Tapering with SGA - math knowledge needed

P.Brewster said:
If you disagree with my math, please draw it in CAD (or work it out with pen/paper) and show me my error.

https://en.wikipedia.org/wiki/Cosine_error
Click to expand...

Okay, 2 ways to do the math. This depends on how the attachment works. In the 1/3 example, if you stack the shims under a hypotenuse at 3 units (like a sine plate), then the angle is 19.47. If you stack the shims on the adjacent at 3 units then the angle is 18.43. From the picture it looks like the bolt is along the adjacent (and fixed) so Devin's math would be 100% accurate. The question is: is it inverse sine (hypotenuse) or inverse tangent (adjacent)?

Also, that middle bolt works great for this. Use the same shims for the reverse side and just put them at the middle bolt. Voila! 2x height, magic.

taperedtang2.jpg
 
LilHoss said:
Okay, 2 ways to do the math. This depends on how the attachment works. In the 1/3 example, if you stack the shims under a hypotenuse at 3 units (like a sine plate), then the angle is 19.47. If you stack the shims on the adjacent at 3 units then the angle is 18.43. From the picture it looks like the bolt is along the adjacent (and fixed) so Devin's math would be 100% accurate. The question is: is it inverse sine (hypotenuse) or inverse tangent (adjacent)?

Also, that middle bolt works great for this. Use the same shims for the reverse side and just put them at the middle bolt. Voila! 2x height, magic.

taperedtang2.jpg
Click to expand...
Thanks for the clarification. And the middle bolt is at 165mm from the pivot point for this exact purpose.
 
Larrin said:
It’s been too long since we had a good math argument.
Click to expand...
I like it. I’m learning a lot. In my line of work (IT sourcing) trigonometry doesn’t get used much, if any.

So it’s back to remembering what you learned in school 25 years ago... and I don’t even remember what I had for lunch yesterday.
 
You have to compare apples to oranges.

OK, guys. Lets just agree that in the very low angles we are using on knives, the shim calculation will be more than sufficient. We aren't manufacturing rocket parts here.

Extreme examples are not scientific. They are false arguments. We are dealing with minute angles and minor tapers. Length to height of any triangle calculation used in tapering a tang or calculation an edge is a factor of .050 ( or less) smaller than 1" in 3". We are talking about tapers of .050" in 3" or smaller.

This is just like the argument on blade bevel calculations on whether to use the blade width as the hypotenuse when figuring bevel angle. On a 2" wide blade .150" thick at the spine. the height ( bar width) and the hypotenuse ( actual side) are within an error factor so small that they can be considered the same. We don't generally use equipment capable of accuracy in 1/100th of a degree, and I have never hear of any surface grinder that can taper tangs having accuracy in the ten thousandths of an inch range.

For those who want to do the math, the difference in the edge angle if you consider the height the same as the sides on the 2" by .150" blade scenario is :
If using 2" as the height as the base of a right triangle, the edge angle is 4.288°
If using 2" as two sides of an equilateral triangle, the edge angle is 4.298°

As you see,either calculation is more than accurate enough - this is a difference of less than .01°, and the actual thickness error for each side taper is .0005"

On a 5" tang being tapered to zero, it would be many times smaller error. Around .003° and .0001" error

Tapered tangs are
 
Last edited:
LilHoss said:
Okay, 2 ways to do the math. This depends on how the attachment works.
Click to expand...

On the attachment, the 330 line pivots, which makes it the hypotenuse - but yes, if 330 is the long side, then proportional math works.

Either way, the matter can be summarized as follows:

A right triangle is composed of 3 sides - the hypotenuse, the long side, and the short side. Proportional math is correct only when we are comparing two sides of two triangles that are the same type - e.g. hypotenuse and short side of triangle A compared to hypotenuse and short side of triangle B.

However, on triangles with very small angles, proportional math will provide very close approximations, despite being technically incorrect. The reason it provides close approximations on small angles is because these triangles have a hypotenuse and a long side that are almost exactly the same length.

Stacy E. Apelt - Bladesmith said:
Extreme examples are not scientific. They are false arguments.
Click to expand...

We've got two approaches here - one that is technically correct and will produce a fully accurate result regardless of the variables, and another that is technically incorrect and will produce results that range from wildly inaccurate to nearly accurate, depending on the variables. It seems a bit baffling - and contrary to the spirit of higher understanding - to dismiss the technically correct method as 'not scientific' and a 'false argument'.
 
Last edited:
This is actually not a triangle but a slope and a parallel line. A tapered tang or a distal taper is simply rise over run. This is simple stuff. No need for a computer or trig. No need to argue or draw a graphic, simply measure how much you want to take off over how long the distance on the tang or blade, divide the rise by the run and multiply that by 330 in this case to get the thickness of the shim.

Hoss
 
Holy crap guys.

Maybe take the math nerd argument to a different thread and leave the simpleton version to us.... simpletons.

I’m not a machinist, have no desire to be a machinist and hate math.

I got all excited when there were very simple, intuitive and cogent explanations, then there was everything else.

Thanks to those that went the simple route!
 
that is too much for me to understand, i only made it to 10th grade :rolleyes: i used the TE method (trial and error). i scribed center lines and kept adding more feeler gauges until it came out right, and wrote down the feeler gauge thicknesses as a starting point for next time. you guys are pretty smart !
 
I'm not trying to argue here, just point out how the math works. Slope math is not fuzzy logic. A rise / run of 1/3 when applied to a knife is an extreme example but is very realistic in other areas (like a 4/12 pitch roof) and must still follow the same mathematical principles.

P.Brewster, even you came up with the two angles. In your reply you said that 1/3 is 18.43 degrees and you followed that by saying 110/330 is 19.47 degrees. It's the same slope ratio and therefore the same angle. Try it in CAD and you'll see that it's (110/330) 18.43 degrees.

I agree, keep it simple. No need to use trig. Just find the slope ratio and multiple that by 330.
 
Yeah, I see what you guys are saying about not complicating things. :thumbsup:

The reason I asked was because I didn't want to continue the trial and error method I've been using so far. My primary use is actually for distal tapering kitchen knives.

Since the taper can be up to 300mm on an already thin stock (down to even thinner) it's good to now have an easy way of setting the angle, especially considering that the margin of error is less than slim on these blades. ;)
 
Don Hanson III said:
I never learned to taper tangs in school, but I do it in the shop on a regular bases, with no math.
Click to expand...

I like it! The less math the better!
I didn’t join the Marines to become a mathematician

kuraki kuraki at least you’re aware it was pedantic. :rolleyes:
Join a math club man, you’re obviously passionate about it. I can’t even, my eyes glaze over and I start to drool o_O
 
You must log in or register to reply here.
Back
Top