After way much googling the closes i found was this Angle = atan(7/24). but in the case with the BT40 holder i cant seem to figure out if its included angle or not.. if its included that would be *2..

Unfortunately, Machinery's Handbook completely glosses over any British Taper standards, mentioning only the BS Standard number (1660) from 1972.

It does go on to mention that the National Machine Tool Builders Assoc. chose a 3 1/2" per foot taper to be standard on milling machines as far back as 1927... i.e. 7/24. so I think we can be reasonably certain that's the BT40 spindle taper.

To get the taper angle, you have to halve the taper-per-foot, so arctan(7/48) should give you the correct result. Do not ask me how I know this...

The above is wrong, see Marv's post below, which is correct!

[mklotz]

If the taper is 7" of diameter in 24" then the INCLUDED taper angle is...

arctan (7/24) = 16.2602... deg

and the taper HALF ANGLE is half this value = 8.130... deg.

Note that this is NOT the same as:

arctan (7/48) = 8.297 deg.

Mathematically,

arctan (x/2) is never equal to 0.5*arctan (x)

[andyf]

Being a bit of a  duffer when it comes to trig, I find this calculator quite useful: http://www.cleavebooks.co.uk/scol/calrtri.htm (if that offers you options, go for Right Angled Triangles).

Put sides a and b in as 3.5 and 24 (it doesn't matter which is which) and it gives you two angles, the one you want being 8.3 degrees. Then if you have an amazing protractor and even more amazing eyesight, crank it up to (say) 7 significant figures and that becomes 8.297145 degrees. The included angle will be double that, of course.

The same website offers other useful geometry in its index:
 http://www.cleavebooks.co.uk/scol/index.htm

Andy

[AdeV]

If the taper is 7" of diameter in 24" then the INCLUDED taper angle is...

arctan (7/24) = 16.2602... deg

and the taper HALF ANGLE is half this value = 8.130... deg.

Note that this is NOT the same as:

arctan (7/48) = 8.297 deg.

Mathematically,

arctan (x/2) is never equal to 0.5*arctan (x)

Oops, you're right of course, my bad.

I think I now understand why I have been utterly unable to cut the taper for my lathe headstock... (see posts I made about 3 years ago).

[BillTodd]

Do not convert ! (it's much easier)

Tapers that are specified in inches per foot are best set-up using a displacement and length. Converting  to angle will always produce an error.

If you are trying to cut the taper with a top-slide, either set the angle with a sine-bar* or dial in on an existing taper.

Bill

BTW - BT40, Int40 , Int30, BT30, BAT50 etc. etc. all use the same 3 1/2" per foot (on diameter), but the drive end will vary (some are designed for auto changers)

* if you are using a 5" sinebar and want a 3.5"/' taper, the spacer block should be  3.5/2 * 5/12 =  0.7292 (OK this produces a tiny error ;))

[Pete.]

If I hadn't read any of this I would have assumed that 7/24 was a ratio. To machine it I would use two dial gauges, one to measure compound slide travel and one against and perpendicular to the work. I'd keep adjusting the angle until cranking the compound 24mm moved the dial against the work 3.5mm

[RussellT]

This may confuse you - but that's OK because at the moment it's just me that's confused.

If the diameter tapers by 7 units in 24, then the radius must taper by 7 units in 48.  So if you're looking to set an angle on a top slide to cut a taper then surely the angle you need is arctan(7/48).  I think it's difficult to use trig on the 7/24 as there is a shortage of right angles and for the reasons given by Marv it's not the same as half arctan(7/24)

Russell

[AdeV]

This may confuse you - but that's OK because at the moment it's just me that's confused.

If the diameter tapers by 7 units in 24, then the radius must taper by 7 units in 48.  So if you're looking to set an angle on a top slide to cut a taper then surely the angle you need is arctan(7/48).  I think it's difficult to use trig on the 7/24 as there is a shortage of right angles and for the reasons given by Marv it's not the same as half arctan(7/24)

Russell - that's what I thought.... but the reason we're wrong is actually relatively easy to show using a diagram; see attached.

The first figure shows the full 7/24 taper. The included angle between b1 and c1 (from an imaginary point at the rear of the taper) is arctan(7/24), or 16.whatever degrees.

If we halve the taper angle, to 7/48; we get the 2nd figure. Now the included angle between b2 and c2 is 8.3 degrees. However, if we rotate the taper so it sits flat on baseline a; you can see it no longer forms a right-angle triangle; and thus, our trig rules go out of the window..

Indeed, it should be clear that if you halve the b1/c1 included angle, you end up with the actual angle of taper; that is the angle between a and b1, or a and c1.

HTH!

[trevoratxtal]

Fascinating topic full of good info many thanks.
One small point
type "7/24".. well that is some ancient way of telling degrees or angle i guess.. 7 inch per foot.. right
this equals 7 inch to 2 feet, twelve  inches to the foot, some times quoted as 3 1/2 inch to the foot but I don,t think it works quite right that way as AdeV says.
Whatever a very interesting topic. 
Trev

[NeoTech]

*sits stares into a wall, smoke coming out of ears...* deargod... they arent making it easy are they, those machinist/invetor standardisation guys....

this is why im asking..

Dont think for 1sec, just coz you know cad, your gonna do it right. 

[BillTodd]

You still don't need to know the angle in degrees ;) 

International taper #40
Gage line (diameter of large end) = 1.7500"
Pilot diameter (parallel at small end ) = 1.000" +/- 0.005
Length  (gage to end of pilot)  = 3 7/8" (3.875")
Taper 3.5" per foot

[edit] Helps if one keeps to fractions

[andyf]

As Bill says, you don't need to know the angle. Not having a sine bar, I would arrange a plunger indicator lying flat and with the plunger at exactly 90° to the lathe bed, and the plunger touching the side of my top slide. I would then set the angle of the top slide so that, when the carriage is moved along by 24 units, the dial indicated a variation of 3.5 units. The greater the distance over which the measurement is taken, the better. My top slide is 120mm long, so I would  look for an indication of 14mm over a carriage movement of96mm. 14:96 is the same ratio as 3.5:24.

It is easier to measure distances precisely than to measure angles accurately, though the dial indicator stil has to be set horizontal and at 90°  to the bed.

Alternatively, you can set the toolholde up to run true on your lathe, and adjust the top slide angle until a DTI mounted on the top slide with its finger at centre height shows no movement as it is run along the taper. The top slide should then be at the right angle to bore the taper in your spindle, using a tool at centre height.

Andy

[BillTodd]

Hope this is clearer.

A spindle nose diameter of  (3.4993 - 3.4986) is also specified

[RussellT]

Russell - that's what I thought.... but the reason we're wrong is actually relatively easy to show using a diagram; see attached.

Ade

I think you were right the first time - I've attached a diagram.

The triangle ABC is the taper we are trying to make.  It starts 7 units wide and over a length of 24 units tapers to 0.  (I've drawn it to a point to make it easier to visualise).  The triangle ABC has no right angles so we cant simply apply arctan.  We need some extra lines.  To get the included angle we could draw a centre line from the midpoint of AC to B to give us two right angled triangles.  They would be the same shape as the triangle BCD and the angle would be arctan(BD/CD).  So the included angle is 2arctan(BD/CD) and the angle to set your topslide to is arctan(BD/CD) or in this arctan (3.5/24).

I hope I've got that right.

Russell

[NeoTech]

Setting the angle with a dial seem to be the "easy" solution here..  I have one of those dual DTI thingys to make sure the spindle head is square to the table.. would that be of any use setting the angle of the topslide? If i square that in a toolholder to a piece of ground roundstock, it should be able to set the angle with it .. wouldnt it? =)

[DMIOM]

Setting the angle with a dial seem to be the "easy" solution here..  I have one of those dual DTI thingys to make sure the spindle head is square to the table.. would that be of any use setting the angle of the topslide? If i square that in a toolholder to a piece of ground roundstock, it should be able to set the angle with it .. wouldnt it? =)

You woud need to know =exactly= the spacing between the two indicators, and you would have to align the beam parallel to a test bar (not just in&out but also in height at both ends so that probes touch on the same point of the circumference) - seems like adding extra complications and extra opportunities for errors to creep in.

Dave

[RussellT]

To address the real issue, rather than just the maths, the most important thing you need is some way of testing the taper without removing it from the lathe - for example a suitable socket.  You set the angle to about 8 degrees and cut the taper, then try the socket and see which end is loose and adjust the angle to suit.  Repeat until it's right. It's fiddly rather than difficult.

Russell

[NeoTech]

RusselT, yeah i have BT40 tool holder that i will use as a "key" to check the fit with. What im gonne cut is the inside cone for a spindle axle. I even considering lapping the final cone if i can get in there. And or have it plated and ground.. but i need to get the angle right before i try to do that. =)

This was very educational anyway.. These ratios for angles is not anything one learn in school anymore.. not me at least.

[andyf]

This was very educational anyway.. These ratios for angles is not anything one learn in school anymore.. not me at least.

If Finland has the same road signs as most of Europe, Neotech, they use a similar system for gradients on hills. For example, "20%" means a gradient where you go 20 units up (or down) for each 100 units you travel horizontally, as viewed on a 2D map.

20:100 could be expressed as 1:5, and life was simpler here in the UK when the signs said things like "Steep Hill 1:5" which was pretty steep. When they said "1:3", things got very steep. Going up a long hill your engine coolant might boil. Worse still, your brake fluid might boil coming down, so you travelled rather fast near the bottom of the hill. Helpfully, the sign for a steep downward hill also said "Engage low gear now".

Andy 

[BillTodd]

These ratios for angles is not anything one learn in school anymore.. not me at least.

You must have use Radians in school surely? 

I suppose that as schools try to teach kids more and more 'stuff', the greater becomes the danger of missing out on the fundamentals . 


I was amused by your description of defining an angle by 'Degrees' as modern 

 Chopping a circle up into an arbitrary* number of degrees is ancient and must be almost as old as describing an incline in terms of height and distance.


Bill

*360 degrees with 60 minutes and seconds was i'm sure chosen by our ancestors simply because 360 factors easily in a  base 10 number system  - a computer would be happier with a 256, 512 or any  n^2 degree circle

[NeoTech]

Well i was taught degress, not.. well angle ratios, or that 16min, 20s, thingy that people uses as well (that is another measurement i dont understand) ;D

[BillTodd]

(that is another measurement i dont understand)

You may be unfamiliar with it, but you can understand it! (because it is so simple)

1 degree = 60 minutes, .0.5° = 30 minutes, 0.25° = 15 minutes
1 minute = 60 seconds  etc etc.

It's primary use is for quick, in-the-head*, maths for navigation. Powers of 60 factor (i.e. can be divided exactly) by 2,3,4,5,6 etc. making division by those factors easy and accurate.

eg. 25° 34' 18"  divided by three =   8° 31' 29" 

The same units are used on charts and navigating instruments so there's no need to convert.

In general, converting units should be avoided (or left to a computer ;)) as can easily lead to rounding or typographical errors, so if the standard says "inches per foot" stick with it 

Bill

*fractions (e.g. of an inch) have some of the same 'magic'  which is why they been around so long - It's only the wide spread use of calculating machines that makes metric/decimal systems attractive.

[mklotz]

(that is another measurement i dont understand)

It's primary use is for quick, in-the-head*, maths for navigation. Powers of 60 factor (i.e. can be divided exactly) by 2,3,4,5,6 etc. making division by those factors easy and accurate.

eg. 25° 34' 18"  divided by three =   8° 31' 29" 

Easy, yes, except

(25° 34' 18")/3 = 8° 31' 26"

[BillTodd]

    

(You know why ? -  I was going to divide it by two  then thought "no, I 'll be clever and divide by three" ...)

Where's the kicking-oneself-up-the-arse emoticon ?

Bill (:LOL:)

[mklotz]

Murphy's extended law says that, if one is to make a computational error, it will always be in a widely viewed public venue.  (It's a corollary of the buttered toast lemma.)

Knowing this, I check every computation in a forum response three ways to Sunday yet I still manage to bugger up a few.  It's the math gods' way of keeping us humble.

[dsquire]

   

(You know why ? -  I was going to divide it by two  then thought "no, I 'll be clever and divide by three" ...)

Where's the kicking-oneself-up-the-arse emoticon ?

Bill (:LOL:)

Bill

No need to feel to bad, We all make them once in a while.

I was sitting back at the original post trying to figure out in my head how to do it and couldn't get it to work out. I figured that I must have forgot how to divide degree's. Now I know it was a trick question just to see if we were all paying attention. 

Cheers 

Don

[PekkaNF]

Old thread but same type problem.....

Good old 4.500" TPF.

I'm trying to get my head around what it actually means and how do I make it meaningfull:

1) Aproximate angle in degrees to make rough check (included angle)

2) Exact numbers to make the taper in the lathe (top slide method = half angle)

So, this is the bugger: 4.500" TPF big end is 0.87"
http://www.wmsopko.com/images/D-00108-R.jpg

Great. I assume that 4.500" TPF stands here for included angle, which ratio is 4,5" for 12,0" and half angle would be 2,25" displacement at 12" distance? Is this correct and if not what actually then this 4.500" TPF stand for and what it is when setting the machine to do this taper?

I think I know but after reading this thread and being raised on ratios of half angle I am having this WTxxxx moment.

Pekka

[seadog]

Hi Pekka, Abom79 shows an accurate way to set up for a taper. He has a taper attachment but still uses a method to check accuracy which is valid for setting a compound too.

3Gcng

Graham

[seadog]

It starts at aound 18 minutes in.

[AdeV]

Great. I assume that 4.500" TPF stands here for included angle, which ratio is 4,5" for 12,0" and half angle would be 2,25" displacement at 12" distance? Is this correct and if not what actually then this 4.500" TPF stand for and what it is when setting the machine to do this taper?

Sounds right to me. If it makes it any easier, 4.5" TPF is the same as 45mm per 120mm...

[PekkaNF]

Great. I assume that 4.500" TPF stands here for included angle, which ratio is 4,5" for 12,0" and half angle would be 2,25" displacement at 12" distance? Is this correct and if not what actually then this 4.500" TPF stand for and what it is when setting the machine to do this taper?

Sounds right to me. If it makes it any easier, 4.5" TPF is the same as 45mm per 120mm...

Great, I didn't want to change the language to make matters even more confusing

Basically my question is that is this TPF just a straight ratio on orthogonal coordinates, no funky stuff here? The funky part is that it is included angle i.e divide by two to get something usefull?


Graham, Abom79 vids are allways worth checking. Thank you. Another person that makes a lot of stuff very clear is that al-bundy looking guy that sounds like my physics professor...what was his name Piezak or something. Spot on, without too many practical omissions.

Pekka

[seadog]

Joe Pie, as he jalls himself bejause noone can pronounce Pieczynski   

I thought a video might be worth a thousand words.

[Pete.]

I only watch some of the youtube lot, I get bored with some where they waffle on with a bit of practical in between but I like Joe P, he does short and to the point vids.