0.001" Measurements at Home With Geometry!

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Whenever people talk about simple machines, they almost immediately talk about mechanical advantage, work, forces, etc. With simple machines, you convert large easy movement into small, stronger movement, and don't get me wrong, that's pretty nice. Archimedes once said something like, "Give me a lever long enough and a place to stand, and I can bench more than you" and proceeded to flex on the Romans with awesome inventions. But let's take a step back. We can convert larger movements into smaller movements using simple machines. As great as it is to apply large amounts of force, it'd be even better to apply the force accurately. Let's do that instead.

Note: What I built is more of a proof of concept prototype than a precision instrument. While my tool does move with a high degree of precision, the setup that I demonstrate is not robust. It is not practical. The concept DOES hold up though and I hope you enjoy as we design and build a linear stage actuator! Or at least, the first iteration of one!
If anyone has any suggestions for easy to assemble parts for the rails and stage, I'd love to hear it

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Step 1: Math, When Applied Correctly, Is AMAZING!

Alright, I admit it. My title might have been misleading. This tool, which I will be hesitantly referring to as a linear stage actuator, if it weren't for my sloppy implementation, would be far more accurate than 0.001"

Alright, how does it work?
It's surprisingly simple. The key is a 1/4" bolt. 6" long, and 20 threads per inch. Now, metric fans, stay with me. I've got something for you in a bit. I'd do conversions, but it's pretty important that the TPI (Threads per i-I mean unit) is accurate.

Now, let me address the elephant in the room. Why isn't that equation in the picture simplified?! Well, I put a lot of thought into making it that way. I don't remember any of those thoughts, but I did think about it.

20 threads per inch. What does that mean for us? It means that it takes 20 rotations to move the nut 1 inch. You know where I'm going with this right? 1/20 = 0.05 inches, so we're not to our 0.001-inch goal, but that was pretty easy so far. Now we just have to divide each rotation into 50 equal parts. If we put a 2-inch knob on the end of our bolt, and we divide that by 50, we end up with each mark just a little more than 1/8" apart along the circumference. Not bad. Not bad at all. Most Imperial rulers are historically looked at... Oh, I mean most Imperial rulers are divided into 16ths of inches. That's 0.0625" apart vs our knob's divisions at 0.1257" apart. So we've done it! Now, let's start making those marks!
...Oh...
They're 7.2° apart... That's a pain...

Rather than trying to measure out and draw 7.2°, you can try your luck with a geometric construction. I tried a couple of times to divide a circle into 5 equal parts, but I think the method I used compounded errors to the point where it wasn't worth it. We won't be doing that.

Step 2: Back to the Drawing Board!

So the marks we're trying to mark are too small to do accurately. That is a problem.

But wait! Those words sound like those other words from the intro!

"...it'd be even better to [do the thing] accurately. Let's do that instead."

If we increase the mechanical advantage, we can reduce the effects of our errors!
Not that you make any errors, dear reader. You're perfect, just the way you are...

But I doubt you can consistently and accurately mark out 7.2° lines, so let's give ourselves a fighting chance.
We will talk about a tool I wish I had in a moment, but first, how did I plan on solving the problem?

I just used a really really big circle.
It works.
I used a compass to draw the circumference of an 8 3/4" diameter circle. We're going to pretend it was 8". You'll understand later. I was going to first use a template or some sort of stencil of a 14.4° angle (1/25th of a circle) and then use my compass to construct an angle bisector. The angle bisector would be very accurate, but if I used a stencil, the errors would compound on every mark I made.

While I did think of a way to accommodate for that, here's a better method.

https://www.blocklayer.com/circle-dividereng.aspx

That website lets you design the template we want, then save it as a pdf. I'd have included a download for the proper pdf, but I figured I'd try to send them some traffic for their good work. Plus, I figured those of you who use metric would find this far more useful. You could then cut it out, laminate it, or/and put it behind a sheet of acrylic attached to the wheel, and you're done.

Because of the printed template, we have our 50 divisions just over half an inch apart. We have enough room to instead divide it into 100 equal segments, and still be more visible than a ruler, measuring down to the half thousandth of an inch!
Why not?
Because increments of .0005" is not so user-friendly, but come on! That's really cool! And we did it with what people consider a drawback of simple machines!

However, since we're printing the template, we can't easily make the circle bigger than a piece of paper. Standard paper here in the US is 8 1/2" X 11". 8 inches is easy enough to work with.

Step 3: Backlash

Backlash is, for these purposes, the slop in the mechanism. If you put a nut on a bolt, then you shake it, you'll hear the nut rattle just barely. if you turn the bolt one turn forward, then one turn backward, it won't be in the same place (this only occurs when you change directions). This is a problem for us. So let's ask a professional how a commercial linear stage actuator deals with it!

"Well, that's uhhh... Hmm. How to put this, We usually just let the user figure it out. They're a machinist, they know what to look out for! Whenever you switch directions, you simply turn the wheel, and only start counting when you feel the weight of the stage moving."

That's definitely... an answer... It's a crummy one, but I guess you just have to be aware of it...

I might be wrong about that, to be honest. I'm just an unemployed student. But I have some potential solutions to test towards the end for those interested.

In the meantime, the build!

Step 4: I Made a Bad One So That You Wouldn't Have To!

We're going to skim over the actual build because it's bad. I didn't want to fork over the money to make it easy and good. That would have been perfectly fine, if I took my time, flattened all the surfaces, checked for square... Here's what I did!

You see that metal bracket! I used a lot of those. They weren't what I wanted, but they were \$0.38 apiece, compared to \$15-30 for the things I planned on using. I ended up buying 8 brackets, but I didn't use them all. You, dear reader, shouldn't use any of them!

The main parts of the stage actuator are the guide rails/tracks, the stage support, and the screw system.

I used 4 of the brackets to make a rail system. 2 of them, closest to the camera in picture 3, furthest in picture 4, were effectively just spacers. Now we have a rail system.

I soldered the nut to a fender washer, then the fender washer to a hole in another bracket. The only advice I'll give for soldering is to use flux for a good product, don't use a can as shown in pictures 2 and 3, because the thermal expansion might cause it to spring hot acid at your face, and of course, use eye protection. The part I made in this step is what is being advanced by the screw.

The moral is to be careful. It doesn't take more than a few seconds to put on goggles. I'm sure glad I did.

Oh yeah, and feel free to use a different adhesive for this step.

Step 6: And That's All There Really Is to It!

The final product was... Ugly...

I should mention, the main bolt that this whole thing is based on should be a carriage bolt. That way it can create its own square pocket to sit in when you tighten it up. I drilled a partial hole a little bigger to avoid cracking my MDF, and it seems to have worked perfectly.

In the last picture of this step, you can see a fender washer. I drilled a hole to screw that in place to act as a rail, but my hole wandered on me. If done correctly, that washer would help guide the sled. In my case, it hardly comes into play.

Step 7: Backlash 2: Electric Boogaloo!

I mentioned dealing with the backlash earlier. Here are my (untested) thoughts.

If you put springs on the rails to push the sled.. thing... you might be able to get it to function while only riding on one edge of the screw threads. This would, in theory, eliminate the backlash from the part of the system we care about. It wouldn't be easy with my setup, so if you're trying to understand, look at the computer-generated image at the beginning.

Also, Ideally the wheel system would have a pointer instead of a mark on the edge as seen in this photo. If there was a pointer instead, you also might be able to get it to mechanically move to account for the backlash when the system changes direction.

I hope you all enjoyed, learned something, or at least had a chuckle at my bad jokes. Constructive criticism is encouraged!

Btw, the reason I don't have the circle divided is that my printer apparently hates refills.

Step 8:

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