First results

[Herman Trivilino]

Thank you very much for the drawing. It depicts the situation exactly .

When you are using a tool to cut into a piece of wood, it's the edge angle that determines how well the tool performs. This is the angle between the two planes of steel that meet to form the edge. The angle that the chord makes with the back of the tool is far less relevant; especially for thinner tools, smaller edge angles, and larger radius grindstones. It is therefore the edge angle, the angle measured by the Tormek Angle Master, that's of primary importance. The difference between this edge angle and the angle the chord makes, what you are calling ß and Lee is calling α, is not a measure of any "error".

[wootz]

Rolf's post echoed in my memories more info on edge angle reduction from the Experiments on Knife Sharpening by Verhoeven.
"The predictions of this equation are shown in Fig. A9 for the 10 inch Tormek wheels on blades having thicknesses of T = 1/8, 1/4 and 1/2 inches.
It is seen that the values of ∆β become fairly significant for common grinding conditions used on chisels, with β angles of 20 to 40 degrees and blade thicknesses of 1/8."
[and an example]
"The iron shown above has a thickness of 3.7 mm, the stone diameter measures 248 mm, the angle setting is 25°. Angle deviation comes to about 2° - resulting cutting angle = 23°."

[Herman Trivilino]

Rolf's formula is surprisingly simple, and so I am wondering, if it is exact expression or an approximate one.

I have verified that it's at least approximately correct. I'm still working on trying to prove that it's exact. No luck so far, and Lee is no help.

[Herman Trivilino]

Herman decribes the essential things. The chords angle is what is set with the WM 200 but the tangent line sets the angle of the of the bevel at the very edge.

Rolf, the WM-200 Angle Master sets the angle of the bevel at the very edge. This is what Tormek calls the bevel angle ß. Note that in the case of a chisel or plane iron, as discussed here, it's equal to the edge angle. For a traditional knife ground on each side with a bevel angle ß, the edge angle is 2ß.

[Herman Trivilino]

"The iron shown above has a thickness of 3.7 mm, the stone diameter measures 248 mm, the angle setting is 25°. Angle deviation comes to about 2° - resulting cutting angle = 23°."

Ok, so let's check these values in Lee's formula. We have D=248 mm, T=3.7 mm, and Θ=23°.

Thus T/(D sin Θ)=(3.7)/[(248)(sin 23°)]=0.03334.

Therefore sin α = 0.03334 and α=1.9°. So that's a match!

Note that we still cannot tell from this whether Lee's formula is exact. The round off error could have already been present.

[Elden]

Vadim (Wootz),

   Please purchase an inexpensive machinist square (or an expensive one if you rather) and measure the chord angle of a chisel ground on a Tormek that was set to 25° with the Angle Master. One such as shown in this link is rather inexpensive here in the USA.

http://www.homedepot.com/p/General-Tools-Steel-Protractor-17/100349259?cm_mmc=Shopping%7cTHD%7cG%7c0%7cG-BASE-PLA-D25T-HandTools%7c&gclid=CjwKEAjwg6W6BRDn6v__7vzN9QkSJAC9l9C3l5R7yevUDSjADj53hSy9F-EX8FvFEs3evBjxxoRPmBoCddnw_wcB&gclsrc=aw.ds

I believe you will find out as I did that the chord angle will be greater than 25°.

The following link records the end of my measuring experiment at that time.

http://forum.tormek.com/index.php?topic=2413.msg12078#msg12078

[Jan]

Wootz,

Thank you very much for posting the graph and sketch concerning the reduction of the edge angle due to the grindstone curvature. They perfectly illustrate the difference in edge angles grinded flat and hollow. 

Please let us know if a formula for Δβ is available in the quoted source.

Jan

[Jan]

Already quoted Stig's statement concerning the Angle Master reads:
"The angle master WM-200 is very good and are measuring the tip of the edge. It is compensated for the hollow grind. If you use a regular "angle setter" or a protractor you will find that the degree will indicate that it's not the same as the anglemaster. A protractor are not compensated for a hollow grind but for a flat surface."

In my understanding it would be more accurate to say that WM-200 is compensated for the reduction of the edge angle due to the grindstone curvature. (See the sketch posted by Wootz)

Rolf (Hatchcanyon) is correct, when he mentions "The WM 200 only sets the angle desired and does not know anything about the thickness of the iron it is rinding on."

Rolf is also correct when he says "...the tangent line sets the angle of the bevel at the very edge."

The WM-200 works in the following way: it sets the selected bevel angle for the very edge of a tool. Via the Diameter compensator the WM-200 compensates for grindstone curvature.

The WM-200 has no info about the tool thickness and so cannot do predictions beyond the very edge.

I think that Rolf started very fruitful discussion which helps to deepen our understanding of this key issue for each Tormek sharpener. The question, how was the Lee's formula derived, remains still open.

Jan

[Hatchcanyon]

Additional facts:

Measuring the bevel angle is not that easy with relatively thin irons. As a solution working for me I used a 1/2 inch thick mortising chisel.

Draw a straight line longer than the Iron on a sheet of paper, position the back side of the tool along the line.


Aligning the backside

Use a triangle:


Triangle across the bevel


Bevel angle is 28°

With this result I set the WM 200 to exactly the same value. The iron itself is mounted into the SVD 186 jig. This is only to hold the iron statically and not intended to use it for grinding (might be possible?). Positioning the iron is done with the universal support fine adjustment.


Aligning WM 200 and the iron

And now comes the insight!
Only the edge is riding on the stone. That means the angle setting of the WM 200 is always correct. Hollow grinding doesn't lead to a decreasing angle, it produces an increasing angle!

Guys, many thanks for the comprehensive discussion. I learned a lot!

Rolf

[Herman Trivilino]

Note that ß-Δß is bevel angle set by the Tormek WM-200 Angle Master. Note that the ß shown in that figure is not the the same as the ß shown in the Tormek literature.

[Jan]

Guys, many thanks for the comprehensive discussion. I learned a lot!

Rolf

Rolf (Hatchcanyon), you are welcome! 

Jan

[Herman Trivilino]

Rolf's formula is surprisingly simple, and so I am wondering, if it is exact expression or an approximate one.

The formula is exact, Jan.

Just keep in mind that the bevel angle ß as defined by Tormek and measured on the WM-200 Angle Master appears nowhere in that formula.

sin α = T/(D sin Θ).

Tormek's bevel angle ß = Θ - α.

[wootz]

Wootz,

Thank you very much for posting the graph and sketch concerning the reduction of the edge angle due to the grindstone curvature. They perfectly illustrate the difference in edge angles grinded flat and hollow. 

Please let us know if a formula for Δβ is available in the quoted source.

Jan

Jan, Experiments on Knife Sharpening by Verhoeven can be downloaded as PDF from https://www.wickededgeusa.com/wp-content/uploads/2012/10/knifeshexps.pdf
See Chapter 5 on Tormek.

[Jan]

Rolf's formula is surprisingly simple, and so I am wondering, if it is exact expression or an approximate one.

The formula is exact, Jan.

Just keep in mind that the bevel angle ß as defined by Tormek and measured on the WM-200 Angle Master appears nowhere in that formula.

sin α = T/(D sin Θ).

Tormek's bevel angle ß = Θ - α.

Thanks for your message, Herman and congratulations to your successful reverse engineering of the Lee's formula. 

It is nice to know that the formula is valid and exact. I like simple and elegant formulas. 

Jan

[Jan]

Thank you very much Wootz, for posting the link to the valuable study of professor Verhoeven from Iowa State University. I look forward  to go through it carefully. 

From the appendix 2, page 52 it is now clear how the parameters in the formula were defined and it is also clear that the formula is exact, as confirmed by Herman.



Having this sketch and explanatory text it is relatively easy to derive the formula. The back engineering task performed and reported by Herman was much more difficult! 

Jan