The Antikythera Mechanism 5: Back Gears

The back gears lie between the large b1 gear and the outputs on the back, consisting of the Metonic and Saros spirals and the Callippic, Olympiad and Exeligmos dials. There are two slightly different versions of the configuration, from the Freeth 2012 and 2021 papers. I used the more recent one, which differs from the other only in swapping the position of the gears on the n-axis. The back gears are better supported than the planet gears, as most of them are on axles running between the midplate and the backplate. However, as we will see, there are some issue with the stability to deal with.

The following diagram, S20 from Freeth 2021's supplement, shows the arrangement of the gears.

Some co-axial sets of gears such as f1 and f2 can often be combined into a single printed object. The f-axis is also an example of an axis with no output. The diagram shows its axle as not extending to the backplate. By extending it and adding a spacer, we can make it more stable. Similar considerations apply to the h and p axes. The m axis can't be extended in this way as there isn't enough space around n3. Some axes extend through the midplate, for example d and m. For these, the gears are printed as separate objects, connected by a hexagonal extrusion on one which engages with a hexagonal socket on the other. I experiments with setting the size of the extrusion the same as the socket and 0.1mm smaller. Very close tolerance like this are a bit of a guess with 3D printing. Different printers, slicers and filament may end up with the fit being a bit different. If they don't fit, they can be sanded down until they do, and if they are too loose, they can be glued. I have this value set as a variable in Fusion, so it can be changed across all the parts that use it. I also noticed that the fit varies from one orientation to another. When I initially printed the m1 and m2/3 gears, they fit was quite loose, but rotating them 30 or 60 degrees relative to each other gave an adequately tight fit. In the end all the gears with hex joins seemed to work fairly well, but for the m and d axes, I allowed for a M3 screw to help hold them together, if it should be needed.

The axes which come through the backplate to form the output have a hex socket on the end, which a pointer will plug into. Most of these axes consist of two parts: the gear with an extended hub meeting up with the midplate, and an axle plugging into the gear and extending through the backplate. The n-axis is a bit different due to an extra gear, and the i-axis has the axle going to the midplate rather than the backplate due to the tight space. In some cases, I added a ring round the axle of hub to keep it stable. Here is an example of the entire n-axis in cross section to illustrate:

In this version, the hub of gear where it passes through the midplate is printed. In a later revision, I replaced this with a dowel pin.

Both the midplate and the backplate are large, much bigger than will fit on my print bed. The design by FizzyChickens splits these into smaller pieces which are clipped together and glued. It's still a lot of material to print, and large flat pieces like this are prone to warping. I decided on a different design, where I made a skeleton with just the necessary fixing points and enough mechanical support to make them rigid. These still have to be printed in two large parts, plus a circular bracket on top and two smaller brackets. The result is quite robust and also takes a lot less print time. Here is what it looks like:

A few M3 screws hold it together. Alternatively, it could be glued, maybe using M3 screws temporarily to keep everything aligned until the glue takes effect. There are two small brackets, intended to go underneath the frame. I later got rid of these as I don't think they are needed and they take up valuable space. The external back panel will need to be made from a complete panel. Maybe laser cutting it would be best.

The e3 gear is too large to fit on the print bed. Initially, I intended to split it up the same way that I split the large gear in the Swingtime clock, by dividing it in half and gluing the parts together with metal pins to help the alignment. The e3 gear is attached to another large gear called e4, and this is small enough to print as a single piece. So I decided to split e3 in two and attach it to e4 with screws along the spokes, plus a few extra ones on a tab. I arranged the position of the split so that it does not divide the hub in two, which makes for smoother movement. The three pieces look like this:

In a later revision, I also added some horizontal pins to the halves of e3 for alignment and extra stability.

Did it work?

The prototype back gear assembly worked somewhat, but not very well, as you can see in this video:

The calendar and eclipse prediction gear trains work well individually. They are supposed to work together, connected via the e-axis gears, and driven from b2, which is attached to the back of b1. In the last segment of the video, you can see that the mechanism seizes up. This is caused by (I think) a combination of friction and the larger gears tilting. In particular the very large e3/e4 gear and the d2 and m1 gears tilt a lot. The friction can be reduced a bit by lightly sanding the gear teeth as well as the parts of the gears that run through the frame.

The gear tilting is the bigger problem. The historical Antikythera mechanism includes supports near the rim of these gear which help keep them level. Although this increases friction locally, the net effect is to keep everything in alignment. I decided to try this out in two ways. For e3/e4, I added small brackets to the frame which lightly hold the rim of the gear. For d2 and m1, I added a thin flange to the gears themselves. In the HAM, supports attached to the midplate are used for both. The results are a bit better:

It still has a way to go, but is getting closer.

More changes

I made most of the gears 3mm thick in this version with just a few either 4mm or 5mm. I think it's better to make more of them at least 4mm. For the gears with a hub through the midplate, but which do not have a gear on each side (for example the n-axis), I had made the hub 10mm in diameter. There isn't really a need to do this, and for the next iteration, I replaced the hub altogether with a 3mm dowel for most of these. The e3/e4 brackets in the video hold the gear on both sides, when one side will do.

I broke the b3 gear tube during testing. It will eventually be much longer and go all the way through the planet mechanism. I'm not even sure this can be printed reliably and so I will probably replace it with a brass or steel rod. Similarly, I might replace some of the printed tubes in the planet mechanism with brass tubing. I've not decided about this yet.