Geometry and Kinematics of Guided-Rod Sharpening Systems

[Lagrangian]

Hi Everyone,

I've performed a very detailed mathematical analysis of the Edge Pro Apex (EP-Apex) and also the Wicked Edge Precision Sharpener (WEPS). I wanted to share the results with the general knife community.

As someone who loves mechanisms, I wondered if there are any ways to improve the EP or WEPS. Of course one could improve the precision of the EP and WEPS mechanisms with more accurate parts machined to a finer tolerance, etc.

However, after careful thinking, I realized that even if these mechanisms were _perfectly_ precise and _infinitely rigid_, that they would not always grind a perfect dihedral angle (informally known as a "V-edge"). That is, if we used a perfect EP or WEPS to sharpen a tanto knife, then the knife edge would not have a perfectly uniform dihedral angle. There will be a tiny variation in the included angle of the knife bevel.

How big is this variation in angle? To study it, I coded a computer program calculate the geometry and have written up the results here.

Topics in the report include:
(1) Tiny variations in sharpening angle in the Edge Pro Apex and also the Wicked Edge Precision Sharpener.
(2) A detailed analysis of the "Stop-Collar Trick" and how it is an approximation.
(3) For WEPS sharpening, a discussion about where to clamp the knife so that the sharpening angle is as uniform as possible over the entire knife edge.
(4) Using belt sanders to sharpen V-edges.

To pique your interest, here are some figures and one animation from the technical report.

Sample Figures:
http://imgur.com/a/1J8GP#0








Sample Animation on YouTube:

https://youtu.be/Lg3dK_n49Gw

The full report is currently an initial draft (version 1.0.beta14), and it can be downloaded here:
https://drive.google.com/file/d/0B8rQYh ... Q2MlRFbTA/

Alternate Download Link:
http://www.mediafire.com/download/2flrq ... eta17).zip

The link is to a .zip file on Google Docs. The .zip file contains:
(1) A PDF file with embedded videos.
(2) Separate video files in a directory named "Movies"
(3) README file
(4) Creative Commons License file

The .zip file is huge because it contains many animation videos and figures. To view the animations within the PDF file, it is recommended that you use Adobe Reader version 9.0 or latter. Other PDF viewers will probably not play the videos. Also, you may have to give Adobe Reader permission to play the videos. If your PDF viewer cannot play videos, you can still view the .mp4 files in the folder "Movies." These files are .mp4 videos (MPEG-4/H.264) and can be played with free software such as Apple's QuickTime Player and/or VideoLAN, etc.

Please note: the slight variation in sharpening angle is VERY SMALL. In fact, it is typically around 0.1 degrees or less. In the worst case that is plausibly realistic, it is at most 0.5 degrees. In the Sample Figures above, we only really care when the sharpening stone is close to position X=0 inches, that is -1" <= X <= +1". These TINY changes in sharpening angle are virtually undetectable in practice.

Therefore this report is only interesting to:
1. Knife sharpening fanatics who like V-edges.
2. Engineers who like to study mechanisms.
3. Those of us who are insane. :)

This is still a beta version draft, so feedback is welcome.
If you have any questions, suggestions, or constructive criticism, just post to the discussion.

Sincerely,
--Anthony "Lagrangian" Yan

[chuck_roxas45]

Awesome! That is great knife math Laggy.

[dbcad]

Nice analysis Anthony :) Who says math isn't fun

[Jazz]

Nice - that's why I sharpen freehand.

[Clip]

Lag, thumbs up! Very interesting read.

[Evil D]

Nice - that's why I sharpen freehand.

Are you suggesting that your own free hand ability is at or better than .1 degree of accuracy?

[chuck_roxas45]

Are you suggesting that your own free hand ability is at or better than .1 degree of accuracy?

Hehe. :)

[senorsquare]

Great work there. I think you may want to double check the geometry in Figure 4.1 (D) Stop Collar Trick for Edge Pro. The surface of the guide rod housing and the stop collar are both perpendicular to the vertical rod. Seems to me that dimension B would be measured perpendicular to the vertical and not on the parallel to the guide rod. The change in height when adjusting with the stop collar trick is the actual change in stone thickness. I have no idea how this translates to a change in angle where the stone meets the bevel


Capture by senorsquare, on Flickr

[Blerv]

Are you suggesting that your own free hand ability is at or better than .1 degree of accuracy?

Or he would rather skip all the math.

[Jazz]

:) No. It's just easier to get around the front of the blade and keep the angles where you want them. I have no doubt those sharpeners work pretty well. I'm not gonna lie, though - I can do pretty well, only after years of practice and learning. And no, I'm not bragging either. I've sharpened lots that I thought were sharp, but as you do more, you learn that no, they really weren't. Wayne Goddard wrote that you turn your body into a sharpening jig, keeping yourself rigid through the strokes. He explained it better, but it works for me.

[Lagrangian]

Hi sensorsquare,

Not 100% sure I understand your question/point, but I think I can answer it. In the figure, the length labeled by "a" is the actual change in stone thickness. Since we changed to a thicker stone, we need to raise the pivot point on the vertical rod.

If we raise the vertical rod by the distance "b", then everything works correctly. In the figure, "b" is trapped between two parallel lines and is straight up and down. So you can slide "b" until it is over-lapping the vertical rod. It will then cover the same distance between the two red lines you drew. (Another way to think about it is to see a parallelogram where "b" is one side of the parallelogram.) I "b" drew it to the side, but maybe I should draw it on top of the rod.

In any case, you can see that "a" and "b" are not the same length. In the stop-collar trick, we approximate "b" by "a". However, typically, "a" is too small of a correction (a <= b). You can draw some triangles and do some trigonometry to find out how different they are. (Details are in the chapter about the Stop Collar Trick.)

Hope this helps?

[Cheddarnut]

With math like that i would assume youve mathematically solved a way to never have obstacles in your path that require a knife, rendering sharpening methodology moot.

[mainaman]

There will always be some flex in the system and thus never be able to truly achieve a V bevel geometry.
The less the downward pressure while sharpening the closer to perfect V the geometry will be.
Another variable is the dishing of the stones used to sharpen. All synthetics dish, the lower grits a lot faster than the higher grits. That also will convex the edges.
If that variable it to be eliminated as much as possible one has to consider using diamond plates instead.

Cool analysis.

[Blerv]

With math like that i would assume youve mathematically solved a way to never have obstacles in your path that require a knife, rendering sharpening methodology moot.

Very Matrix, Cheddar :) .

[razorsharp]

Jesus Antz. lol

I was waiting ages for this, now that its here, Its flown over my head LOL.

I think Ill drink a few coffee's before I attempt to look at this :D

[Evil D]

Yeah I think I can live with .1 degrees off. To me that still counts as removing human error, since even if you were able to either hold the right angle or even find the right angle freehand, you're much more likely to lose either one by at least .1 degrees. All the issues with dishing stones also apply to freehand too. Anyone who is truly concerned with .1 degrees better never go near a strop lol.

[jackknifeh]

Very cool analysis. I admit I haven't read it thoroughly yet cuz I have to go but will read it soon. There is one other sharpening system I think deserves attention. It is not mass produced AT ALL. These guys are hand made to order. The system can use almost any size stone so stones you already have up to 8" I think will mount to the system. I learned about this system a few years ago but had forgotten about it until recently. On another forum the inventor/manufacturer posted he can put multiple bevel angles on an edge with a difference as small as 1/100th of a degree. Read it again. 1/100th of a degree. NOT a typo. First of all he admits this plays absolutely not part in cutting performance. He has pictures of different bevel angles beside each other on a knive blade with the scratch patterns going in different directions making it easier to see the bevel angles. It resembles the side of a Chaparral 2 handle. The system is called the EDGE PAL. http://www.edgepal.com/

I don't know exactly what pictures are on the site. If you don't believe me I suggest you contact the inventor. When I first saw his claims on a forum I thought he had typed the angle measurements wrong. He is in Europe somewhere I think. If I had more money I'd buy one of these things just because. It is more expensive than the EP or WE especially considering it comes with no stones because by design it's intended to use the stones people already have. It's more like an EP than a WE in design.

If anyone is interested check it out and please contact the manufacturer to question any claims he makes. I didn't believe the 1/100th degree angle setting difference at all when I first read it. I replied in the forum and also contacted him via the web site. Until I see one and prove him wrong I believe it. I do think that like the EP it may take practice with the tool to master it to the level he can. I get much more accurate results from my EP now than when it was new (just like any tool).

About the stop collar being an estimation. I agree 100%. I noticed this when using the really fine grit stones. Using the collar to set the angle for a new stone of different thicknesses is ok with the coarser stones because any VERY SLIGHT inaccuracy is made up for by the stones removing enough steel to actually create a new angle. However when going from a 4k to an 8k stone after a couple of strokes I look at the bevel and see that the stone is only contacting the top or bottom of the bevel. Such a small amount of steel is removed per stroke by these grits that it would take many strokes to remove enough steel to create a new angle (of maybe a 1/100th degree??? :) ). Therefore I always double check the contact area on a bevel after my 2k stone. Then I almost always have to make micro-corrections for the contact area of stone to bevel hit the entire bevel. Repeat: The coarser stones do the same thing I'm sure but after a 500 grit the 1k grit stone removes enough of the top of the bevel (for example) in one or two strokes that the stone is also touching the bottom of the bevel because the angle has changed from 18° per side to 18.01° or 17.09°. Does this make sense?

Jack

[Lagrangian]

@ Jackkneifeh:

I had some discussions with the owner of EdgePal in the forums. I had some trouble communicating with him due to a language barrier (he is not a native English speaker from what I understand). From the photos on the website for EdgePal, I don't think I see anything particularly unusual. The main advantage of his design, is a very fine adjustment screw for setting the angle. This is why he says 1/100th degree angle setting. That is, the EdgePal might have a 1/100th degree _resolution_ in setting the edge (ie: precision). However, that is somewhat different from 1/100th degree accuracy. I'm using the words "resolution", "precision", and "accuracy" in the technical sense. These are terms from scientific measurement. Wikipedia has a reasonable description of these:
https://en.wikipedia.org/wiki/Accuracy_and_precision

I like the EdgePal design that uses a spherical bearing. The other designs, I feel unsure about how repeatable they are, due to potential "play" in the way the rod is guided by a circular metal loop. It seems to me that a simple metal loop might have too much "slack" or "play" for guiding a rod. But it is hard for me to say, since I haven't actually tried one in person.

In theory, there is no reason an Edge Pro, WEPS, or other sharpener couldn't use a fine adjustment thread just like the EdgePal. These designs are all very similar from a theoretic point of view. In practice, all sorts of real-world effects matter: accuracy of parts, rigidity of parts, "play" etc.

As for the stop-collar trick, I'm trying to think of a way to modify it so that it becomes perfectly accurate in theory. So far, I haven't thought of a design that I like (ie: simple, mechanically robust, and perfect in theory). From the chapter on the Stop Collar Trick, you can see the source of the error. If you think of a good way to fix it, please let us know! :)

[jackknifeh]

@ Jackkneifeh:

I had some discussions with the owner of EdgePal in the forums. I had some trouble communicating with him due to a language barrier (he is not a native English speaker from what I understand). From the photos on the website for EdgePal, I don't think I see anything particularly unusual. The main advantage of his design, is a very fine adjustment screw for setting the angle. This is why he says 1/100th degree angle setting. That is, the EdgePal might have a 1/100th degree _resolution_ in setting the edge (ie: precision). However, that is somewhat different from 1/100th degree accuracy. I'm using the words "resolution", "precision", and "accuracy" in the technical sense. These are terms from scientific measurement. Wikipedia has a reasonable description of these:
https://en.wikipedia.org/wiki/Accuracy_and_precision

I like the EdgePal design that uses a spherical bearing. The other designs, I feel unsure about how repeatable they are, due to potential "play" in the way the rod is guided by a circular metal loop. It seems to me that a simple metal loop might have too much "slack" or "play" for guiding a rod. But it is hard for me to say, since I haven't actually tried one in person.

In theory, there is no reason an Edge Pro, WEPS, or other sharpener couldn't use a fine adjustment thread just like the EdgePal. These designs are all very similar from a theoretic point of view. In practice, all sorts of real-world effects matter: accuracy of parts, rigidity of parts, "play" etc.

As for the stop-collar trick, I'm trying to think of a way to modify it so that it becomes perfectly accurate in theory. So far, I haven't thought of a design that I like (ie: simple, mechanically robust, and perfect in theory). From the chapter on the Stop Collar Trick, you can see the source of the error. If you think of a good way to fix it, please let us know! :)

The "play" you are referring to on the EP Apex is a VERY REAL issue when using it AND wanting very accurate results. Here is how it works. With the stone in the EP, lay the stone on the end of the blade table with the end of the stone toward the vertical post on the table. Then gently tap the stone rod toward the rear of the unit. When the balance point on the stone is passed you will see the rear of the stone drop a tiny bit. This is because there is excess room inside the hole where the stone pivot rod slides. 95% of the time we are using enough downward force on the knob that it keeps the rod at the top of the pivot hole. But when using VERY light strokes (WHICH IS ESSENTIAL FOR PRECISION STROKES IMO) it's easy to allow the stone to teeter-totter- at the balance point. This will happen anytime only the weight of the stone is used. I have limited my stone area of contact to the front end of the stone to keep the rod at the higher angle. I have also used a finger laying on the top of the stone or rod to hold the stone in the position I want. Without dealing with this issue I'm a firm believer I could not get the delicate results I'm getting. Also, if this issue is not addressed the EP can be blamed for not being able to deliver extremely accurate results. Actually, the EP is NOT capable because of this issue if the person using it doesn't realize the issue and overcomes it. I believe Madrookie developed a new pivot rod for the EP Professional model that eliminates this issue.

I don't want to sound like a know-it-all but any EP users out there looking for the most accurate and sharp edge possible in all the world please re-read this post. Without this issue being recognized and a workaround is found you may find it hard to get consistancy and not know why. I just put the angle cube on my EP and I get a .5 degree angle difference whan putting downward pressure on the rod handle (which is normal during use) compared to putting downward pressure on the rod on the other side of the stone holders. By putting pressure there it's the same as when the stone rod is past the balance point with the rod pushed toward the machine. Different EP's may have different tolerances. Mine has been used a lot in the past 3-4 years.

If I didn't make this clear please ask questions. Or if you think I'm wrong or have another way to deal with this issue.


Jack

[Lagrangian]

@ Jackkneifeh:

I do like MadRookie's EP modification with a spherical joint (spherical rod end). It is discussed in the report, on the chapter on the Edge Pro Apex. In that chapter has a photo of MadRookie's setup.

Mostly though, the report is about the idealized design. From a theoretical point of view, using a spherical joint increases the accuracy. (See chapter on Edge Pro Apex.)

Your description of the "play" in your EP sounds disappointing to me. I was hoping they would have a finer tolerance.

I think what you are saying is that in your experience, using a spherical joint also improves the accuracy due to parts made with a finer tolerance? There is an almost infinite variety of spherical rod ends which are manufactured by an almost equally infinite number of manufacturers. Some designs have almost zero play. I say "almost" because we have to be careful about what this means. The so-called "zero play" rod ends use pre-loaded pressure on the pivoting ball. Typically, what this means is that there is a polymer liner that holds the central ball, and this polymer liner is actually under compression. So the center ball is being squeezed all the time. So in this sense, there is zero play. However, if you push the ball hard enough, then the polymer liner is likely to deform. Other rod-ends use metal liners are less likely to deform significantly under normal load. (Of course, everything flexes under any load, and if you load it enough it will bend and then break.) Not all polymer lined rod-ends are "zero play"; I think most are not. Hope this makes sense.

I very much like the idea of using high-precision and high-wear parts. If you're interested, here is a sharpening rig I built using spherical rod ends. The rod ends I used are manufactured by Heim, and use an oil-impregnated bronze liner. The bronze is porous and the pores are filled with oil. I found these particular rod-ends to have slightly axial play, but virtually no radial play. I was unable to insert a piece of aluminum foil (16 micron thickness) between the ball and the liner, so I think they're pretty good. When I have time, I hope to do a "bump" measurement to see how large the play actually is, as measured by a dial indicator.
http://www.bladeforums.com/forums/showt ... Pict-Heavy

I know that ball bearings are typically made to incredibly high accuracies. It is easy to find ball-bearings manufactured to the G25 rating, which means they are spherical to within 25 micro-inches. So that's 0.000,025 inches. What I don't know is if the balls in spherical rod-ends are also manufactured to such high accuracy. Most rod-ends I have seen do not have an accuracy rating for the ball itself. In fact, I haven't seen any. If you ever do see a specification on the ball accuracy in a spherical rod end, please let us know.
https://en.wikipedia.org/wiki/Ball_(bearing)#Grade

I think mechanisms are very interesting... And I'm happy to discuss both at the theoretical level and also in the real-world. So let's talk some more! :)