Gear Tooth Geometry.

I feel like, with making the first two “Introduction to Digital Computers” videos I’ve gained a superpower. And I’ve used that power to help me visualize and solve another problem I was having, figuring out the shape of spur gears (and eventually, internal spur gears, the next major feature I’d like to add to the Kythera application).

Which led me to making this video.

As always, show notes give further details.

Let me know if you find any problems or have any questions.

And as always, thanks for watching.

A second pass at my earth/moon orrery.

I’ve come to realize making something from scratch winds up building a bunch of prototypes as you both perfect the plans and figure out how to make the parts.

And in my case, while the plans may be great–my ability to machine parts is… shall we say questionable?

At any rate, I have a second pass at the Earth/Moon orrery.

A few observations about my design.

First, the gears don’t quite have the axle holes dead center. It’s not that I don’t own a set of mill center drills, which are useful for starting a hole at the exact spot you want it at (rather than trying to use a drill bit and getting a hole wherever the damned bit bends to), it’s that I forgot to use them when I made the center holes of the gears.

So I have to remember when cutting a gear to use the following steps, which I document here because someone else may find these useful. (I’d make a video, but I’d need to license the Benny Hill Yakety Sax music.)

1. Measure and cut the blank for the gear from 1/8th or 1/16th thick sheet metal. (I’m using aluminum because it’s cheaper than ruing hundreds of dollars of brass.) The blanks should be slightly bigger than (N+2)/32 inches (for 32P gears, which is the size I’m cutting) in diameter. (I made the mistake once of thinking “radius.” That wasn’t good.) The blank should be drawn around a divot in the center made with a hand punch.

2. Set up an arbor for the gear. Take a piece of aluminum rod close in diameter to the gear we’re cutting, mount and center on the lathe, and face the arbor. This causes the end of the arbor to be flat relative to the cutting surface we’re cutting on.

Note that all of this is taking place on the little Sherline, because one of the steps will completely screw up the center otherwise.

3. Attach the blank to the arbor. Using a center on the tail stock and a dash of superglue, superglue the gear blank to the arbor. Use the tail stock center to help center and push the gear blank on the arbor, and allow five to ten minutes for the glue to set.

4. Cut the gear blank. On the lathe, turn the blank to the desired diameter. For 32P gears, this is the Outer Diameter of the gear, and is (N+2)/32 inches in diameter for a gear with N teeth.

5. Using a mill center drill bit, start a center for the axle. This is the step I kept missing, and that caused my gears to be slightly off center. The end result is a tiny little divot at the exact center of the gear blank, which then can be used to drill a hole without having the drill bit wander too far off center.

6. Drill the center hole the desired diameter. Enough said. In the case of the orrery, I have to remember one of the 52-teeth gears has a center hole of 9mm. (At some point once I’ve perfected the plans, I’ll upload the instructions for making this device.)

7. Re-mount the gear blank on the mill. I assume the mill has been set up for cutting gears, with the appropriate cutter mounted on the cutting arbor, and the rotary table set up at a 90 degree angle. At this point you’ll need to then verify the gear cutter is centered in the gear to be cut.

8. Start cutting gears. What I’ve been doing–to great effect–is to center the gear across the cutter, so turning the Y axis knob in front slides the gear into the cutter. The moment I hear the tell-tale sign of the gear cutter starting to cut the gear, I note the depth on the knob (all my mills and lathes are marked in inches), rotate the X axis to get the gear out of the cutter, and turn the knob into the cutter 0.0625 inches. This is the desired depth of the teeth.

(We get 0.0625 inches from the observation that if the outer diameter of a gear grows by 2/32″ for a 32P gear, this means the inner diameter must shrink by a corresponding 2/32″ as well. This means the diameter difference between the outer and inner diameter differs by 4/32″–and the tooth depth, the delta of the radius–must be half that, or 2/32″. That is, the tooth depth is 2/32″ = 0.0625″. By listening for the cutting, my guess is that I’m cutting the teeth a hair wider and a hair deeper than needed. On the other hand, this works out fairly well, given that the final mechanism I built, pictured above, rotates pretty freely with center holes precisely at the locations my Kythera program predicted.)

Once the gear depth is set, lock the Y axis, and start cutting gears by turning the rotary table the desired amount and sliding the gear blank through the cutter using the X axis.

I don’t have any CNC motors attached to my mill, only to the rotary table–so this gets pretty boring pretty fast.

9. If all goes well, separate the blank from the arbor. The technique I’ve seen which works well is to use a blowtorch and heat up the gear blank until the superglue releases. Try not to do this on cement, because you can cause the imperfections in the cement to pop, throwing small bits of cement at your face.

The different gears that are supposed to be attached to each other, by the way, I simply superglued together. This is probably not a great long-term solution, but in the short term it works very well.

I think with a little better technique I can cut the gears and mount them and have a functioning orrery. But the last gear, the thing that rotates around showing the position of the moon–that leaves a whole lot to be desired.

Originally I had built a 14 tooth gear using 1/8th inch thick aluminum, and then cutting a separate component:

Moon mount

This I then superglued to the 14-pin tooth, and I cut the 6mm end with a 6mm thread, which I then screwed a cap on top that holds the pin that will eventually support the moon.

And this went… poorly.

The stupid part is that in retrospect this should have been cut as a pinion:

Moon pinion

And the moon should have been mounted not by screwing the top piece on, but by using a grub screw.

Well, in a few days I’ll go back to cutting parts and seeing if I can’t machine something better. Meanwhile, version 2 of my orrery prototype.

Finally, at long last, I got my two 52-tooth gears for my Orrery!

It took another couple of tries, and the trick for these gears was to get the outer diameter to within 0.002″. Further, it required depthing the teeth to 1/32″ == 0.625″ (the precise measurement of the depth of the teeth) rather than cutting them a hair deeper. And I’m suspicious the Sherline divider may not be stepping the precise number of steps–but I have no way to prove that, and usually when I think equipment has gone south, it’s actually user error.

(Since, as I’ve said several times before, I have no idea what I’m doing here.)

And how did I get the blanks cut to the right diameter?

On the Sherline lathe, of course.

Because in the middle of the night last night I had a brainwave. I realized you don’t have to cut with the cutters straight on; you can turn the cutter so that it cuts off the carriage. Meaning if the cutting head is turned, the pointy bit that does the cutting is not over the carriage. Each wheel is only 1/8th to 1/16th of an inch thick, so it doesn’t take much. And of course be aware that you can crash the carriage into the wheel.

But that gives me about 3 inches and change of useful circle cutting power now–and that implies gears of up to 100 teeth at 32P. (For bigger wheels I’ll need to do something different–but my goal here is to design my own Orreries, and nothing I’ve designed so far require more than 80-something teeth.)

Anyway, good news, everyone!

Tonight will be some TLC with OpenSCAD, designing a new harness for the gears (so I can test to see how well they spin), and soon cutting the harness out of aluminum.

No, I haven’t forgotten about the gear cutting. I just haven’t been having a lot of luck.

It probably doesn’t help that I have no idea what I’m doing. Though I do have all my fingers and toes. So there’s that.

So today, after getting the M3 to M2 adapter, the M2 to Sherline adapter, and hooking up the Sherline 4-jawed chuck to the larger 7″ lathe, I went to round the blanks and get them ready for gear cutting. (While gears smaller than about 32 teeth or so make sense cutting on the Sherline, anything bigger than about 48 teeth just cannot clear the carriage. (The distance is 0.9 inches or so, which means the maximum OD of a gear that can clear the carriage is 1.8″–which implies a 55 tooth gear. That’s because

Max Sherline OD

(The OD of a gear of N teeth with a pitch diameter of D is (N+2)/D.)

And while I have two gears that are 52 teeth, my ability to cut blanks sucks. Meaning what I get aren’t circular blanks; what I get are jaggedety things that will inevitably crash onto the carriage. Now I’m sure I can probably reorient the carriage on the Sherline to cut the bigger blanks–but I figured “hey, I have a larger lathe; let’s try that.”

And then I got the first gear, the one on the right in the photo above.

Notice anything about the gear?

Here, let me show you a more detailed image:

Detailed Image Gear 1

Yep, the teeth at the top are nice and pointy; the teeth at the bottom are flat. That’s because in the process of transferring the gear and mount from the adapter on the 7″ lathe to the mount for gear cutting, the center was not preserved.

So the teeth on the top of the image were cut deeper than the teeth on the bottom.


This is, by the way, when I pulled out the measuring jigs and discovered that my 7″ mini lathe wobbles. By about 0.005 inches. Combine that with the error in the M3-M2 adapter (roughly of the same magnitude), and that explains the error in the shape of the teeth.

I could bitch about “cheap Chinese crap,” as have many on the Internet–but let’s be serious: I don’t know what I’m doing. Blaming the equipment seems a little premature here. Certainly dismounting a jig from one thing and mounting it on another–unless it’s very expensive, precision equipment that has been carefully calibrated by someone who knows what they’re doing–is probably not going to go very well.

And that tells me not to assume the precision of the equipment I’m using, especially if I need to cut stuff to tolerances of 0.001 inches.

It also probably doesn’t help that I’m using soft aluminum, which, if breathed on strongly, will easily deform 0.005 inches on its own.

So plan two.

Mount the arbor on a four jaw chuck on the 7″ lathe, turn, drill the center, then place in the four jaw Sherline chuck. And yes, the moment you unmount anything from a chuck you will never get it centered again unless you use a lot of fiddling.

It took me an hour to get the wheel centered on the second 4 jaw chuck for turning. Because newbie.

(To be clear, it was centered to within 0.002 inches. Since the teeth are cut 0.0625 inches (2/32 inch) deep, I figured an error of around 3% on the depth of the teeth was acceptable for test purposes. Besides, I’ve been cutting my wheels a hair smaller.)

For centering purposes I used the stepper motor on the rotary table and went back and forth for what seemed like forever.

Then I switched modes on the stepper motor divider (without resetting the device by turning it off and on), and started cutting teeth. And got the gear in the top image on the left.

Noice the problem?

Here, let me zoom in:

Detail Image Gear 2

See what looks like half a tooth?

Well, it appears the stepper motor and wheel assembly turned around slightly more than 360° when rotating the circle. Which left me with a useless gear, since the teeth are not cut 360/52 ° per tooth, but slightly more.

And I don’t know if it is because my arbor slipped or because I mistreated the stepper motor/rotary plate. Certainly I should have reset the thing.

So I’m left with a few possible plans for cutting the larger gears.

One is to somehow mount the cutting bit on the smaller Sherline lathe so I can cut the wheels without them crashing into the carriage. Without the carriage there is easily several inches of clearance–meaning I could cut gears with nearly 100 teeth fairly easily.

Another possibility is to build some sort of mount which I can use to mount the Sherline 4 jaw chuck to the larger 7″ lathe and center everything with precision. I’ve ordered a 3/4 16 threaded rod and some hardware to try this option.

Another possibility is to revisit plan two, but make sure the stepper motor and divider have been properly reset by power cycling the whole thing and letting it rest for a few moments. However, I sort of suspect the problem was not the stepper but the possibility that the cutting forces on my arbor caused it to twist in the mount as I was cutting the teeth. After all, my arbor is, well, rather soft aluminum. And cutting teeth certainly is more force than breathing on it.

But all of that is for another day.

On the other hand, I now have a much greater understanding of the shape of the teeth for gears–and now that I’ve gained a superpower in doing animated visualizations, I think I can visualize the math behind the shape of the gear teeth.

That means I think I can also visualize the math for the inner teeth of a planetary gear.

And I may put that together in the following weeks–to help explain why the teeth of a gear are what they are, and so I can rework my Kythera app to do planetary and inner gears.

Another day, another four gears. And tales of woe.

So one downside of the superglue arbor is that you really need to clean up the end of the arbor holding your gear–and you need to apply enough glue to get the gear to stick to the arbor. More than once today I had wheels fall off the arbor–and that led to… well, complications.

I also learned the amount of space above the swing arm of a Sherline lathe is… perhaps slightly less than an inch? This means it’s okay for doing small parts–and in fact, is much better at very small parts than the gigantic “Mini Lathe” that I also have. And in the future when I need to machine set screw spacers for my 2mm rods, I’m definitely going to do it on the Sherline.

But I have two gears that have 52 teeth; the outer diameter is (52+2)/32 = 1.6875″. Which means it can’t be cut to the proper diameter on the Sherline, since I’ve been cutting the blanks just a hair larger.

A quick search of the Internet coughs up a 3/4″ 16 threaded end which fits the Sherline 4-jaw chuck on one end, and has a MT-2 tapered end to fit the MicroLux 7×16 mini lathe on the other.

The good news is that I should be able to duplicate my process flow for bigger gears, but just use a bigger lathe.

And if it turns out the Sherline mill is too small for bigger gears–I do have the MicroMark “Miniature” milling machine.

Though after spending time with the Sherline, it’s hard to see the relatively gigantic MicroMark lathe and mill as “miniature”…

And I have all the other gears for my Earth/Moon Orrery “proof of concept.”

I may still not quite know what I’m doing–but at least I got a gear.

So I drew and cut out several blanks from the 1/8th inch aluminum (and I confess I had to do this several times, by the last blank I was very close to the line) using a scroll saw and a jeweler’s blade.


I then perfected them on the Sherline Lathe to the correct diameter of (29+2)/32 inches, and drilled a 2.5mm hole in the middle.


I made two mistakes; the first was that I got the depth of the teeth wrong; I figured the depth of the teeth would be 4/32 inches–which is the difference between the outer diameter and the inner diameter, not the difference in the radius. (So the teeth should be 2/32 inches deep not 4/32 inches deep.)

Go back, make another blank.

Second mistake: the three jawed chuck (shown in the photos) can’t really hold the load during cutting. The gear and arbor got loose, screwing up the gear. Replace with 4 jawed chuck. Re-cut mounting arbor (since old one was trashed).

Go back, make another blank.

But finally, I got my 29 tooth gear. And it even meshes properly with a commercially purchased 32p gear I bought from SparkFun:

IMG 4578IMG 4581

Only 6 hours of my life, too!

(But now I know the process I think I can shorten the time somewhat…)

One down, 6 to go for my Earth/Moon orrery.

I’ll be the first person to confess I have no idea what I’m doing here.

First, let me observe that I’m fascinated by all things which calculate. That means computers, of course–from a desktop computer to mobile devices to Arduinos and other 8-bit computers. But that also means wiring transistors which can calculate things, and that means mechanical calculators: Orreries, Tellurions, mechanical adders–all of these things fascinate me as well.

This is why I put together the Kythera application, and why I bought several books on computationally calculating gear dimensions: so I can eventually expand the Kythera application to cover other types of gear mechanisms and gear trains.

It’s why I’m putting together a video series on digital computers, building one from the ground up, from transistors.

And it’s why I’ve been slowly accumulating the stuff necessary to cut gears–so I can build my simple Tellurion, which I 3D printed earlier this year.

Now if you do machine shop stuff, if you know how to use a lathe and a mill in order to cut metal–well, I’m sure the next section here is going to be pretty cringy. Because I don’t know what I’m doing.

We all start from ignorance, and learn through experience.

So let’s go through the steps I performed today, and why I only have the broken bits of a gear to show for my efforts.


First, starting with a 1/8th inch thick sheet of aluminum (because I want to practice on something relatively cheap before I destroy a whole bunch of expensive brass), I used a DeWalt scroll saw, and some Pike Jeweler’s saw blades and rough cut a 1 inch disk of aluminum. I set the scroll saw on the slowest speed it would do–the theory that it would be easier to allow the machine to move the blade back and forth than to do it by hand.

I went through four blades before I stopped breaking them; the trick is to allow the blade to do the work rather than force it. (Fortunately the blades come 144 to a set. Unfortunately the bloody things are so microscopically small I had to use a hand magnifier just to identify which side had the teeth, and to figure out the orientation of the teeth. Hint: if you lightly rub your finger along the side without the teeth, it’s smooth to the touch. Along the side with teeth, it should ‘catch’ when your finger is running upwards; that means the cutting motion will be downwards as the blade moves down through the hole in the table.)

Once I got a rough blank, I then super-glued it to my super-glue arbor made a couple of months ago.

And this is where things went south: I didn’t properly center the disk on the superglue arbor. Had I had used a tail center and the tail stock to hold the wheel in place as showed in this video, then I would have things centered correctly.

Live and learn, so I took the arbor and wheel out to the driveway, heated the whole thing up with a butane torch, and tried again.

IMG 4547

As an aside, I’m cutting a 29-tooth gear that is slightly less than 1 inch in diameter. And for this operation I’m using a Sherline lathe to turn the gear. Yes, I could use the 7″ lathe, but for smaller parts I wanted to try the Sherline that I inherited earlier this year to see how well it works out.

I managed to turn the wheel down to the “correct” diameter (*cough*) without any real incident.

(As a note, the outer diameter for a gear is typically 2 plus the number of teeth; divide by the Diametrical Pitch and you get the correct outer diameter. So for our 29 tooth gear and 32DP gears, you’d calculate (29+2)/32 = 0.969″ for the diameter of the blank. Note that the teeth are then cut between 4/32 and 4.5/32″ deep.)

Next came mounting this on the mill to cut the gears.

So what I’m using is a Sherline mill which I inherited (but which had some rusted pieces I needed to fix). To that I’ve added a CNC Rotary Table Indexer (because hey, 29 teeth), a tilting table set at 90 degrees, a mill cutter arbor (specifically part #3236 for 1″ diameter cutters), which holds a 32DP gear cutter from these guys.

IMG 4542

Of course this process requires that I detach the 3 tooth jaw from the lathe and attach it to the rotary table–and I’m sure there are a bunch of machinists out there who are saying “no, no, no!” because once you detach something from the lathe, it will never ever center again.

Like I said, I’ll be the first person to confess I don’t know what I’m doing.

However, in this case, it appears to work since the 3 tooth jaw screws onto the lathe and screws onto an attachment on the table–so it’s “centered enough.” At least I hope it is.

Now of course my second mistake was to put the rotary table on the right. The gear cutter spins so it cuts from right to left–which means it is putting pressure on the back side of the gear, essentially “pulling it” off of my superglue arbor.

Which–six teeth into my cutting process–it did, throwing the gear part ways across the room.

The results… well, they’re not pretty.

I haven’t cleaned the superglue off the back; that’s the rings. The gear teeth are not quite right; turns out I turned the wheel down to the wrong diameter. (I turned it down to (29+1.5)/32 inches OD, on the theory that “if it rattles it runs”–but I think I should stick to the tried and true first and if I fudge the OD, perhaps fudge it more like by 0.005 inches and not by 0.015 inches.) And of course my little coin-sized gear is pretty rough; I didn’t bother cleaning it up because after this picture it’s going into the scrap bin.


But it did mesh correctly with a 32DP gear I bought from SparkFun. And it undeniably proves I’m at least on the right path.

Tomorrow I’ll try again.

The Earth-Moon Orrery: another update.

So the biggest problem, apparently, when it comes to cutting gears, is just getting all the fiddly bits together to make your gear cutting system.

In my case, I’m starting with a Sherline mill, and I’m adding a CNC rotary table, a 90 degree table, and a 1 inch arbor to mount the gear cutters with. The gear cutters have showed up; and I bought gear cutters from #2 to #7; #1 is for the tiniest gears I will probably never need to cut, and #8 is for gears with 135 teeth and more. If I decide to make a clock, well, then I’ll buy the other cutters as needed.

(The CNC rotary table is for the simple reason that, unlike folks making clocks, an Orrary–especially one of my own design–has some rather odd sized gears. A dividing table that can help cut a 29-tooth gear or a 52-tooth gear is probably not all that common. But for a CNC computer, 29 or 52 is just another input.)

Well, all the parts have arrived, I’ve mounted the Sherline mill, everything fits, everything appears to be in working order–and I’m ready to cut gears, just in time to be out of town for a week.

Which is just the way things work out, I guess.

But I’m all set up for my gear-cutting debut–and when I get back we’ll start cutting the gears for the Earth-Moon Orrery.

I promise I’ll start cutting gears soon–just as soon as life stops interfering. And when I do, I’ll walk through all the steps I took–including the missteps–and I’ll also include a list of the stuff I ordered, so if you want to try to reproduce what I’ve done, it’ll be easier than squinting at otherwise well produced videos to try to divine the setup.

The Earth-Moon Orrery: a quick status update.

My ultimate goal is to recreate the 3D printed object in brass, creating a small little tabletop Orrery for display. That means, of course, cutting gears.

So there has been a little bit of a pause while I order the correct things for gear cutting. Ultimately my goal is to set up a Sherline mill as a gear cutting system. I have 32DP gear cutters on order–which means I’m going to be resizing the machine I make even smaller than its current 3D printed size. And that means a lot of waiting and tinkering–since I’m a complete noob when it comes to metal working. But my goal is to learn along the way.

I also have some plate aluminum on order; my theory is learn on the cheap stuff–though I bet I’ll waste a lot of (more expensive) engravers brass when it comes time to cut the final gears.

Meanwhile, I do have a 7″ lathe and mill, both from these guys, which I bought a couple of years ago (but circumstances in my life meant they got to sit around gathering dust), and it’s time to cut the gear arbors.

I’m using something I first saw on Clickspring. The principle of cutting a gear is more or less the same: cut a roughly circular blank out of sheet metal, super-glue it to a gear arbor, then true the blank to the correct diameter. Mount on the Sherline (and yes, I know; this means a lot of fiddling to re-center the blank), align the cutters, and start cutting gears.

So today I’m making my superglue gear arbors.

For our orrery we need three arbors: one that is 1.5 inches in diameter (to hold the 52-tooth wheels), one that is 0.75 inches in diameter (to hold the 26- and 29-tooth wheels), and one that is 0.375 inches in diameter (to hold the 14 tooth wheels).

Each are made from aluminum stock I have on hand; the 0.375 inch arbor was turned on a lathe down from 1/2 inch diameter aluminum stock. And while my arbors may not have the same fit and finish as the Clickspring arbors, hey; I’m still learning how to do this stuff. My hope is that by the time I get to working with brass, I’ve learned enough from working with aluminum to get the fit and finish right.

As a footnote, since I’m using 32DP gear cutters (which has the nice property that they should work well with stock gears from Sparkfun,) this means in the future if I get interested in doing some robots work, I’m well equipped to cut my own custom gears in case I need a custom tooth ratio.

But it does mean going through and resizing the gear parameters in Kythera for the Earth-Moon Orrery.

After doing a little research I decided to cut the gears from 1/16″ thick brass sheet and 1/8″ thick brass sheet. A little fiddling with the settings for the document, but setting the gear and layer thickness to 1/16″ and 1/8th”, respectively, and I’ve come up with a tentative update to the Earth-Moon orrery design, which you can download here.


The test print of the Earth-Moon Orrery

So after a couple of days of printing, and a little sanding, some assembly, and not a tiny bit of cursing, I have a printed orrery gear assembly for our Earth-Moon Orrery.


It’s a little stiff, but that’s to be expected since I haven’t perfected the gears through sanding. (That means carefully going through each tooth with a tiny little strip of sandpaper and polishing down all the little rough edges.) Of course the mechanism also needs to set for a few days; I’ve found 3D prints out of the FormLabs Form 2 printer generally take a couple of days to set before the parts seem totally cured. (Of course I could buy the system FormLabs sells to harden parts–but I haven’t.)

The ideal way to build this mechanism, of course, is out of brass–and the next thing for me to do is figure out how to cut brass gears and replicate this mechanism in aluminum and brass.

But as of now, our very simple little mechanism works.