How do I determine a taper angle, given length & width at both ends?

cazclocker

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I'm fabricating a replacement for the part shown in my attached sketch. The severely-worn part is my locking cam lever, used to secure the tailstock of my watchmaker's lathe to the lathe bed ways. I've got the main body made (pretty simple, really) done - now it's time to turn a new handle.

So...my question is, what is the formula to figure the half-angle for the tapered part (obviously, minus the ball-end)? The length is 0.758" (but I'll call it 0.760" for simplicity's sake), and the diameters are 0.146" at the narrow end, and 0.218" at the larger end.

Instead of you guys just feeding me the answer (although that would be nice, too, just so I can get going on the project!), I would like to know what the correct formula is, so I can figure similar equations in the future.

Thanks!
...Doug

CamLever_003.jpg
 
Last edited:
You will need to use the Arctan function on a calculator or look it up in tables. Sometimes the button is called ATAN or TAN[SUP]-1

[/SUP]Anyhow, the formula is Arctan (change in height / distance).

In your case this will be Arctan [ (0.218 - 0.146) / .758 ]. This is 5.43 degrees. Since you are measuring diameters, this is the full angle so we still need to divide by 2 to get the half angle you requested thus, it's 2.72 degrees.


Ray

EDIT: On some calculators Arctan can also be called Invtan or TanInv. The number sequence on a typical calculator is like this:

0.218
-
0.146
=
(the answer will show up as .072)
/
.758
=
(the answer will show up as 0.095)
ArcTan
(the answer will show up as 5.4261....)


PS: Please don't be offended by the explanation of the calculator. A long time ago, I had a post which needed the calculator and I received many PMs on how to enter the numbers so, every now and then, I cover the material again.
 
You will need to use the Arctan function on a calculator or look it up in tables. Sometimes the button is called ATAN or TAN[SUP]-1

[/SUP]Anyhow, the formula is Arctan (change in height / distance).

In your case this will be Arctan [ (0.218 - 0.146) / .758 ]. This is 5.43 degrees. Since you are measuring diameters, this is the full angle so we still need to divide by 2 to get the half angle you requested thus, it's 2.72 degrees.


Ray

Thank you, Ray. I am not strong in trig at all, but I have a pretty good scientific calculator with a TAN(superscript -1) button, which I guess is the function you're referring to. I still don't know how to apply the formula you supplied to the buttons on my calculator, but fortunately I still have the manual! So hopefully a little studying up will be in order...
...Doug
 
I edited the thread to provide some calculator instructions. Have a look again. Or, you can tell me the type of calculator and I can walk you thru.


Ray


Thank you, Ray. I am not strong in trig at all, but I have a pretty good scientific calculator with a TAN(superscript -1) button, which I guess is the function you're referring to. I still don't know how to apply the formula you supplied to the buttons on my calculator, but fortunately I still have the manual! So hopefully a little studying up will be in order...
...Doug
 
I edited the thread to provide some calculator instructions. Have a look again. Or, you can tell me the type of calculator and I can walk you thru.


Ray

Hi Ray! I followed your revised instruction and it works great! Thanks! Good grief, I am not offended at all - quite the contrary, what I needed was for someone to literally grasp my index finger and make me hit the right buttons. Short of that, I can follow instructions very well, and yours are very clear...step-by-step! My calculator is a pretty old solar-powered Casio fx-85, and the arctangent button is marked with "tan(superscript -1)" .... by the way, how do you get the actual little "-1" miniaturized and up in the air like you do? Anyway, my arctangent button doubles as a tangent button, so I have to hit the INV button to make the tangent button do the arctangent function.

The important thing is, I now know how to arrive at the same result that you supplied, so I must be doing it right. So now, if I want a slightly different angle all I have to do is duplicate all the moves but substitute different numbers.

Thank you Ray!!
...Doug
 
Glad the slope/taper problem had a happy ending and properous looking future! And by the way, that same formula is used for any slope problem such as for calculating the slope of a staircase, hill, mountain, wheelchair ramp... Just gotta watch out if you need the half-angle. For a wheelchair slope you are measuring the heights from a flat centerline so, just subtract the heights and divide by distance and hit arctan. No need to divide the answer by 2 in that case. Easy Spheasy...


When I press the reply button, I get 3 bars of editing options just above the editing window. On the 3rd row, there is X[SUB]2[/SUB] and X[SUP]2[/SUP] for subscript and superscript.

So... if you want to write Tan-1 just write it then, highlight just the "-1" and press the sub or super-script button. Now watch-out for something. You need to leave a space character after highlighting the "-1". If you don't everthing you type afterwared will show-up as scripted.

Ray

PS: I get the biggest kick out of your icon picture. There's another one like that with a kid sticking a screwdriver in a socket.



Hi Ray! I followed your revised instruction and it works great! Thanks! Good grief, I am not offended at all - quite the contrary, what I needed was for someone to literally grasp my index finger and make me hit the right buttons. Short of that, I can follow instructions very well, and yours are very clear...step-by-step! My calculator is a pretty old solar-powered Casio fx-85, and the arctangent button is marked with "tan(superscript -1)" .... by the way, how do you get the actual little "-1" miniaturized and up in the air like you do? Anyway, my arctangent button doubles as a tangent button, so I have to hit the INV button to make the tangent button do the arctangent function.

The important thing is, I now know how to arrive at the same result that you supplied, so I must be doing it right. So now, if I want a slightly different angle all I have to do is duplicate all the moves but substitute different numbers.

Thank you Ray!!
...Doug
 
... And we all make mistakes -and I failed to mention a longer (but more precise) method of calculating angles in the case of the original post.

Sit tight, I need to draw some pictures to show things properly. Don't despair, the quick method is generally correct in the 1st and 2nd decimal place and perfectly suited for things of a "ornamental" nature. BTW: An angle represented out to one decimal place is generally hard to machine and two decimal places is ten times harder. Three decimal places? Probably ain't going to happen on homeshop equipment...

I'd like to thank Terry for reminding me offline that I should show both techniques for the sake of completeness...


Ray
 
It's important to recognize the difference between these two examples of taper. It's not a good idea to think that all tapers look enough alike to think that one formula fits all.

Gene

TAPER.jpg
 
Correct. For the figure on the left you'd use ArcSin instead of ArcTan.

However, this is not where the additional complexity comes into play. I'm making some drawings to illustrate...


Ray


It's important to recognize the difference between these two examples of taper. It's not a good idea to think that all tapers look enough alike to think that one formula fits all.

Gene
 
OK, here's "the rest of the story"...

Please take a look at this example. The geometries are chosen to really show the problem in the worst case.(You can download the file to see a better version).


If you do the quick & dirty method, you simply calculate Arctan (8-0.5)/15 as 26.565. The half angle is 13.2825.
The correct (most accurate) method is is to subtract the differences of the halves which is Arctan (4 - 0.25)/15 = 14.0362. This time, we do not divide by 2. There is a difference of 0.754 degrees in the answers.

If you calculate the values from the original post using the precise method, the final answer comes to 2.7191. The shortcut method showed an answer of 2.72.

This is kinda my bad because when I glanced at the OPs original measurements, I instinctively knew the results would be about the same...

Ray

EDIT: I've included a larger version of the same file. Hope it's bigger this time.

View attachment document.jpg

View attachment document.jpg
 
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