naming and math help needed

[Jan]

P.S.: Your formula will be in compliance with Dutchman tables only when D'=D. After this correction your formula can be converted to the initial Dutchman's equation F0. This formula is approximate.  The approximation concerns the way how we measure the distance D between USB and the grindstone.

Hopefully exact set of Kenjig parameters is following: P=139 mm, D=78,67 mm, R=125 mm and  β=15^o  .

Do you mean the Dutchman tables give approximate figures? Because I'm not aware of any approximations made in the derivation of the formula I posted.

Good question, Herman, I have to clarify my statement concerning approximation.

Dutchman formula and your formula for D'=D provides the same numerical results. Your formula can easily be converted to Dutchman's formula F0.

The approximation which is used in both is conceptual one. Dutchman and you assume, that D is the distance between the USB and the grindstone surface. This is only approximately true. The real distance D is measured along a line connecting the centre of the universal support rod and the centre of the grind stone. The error introduced by this approximation is less than 2 mm.



Jan

[Ken S]

I go back to the source document of this project. Dutchman's Grinding Angle Adjustment booklet is posted on the forum. Everyone interested in angle adjustment should print it or download it. I have both printed copies and have it in ibooks on my ipad.

Ton's(Dutchman) booklet contains more than tables and formulae. Ton has stated the purpose of this whole project in the introduction. He wanted a simple system which could be realized by measuring and adjusting the position of the universal support and the length of the adjustable jig. His purpose has been my guiding North Star.

I think Ton may have designed his tables to work with a combination square (with a metric blade). I am certain that as he sharpened knives this way, he zeroed in on just a few well chosen combinations.

All I added was a belief that a combination of positioning the universal support and adjusting the jig(s) which would allow single setup which would work for most kitchen knives. The simple jig would also allow rapid and accurate to those settings.

I don't want forum readers to lose sight of Ton's vision, his simple system. I am glad we have several helpful members to make sure this system is accurate as well as simple.

Ken

[Herman Trivilino]

The approximation which is used in both is conceptual one. Dutchman and you assume, that D is the distance between the USB and the grindstone surface.

I didn't assume that. I was told that it was the definition of D.

[Herman Trivilino]

Here is the figure I used to derive my formula:


The large circle is the grindstone of radius R, the small circle is the US rod of radius r.

I simply applied the law of cosines to the triangle shown in the figure, and the trig identity cos (90°+ ß) = - sin ß.

[Herman Trivilino]

Deleted post made in error.

[Jan]

Here is the figure I used to derive my formula:


The large circle is the grindstone of radius R, the small circle is the US rod of radius r.

I simply applied the law of cosines to the triangle shown in the figure, and the trig identity cos (90°+ ß) = - sin ß.

Deleted post made in error.

Thanks Herman for your figure which illustrates your approach. 

You still stay conceptually approximate, because the angle β in your figure is not exactly the real bevel angle, which we will get as the result of knife sharpening using a Knife jig.

I have added a red line to your figure which shows the longitudinal axis of the Knife jig, which is 12 mm above the centre of the universal support rod. The real knife bevel angle has to be measured with respect to the red line. 



The problem is slightly more complex and cannot be solved exactly by one application of the law of cosines. 

Jan

[Ken S]

Looking at Dutchman's tables, if the 6mm radius of the universal support is critical, moving from the 80mm distance column, which shows a 139mm protrusion to the 75mm distance column which shows a protrusion of 133 mm might help.

My unmathematical gut feeling is that this is an oversimplification.

With the error, if we maintain the same physical (and uncorrected) D and P dimensions, would the only difference be the sin? If so, how much would the error be in degrees of bevel angle?

We have not discussed tolerance. Even Weber gage blocks of laboratory tolerances have a tolerance range. The range may be + or - a couple millionths of an inch or equivalent in microns, however, even these are not "exact".

.2588, the commonly used sine for 15°, is rounded off, as is either 3.14 or 3.1416. We introduce many small errors. Even our grinding wheel radius is an approximation.

I believe we should determine a tolerance range which seems generally acceptable. Plus or minus one degree? two degrees? thirty minutes?

The other tolerance consideration should be the tolerance range of the black marker and Anglemaster. Even when used very skillfully, the black marker is dependent upon the accuracy of the existing bevel angle. Tormek has not shared its projected accuracy tolerance of the Anglemaster, with good reason. A machinist, fully realizing the limitation of even the most precise measuring tools, would not be taken aback by a realistic tolerance range for the Anglemaster, nor should he. The average customer, totally oblivious of measurement limitations, might question the value of a very workable tool.

This is leading to the question of how does the accuracy of our tables and formulae, as used with our various forum jigs compare with the traditional Tormek methods? We should be aware of both accuracy and efficiency. The other question is how accurate do we need or want to be?

Ken

[Jan]

Ken, we are not discussing here some negligibly small changes in bevel angles. The consequence of a non-justified simplification or unconscious approximation can be that we sharpen a bevel angle of 12^o instead of the desired 15^o. 

The great importance of the Dutchman approach is that he selected suitable geometrical approximation which lead him to relatively simple calculations with acceptable accuracy.   

Wootz slightly modified Dutchman's approach. He sets the height of the USB above the Tormek housing. This approach works even more precisely than the Duchman's. 

On the other hand, Herman suggested an approach, where the distance D is perfectly defined, but the bevel angle is biased. His corrected formula is trigonometrically correct but the approach is not suitable for prediction of the bevel angle with acceptable accuracy.

The point is to find simple expression which gives predictions with acceptable accuracy, e.g. 0.5 mm in distance measurements and less than 0.5^o in bevel angle.

Jan

[Ken S]

Jan, please understand that I am not criticizing. I make these post questions in order to learn and understand. I believe in this concept of setting bevel angles. I also believe that the fine math work done here will help support a solid concept with very precise control.

I did not have a concept that the combined error might be three degrees. That does need work. I find the tolerance range you put forth of .5 mm in length and .5° quite adequate.

I see the end result as a very easy to follow procedure, well within the grasp of our members, yet supported with very solid trig. Originally I thought this process would benefit new users, users who sharpen infrequently, and very busy, high production sharpeners who must sharpen a variety of knives and tools. I still believe it serves all three groups and is a significant step forward.

I do appreciate the combined expertise.

Ken

[Jan]

Ken, I am not afraid you are criticising the lengthy discussion here. I know you are expecting the outcome which you will integrate into you considerations. Detailed understanding how the knife sharpening works is for us Tormekers crucial.   

I do not need to convince anyone, but when the discussion is open and it is about conceptual questions I can be pretty insistent.

In comparison with the geometrical concept of the TTS-100 setter the knife sharpening geometry is an easier issue. 

Jan

[Herman Trivilino]

I have deleted the contents of Reply #19 because I had not made the error I thought I'd made! Unfortunately Jan has quoted that material in Reply #20.

For clarity, let me state that the correct relations are the ones I originally stated:

D'=D-r
P'=P-r

[Jan]

Herman, if you wish I can delete my quote of your reply #19 also. Let me know. 

To understand you correctly, your formula is now given in reply #11 and the figure in your reply #18 was used to derive it. 

My major objection is expressed in reply #20. Your quantity β is by some 5^o smaller than the real edge angle!

Jan

[Herman Trivilino]

Jan, yes what you say is correct. I had not accounted for the bevel angle ß properly. I will have to go back to the drawing board.

[Jan]

OK Herman, please take your time, we are not in a hurry. 
It is really an essential issue for knife sharpeners.

Jan

[Ken S]

Jan,

In English, we have an epression "a twofer".  "Twofer" is a corrupted spelling of "two for", and means an unexpected second item at no extra cost. In a store, if one purchases one item and gets a second item at no extra cost, one gets a twofer.

For me, this process is a twofer. I began working on this process very shortly after purchasing my Tormek in 2009. This was long before Dutchman posted his tables or the kenjig. My original motivation was to discover a more efficient way to set up chisels and plane blades. I have always marveled at the combination of the TTS-100 setting gage and the SVD-185 (now SVD-186) gouge jig. The combination seems so logical and automated for turning tools. I wanted to achieve that same degree of automation with chisels and plane blades. I used the A and B settings on the TTS-100 as consistent Distance settings. I placed a blank piece of label maker tape in one of the protrusion slots and marked protrusion distances for different bevel angles with a black marker.

These Protrusion settings were established by setting Distance and adjusting the blade protrusion until the Anglemaster indicated the setting was correct. It was very low tech, but quite repeatable and consistent. I was looking for consistency and repeatability, as I had no reservations about the accuracy of either the Anglemaster of the marker method.

I still have no reservations about the accuracy of the Anglemaster or black marker. Many thousands of edges have been happily sharpened with these methods.

For many years I have had a fascination with machine shop tools and measurement. One of the century long standaards for precision measurement is using a set of "Jo Blocks". "Jo Blocks" is the machine shop floor term for Johansson Blocks, named after Carl Edvard Johansson, the Swedish machinist who invented them in 1896.  These deceivingly simple metal blocks are capable of quite precision measurement to .001mm or .0001".  (Look in wickipedia under "gauge blocks" for more information.)

The kenjig functions as a less precise gage block, as do the protrusion stops on the TTS-100. Being able to incorporate some of the accuracy of gage blocks is the unexpected twofer of the project for me. I recognized this when I first read Dutchman's booklet posted on the forum. While I do not have the mathematical skill to produce the work myself, I am aware enough to recognize the potential of incorporating math into the project to introduce much more precise results. We have the very good fortune on this forum of having members with the math skills to produce this work without introducing substantial cost into making the jigs. In industry, going from an accuracy of  .0254mm (or .001") to to .00254mm (or .0001") would involve substantial cost. We can tighten tolerances to a practical maximum with essentially no extra cost. That is much added value.

I find this project increasingly exciting. I do not expect it to work with every blade in the shop. I believe that would be carrying a good idea to the point where it did not work well. I do believe it has the potential to make Tormek sharpening easier and more efficient.

Ken