MĒCHANĒ

MĒCHANĒ is a mathematical mechanism designer that runs in your browser. You pick a mechanism – a pair of gears, a linkage, a cam – adjust its parameters, watch it move, and eventually end up with a set of parts you can send to a 3D printer. Unlike a vase, these parts are expected to actually do something once they’re assembled, which raises the stakes somewhat.

It started with MATHĒMA. Turning equations into vases worked rather better than I’d expected, and once something works, the obvious next question is whether something harder would work too. A vase is a single object that just has to stand there and look good. I wanted to know whether the same approach would hold up for things that move: several parts that all depend on each other, where a small mistake doesn’t just look a bit odd, but stops the whole thing from turning. MĒCHANĒ is me finding out.

So far, it does gears, linkages, an intermittent drive, and cams. The gears are the part I’m most pleased with: spur, helical and herringbone pairs, internal gears, racks, planetary sets, clock-style cycloidal gears, and – my favourite – noncircular ones, where the gears are elliptical or lobed and the speed changes as they turn. You can even draw the speed ratio you want and let it work out the gear shapes. The teeth are cut the way a real gear would be: a simulated cutter rolls along the gear and the tooth is whatever is left over afterwards, undercut and all. On the linkage side there’s the four-bar in all its varieties, the slider-crank, the Peaucellier straight-line linkage (which still seems a little like a magic trick to me) and Theo Jansen’s walking leg. There’s also a Geneva drive and disc cams.

Everything runs through the same solver. You can let a mechanism play, scrub through the cycle, or just grab the input part and turn it by hand. The plots follow along, and clicking a plot moves the mechanism to that point. Every parameter has a help text explaining what it does and what goes wrong when you push it too far – for example why a gear with nine teeth isn’t quite the free lunch it looks like.

Most of the work went into checking whether things actually work, rather than whether they look like they work. MĒCHANĒ places the gears against each other at lots of positions and measures the play between the teeth. It puts the follower on the cam it has just generated to see whether it really moves the way it’s supposed to. It samples the entire motion cycle for collisions, not just the starting position. And if a mechanism isn’t fully determined, it says so instead of making up a motion that happens to look plausible. An early version had the planet gears leave the gearbox after the first turn of the sun gear – the sort of thing I’d much rather discover on the screen than on the printer bed.

It also works backwards, at least for some problems. Give it a gear ratio and it looks for gear trains that achieve it – exactly, where that’s possible. Draw a path and it looks for a four-bar linkage that follows it; that’s always an approximation, and it tells you how close it got. Draw the motion you want from a cam follower and you get the cam.

The practical end is aimed at FDM printing. Parts come out as STL, 3MF or OBJ, or as SVG and DXF outlines, with the fits and clearances for your printer kept separate from the actual geometry. Every exported STL gets read back in and checked before you get it, and there’s an optional ground plate with holes for the axles, so you don’t have to improvise one. If you want to keep track of how a particular design came about, there’s a provenance record that lists what was proven, what was checked numerically and what was only simulated.

A word on the limits: everything is flat – two-dimensional, that is – and there are no forces involved. MĒCHANĒ can tell you that a mechanism moves the way it should, but not whether it will survive being used. A free account lets you save your projects on the server; everything else works without one.

The first version is now available at works.baumann.gr/mechane.

Have a play. Start with a preset, turn something, and see what happens. If a design gets rejected, the reason is usually more interesting than the design would have been. As with MATHĒMA, I’m not in danger of running out of things to improve.