Mini Mill Fine Feed Z Axis Crank

Every time I use the boring head on my mini mill I've told myself I need to make a crank for it.  The stock knob for the fine feed is pretty small so it's a pain to run the cutter down & then back up.  I had held off because I had been thinking about a crank that's similar to the X and Y axes.  Then I realized that I just needed a longer lever (and a handle) to get what I wanted.  After that, I got to work and made this:

The crank is made of flat aluminum stock.  I made the handle out of brass and polished it up a bit using some quadruple-0 steel wool.  I used an off-the-shelf shoulder bolt so the handle could spin freely, and eased the head a bit using my lathe and ball turning attachment.  The crank is attached using a couple of drilled/tapped holes in the fine feed knob.

The ball turner turned into a project as well -- I figured out that its design really required a mod to the lathe so I could manually drive the lead screw (to drive the carriage).  So the first crank I made turned out to be for the lathe.  Of course, the handle isn't nearly as nice because I did not have a functioning ball turner at that point.

I had thought the crank would be in the way too much to leave it installed all the time, but it has turned out to be so handy that I haven't removed it.  When the fine feed is not engaged the crank hangs down, as shown in the photo.  I suppose if it gets too annoying I could come up with something to hold it in some other orientation.  But to date it hasn't been a problem.

Knife Sharpener Prototype (?)

Implementing my knife sharpener took longer than expected, at least partly due to the surprising number of different machining steps it needed.  The knife holder needed three drilled/tapped holes and a 1/2" hole for mounting it on the base.  It also needed a recess milled to accommodate the thickness of a washer used with the mount.  The mount is a 1/2" diameter aluminum rod that was cut down to 7/16" and threaded so it could be bolted to the base.

The mast also required some machining -- in this case, a 1/2-20 threaded portion to bolt it to the base.  The pivot piece was machined from Acetal, and needed some machining so it could be attached to the mast (and also permit the sharpening rod to pivot).

The sharpening rod has two Acetal blocks screwed to it.  They are used to hold the sharpening stone.  I also had to fabricate a handle and place it correctly so the operator could not be cut by the knife.  Making the handle took a number of steps, too, but it turned out pretty nice.  I used brass for a bit of eye candy.

Anyway, here is the result:

The photo shows the knife installed in the holder.  It is not clamped down -- there is a recess on the back of the holder where the back of the blade fits, so it is held securely enough while it is being sharpened.  In use, the sharpening rod rides in the slotted piece that is attached to the mast.  The height is adjusted to get the desired bevel angle (in this case, 20 degrees).

After using this on a couple of knives (starting with an inexpensive one I didn't care about much), I have figured out that I need to make a couple more pieces.  First, I need another rod/stone holder assembly, for a honing stone.  Right now I've got my water stone installed, and after that I hone the knife using a freehand approach.  I also will make another pivot assembly that will be installed above the first one.  In combination with my honing stone I should be able to produce a nice secondary edge that will last a long time.

By the way, the base is a hunk of oak veneer plywood left over from our house project.  The feet are wood dowels that were turned to size on my lathe, then some rubber floor-protector feet were installed over them to protect the counter top.  Since I've got bolts projecting from the bottom of the base I definitely had to have SOMETHING to elevate the base above the counter and this approach seemed the simplest to implement.

At this point I have less than $50 invested in this (including the stones).  That may go up a bit after I make the additional rod/stone assembly.

Anti-Backlash mill modification revisited

It didn't take long for me to become unhappy with the design of my anti-backlash mod.  Due to the large amount of friction, I became concerned about excess wear of the lead screw.  A lot of nonuniform wear could cause problems when traversing to the extremes of the table.  So I started working on a scheme similar to the approach Sherline took on their CNC mill.  Their approach uses an external feed screw nut that is snugged up against the table to remove backlash.  To keep the nut in place, its outer perimeter has a rather coarse knurl.  A thick washer, also knurled on its outer perimeter, engages the external nut.  The external nut is turned via the washer until the backlash is reduced to a low value (while keeping the drag relatively low), then the washer is fixed in place with a bolt thru its center.

Below you can see my version of this:

The round piece is the external nut (fabricated from a feed screw nut I bought from Little Machine Shop).  I drilled a 1/16" hole in the nut, to engage a wire.  The bracket is bolted to the end of the X-axis table.  The slotted piece holds a 1/16" piece of piano wire that was bent to fit into the nut, and is fixed in place with a standoff/bolt combination.

Here's a photo of the installed pieces:

In use, the bolt is loosened so the nut can be rotated clockwise in order to remove any backlash, then tightened.  Since the only force the standoff, slotted piece and wire experience is the frictional torque between the nut and feed screw, they don't need to be very heavy-duty.

With this setup I was able to reduce the uncontrolled table "slop" to about .001", and the mechanical turns dial vs. DRO indicate I have about .002-.003" of backlash.  Before, there was about .005" of slop and about .010" of backlash.

The only thing I would change at this point would be to replace the Philips-style screw with an socket head screw.  Right now, to adjust the backlash I have to remove the feed screw bracket to gain access with a screwdriver.

Knife sharpener geometry -- spreadsheet

I have created a spreadsheet to illustrate the geometry I described in my previous post on the subject.  It can be found at: knife geometry.xls. 

Variables to play with are the radius of the arm (actually, the distance from the pivot point to the knife holder), length of the knife and the rotation angle of the knife.  If the radius is changed you will need to figure out the height of the pivot above the blade, to give you the desired bevel angle.

The spreadsheet has some warts -- some of the parameters have to be entered in several places.  To help figure out what goes where, the spreadsheet parameters are as follows:

Arm radius = 18"
Length of knife blade = 8"
Knife is rotated 12.75 degrees.

To determine the error in bevel angle halfway down the blade, just change the value in cell I50.  Right now it is set to show what the error is at the tip of the blade.  Changing it to 4 inches will show what the error is at the halfway point (in cell B46).

FYI, the angle calculations use the dot product of two normalized vectors.  If you take the dot product of two vectors whose magnitude is 1.0, the result is the cosine of the angle between the vectors.  Vectors can be normalized by dividing each component by the magnitude of the vector, which is given by:  sqrt(x^2 + y^2 + z^2), where x, y and z are the components of the vector in 3 dimensional space.

Dot products are very useful beasts, having a wide range of applications, from calculating the Fourier Transform to computer generated graphics.

Knife sharpening fixture

This entry contains some preliminary work I've done to build a knife sharpening fixture.  No photos as yet, but there probably will be some in a future post.  This time I'm just outlining the geometry of the fixture, variations in the bevel angle along the blade due to the geometry, and approaches to minimize the variations.

Below is a simple figure showing the basic fixture from the side:

 

The vertical mast has a pivot (the circle) whose height can be adjusted.  The line passing through the pivot depicts a rod, whose opposite end has a sharpening stone attached to it (the green line).  The knife is shown in magenta.  Hardware for holding the knife is not shown.  The whole thing is assembled on a base, shown by the thick horizontal black line.

The basic idea is as follows.  The pivot is above the plane of the knife by a distance dictated by the desired bevel angle and distance between the mast and knife.  We can calculate the bevel angle using this:  tan(theta) = H/L, where theta is the desired angle, H is the height of the pivot above the knife, and L is the distance from the mast to the edge of the knife.  If we know theta and L, then H = L*tan(theta).  Caution:  many spreadsheets assume radians, not degrees, are the argument to trig functions.  To convert degrees to radians, remember that 2*pi radians = 360 degrees.  (2*pi/360) * degrees gives you the radians.

There is an interesting aspect of this.  My online searching revealed that most everyone seems to be building fixtures that create a specific half-angle -- not the full angle between the two sides of the bevel.  So, if you're sharpening a knife to a "20 degree bevel", the angle between the two sides of the bevel is twice that -- 40 degrees.  Not that it matters, it's just different relative to the usage found in general machining practice.

Anyway, back to the fixture.  While it might seem to ensure you are sharpening your knife to a specific bevel, that is incorrect.  Take a look at the diagram below.

 The diagram shows the fixture looking down from the top.  The magenta rectangle is the knife and the thin black lines show the position of the rod (plus sharpening stone) over two positions on the knife.  Observation tells us that the bottom line is the shortest line and the top line is longer.  The difference depends on the size of the knife.  This variation in distance causes a variation in the effective bevel angle that will be ground into the knife.  How much of a difference are we talking about here?

Let's do some calculations based on a fixture design I found on the web.  In that design, the shortest distance was set to 10 inches.  Probably to keep the fixture small and easily transported.  If we want a 20 degree bevel, trig tells us that the pivot point must be 3.65 inches above the knife blade.  If we are sharpening an 8" chef's knife, the tip of the knife is further away from the pivot, and as a result the bevel angle is reduced to 15.9 degrees!  Ouch.

How can we improve this situation?  The easiest approach is to increase the distance between the pivot and knife.  Let's increase the distance to 18 inches.  We have to raise the pivot point to 6.55 inches to get a bevel angle of 20 degrees.  This change reduces the variation in bevel angle to 1.6 degrees.

But we can do better than this if we want.  Since the change in bevel angle is due to the increased distance, let's rotate the tip of the knife toward the pivot point to reduce the distance.  It turns out that a rotation of 12.8 degrees will reduce the error to close to zero.  Nice, huh?  Not so fast.  What about the bevel angle at the midpoint of the knife?  With this rotation, the error is .4 degrees.  Still, not too bad.  Zero rotation gives us a MINUS error of about the same magnitude, so that's a wash.

Figuring out the optimum rotation angle of the knife may seem like a mysterious process.  But it's not, and actually is easy to set up.  In the case of our 18"-long fixture, let's use a very large compass to draw a circle around the support mast for the pivot.  The circle will have an 18" radius.  Now, loosely install your knife in the holder and rotate the tip so it just intersects the 18" circle you just drew.  That's the angle you need, because the circle denotes a constant 18" distance from the pivot!  Tighten the holder down and start sharpening, with the assurance that the variation in your bevel angle along the length of your 8" knife is no more than .4 degrees.  Shorter knives will have less variation.  If you are sharpening 10-inchers or longer, maybe you should think about an even bigger fixture, maybe with a baseline of 20 inches or more.  Even so, it still will have a footprint less than 2 feet deep.

The downside of a setup that requires you to rotate the knife has some design complications you may not want to bother with.  For one, you probably need to have a knife holder than can rotate, too.  Then be easily locked into position without engaging in an excessively-complicated procedure.  And for longer blades you probably want some sort of support partway down the blade to keep the blade from flexing (or popping the knife out of the holder).  But to accommodate different blade lengths, the support has to be movable -- the knife rotation angle will change.  Or you will need to fabricate custom holders for each size of knife you have.  Mmm, more tradeoffs.  But that's the fun of design -- addressing problems like this in as elegant a manner as practical.  No, I didn't say "as possible" -- that's not engineering.  The art and fun of engineering is finding the balance between performance and practicality.

 

Was Julia Child really left handed?

My wife and I were watching a very old Julia Child cooking episode on YouTube when we noticed something interesting.  OK, I noticed -- she was oblivious.

This was an episode where Julia was extolling the uses of eggs in things like chocolate mousse.  The video alternated between face-on views and top-down, showing the whisking action when Julia was beating egg yolks and stuff like that on the stove.  I noticed the top views showed Julia using her left hand beating the eggs and whisking stuff etc.  But the face-on views showed her using her right hand.  Finally, I saw a top down view that showed several pots on the stove, and the arrangement was reversed between the top-down view and face-on view.

So, here's an interesting observation.  Maybe it's been noted before, but a quick internet search did not reveal any confusion about Julia's handed-ness.  I think the top-down views were done using a mirror, so the viewpoint was reversed.  Perhaps today's directors are more concerned about consistency from shot to shot so that doesn't happen now....maybe.

I would be interested to know if anyone else has noticed this in the old Julia Child episodes.

Repeating decimals and encoding messages

I have another math nurd post. 

It has a loose relationship with a branch of mathematics called number theory, and a close relationship to a sleepless night.

I was (for no obvious reason) thinking about repeating decimal numbers that are the result of dividing one number by another.  A good example:  .33333.... = 1/3.

I started wondering if would be possible to define two numbers A and B to generate any arbitrary sequence of repeating decimal numbers when calculating A/B.  Oh yes, those dark winter nights!  For a triplet of repeating digits, my initial mental math went like this:

A/B = .WXYWXYWXY.....

where W,X and Y are numbers between 0 and 9

Multiply A/B by 1000 and we get:  1000*A/B = WXY.WXYWXYWXY....

The value behind the decimal, clearly, is the same as A/B so we can write:

1000*A/B = WXY*A/B.  If we divide both sides by A/B we get:  1000 = WXY.  Clearly, there is no sequence of 3 digits that can equal 1000 (999 is the closest we can get).  So it would appear that there is no solution for this. 

Not so.   Let's say WXY = 123.  Now set A = 123 and B = 999.  Get out the old 4-banger calculator and what do we see?  123/999 = 0.123123....

So what is the problem here?  Unfortunately, simple math (remember I mentioned it was a late night).  The real equation for this relationship is as follows:  1000*A/B = WXY + A/B.  Where WXY really equals W*10^2 + X*10^1 + Y*10^0

If we go back to setting WXY = 123, we get:  1000*A/B = 123 +A/B, or 999A/B=123.  I.E., 999A = 123B, or A/B = 123/999.  Do the division again, and you get:  .123123....

A straightforward generalization tells us that we can generate any arbitrary repeating-decimal number by multiplying the sequence of numbers by the power of 10 equal to the length of the sequence and dividing it by 10^K -1, where K is the length of the sequence.

An interesting possibility raises itself in this discussion of rudimentary numerical analysis.  Suppose we encode the alphabet by setting A = 01, B = 02....Z = 26.  If you have a message you want to send securely, encode it into a string of tuples (2 digits each) and treat the whole thing as your repeating rational number.  Unfortunately, not too hard to decyper, right?  Well, now multiply the W-wide number (equal to 2*the number of characters in your message) by another number, large enough to ensure some carries between each tuple.  NOW we have a more interesting thing to decode, because the resulting tuples depend on each other.  It becomes a context-dependent code.  The keys to the code are the multiplier....and the length of the message.  In today's age, the two could be sent separately without too much trouble so time-sensitive information (which would take some time to decode) would be relatively secure.  Attempts to crack the code would be complicated by the fact that the encoded message string is unique due to the context dependency.

Suppose we choose the multiplier to be 11.  This is an interesting example because it is equivalent to shifting the original code over by one, and adding the result to the original code.  It would completely alter the cypher.

Is this completely original?  Hah --- close to a zero chance of that.  But it is an interesting illustration of how easily a simple encoding scheme could be turned into a nontrivial decyphering problem, except for government level expertise.  Also, due to the fact that certain letter sequences occur pretty often, the encoding scheme would be susceptible to statistical analysis -- if long enough.

Relatively simple obsfucation techniques like reversing the digit sequence would still yield to this type of analysis, if the analysis is thorough enough.  Possibilities are endless though -- consider encoding some well-known text in the same fashion and adding its digit sequence to the message.  This is an example of the use of a code book, in addition to the mathematical encoding.  Without knowledge of the code book (and where the code string begins), things get difficult in a big hurry.  The code book string itself could be mathematically obsfucated by using yet a different multiplier.  And maybe reversed (or not) as well.  Ouch!