The bevel family (straight, spiral, crown, hypoid): drive at 90 degrees

Up to now all your gears lived in the same plane: parallel wheels passing rotation between shafts that run in the same direction. But sooner or later motion comes in on one shaft and has to leave on another that's perpendicular to it — a drill press, a differential, a mechanism that turns the corner — and there the flat wheel is no use. You need a gear whose teeth sit not on a cylinder but on a cone. That's the bevel family: the way to turn a drive through an angle. And in FDM it hides two traps that parallel gears don't, one in the part and one in the assembly, and neither of them forgives.

The kinematics are two cones rolling without slipping

A cylindrical gear is understood as two pitch circles rolling against each other without sliding; the teeth exist only so they don't skid. The bevel gear is exactly the same idea raised a dimension: instead of two circles, two pitch cones that roll resting on their lateral surfaces. The geometric key, the one that makes it work, is that the apexes of the two cones meet at a single point: the point where the shafts cross. That's why, when you mount a bevel pair to drive through 90°, the two cones share an apex and their surfaces roll on each other with no net sliding along the contact line. Rotation comes in on one shaft, reaches the common apex, and leaves on the other, turned through an angle.

That common-apex condition has a consequence worth fixing before going on: the bevel tooth is not prismatic. It's taller and wider at the large end of the cone and narrows toward the apex, because the whole geometry of the tooth — module, height, thickness — scales linearly with the distance from the apex. A cylindrical gear you can extrude: the tooth profile is the same top to bottom. A bevel gear you can't; each section along the tooth is a scaled version of the one before. That's the root of nearly everything that complicates it, both to model and to print.

What doesn't change relative to the cylindrical gear is the law of transmission. The ratio is still the quotient of the tooth counts: a 12-tooth pinion against a 36-tooth gear gives you 3

, just as in parallel. The angle between shafts doesn't enter the speed ratio; it only decides the shape of the cones. What does change is how that total angle is split between the two cones, and it's not split equally except in the 1 case. The half-angle of each cone is set by the tooth ratio through trigonometry, not by direct proportion: for shafts at 90°, the tangent of the pinion's half-angle is the pinion/gear tooth ratio. That 12/36 doesn't split the right angle into 22.5° and 67.5°, but into roughly 18° for the pinion and 72° for the gear. Only when the ratio is 1 — two identical cones with a 45° half-angle, the case of miter gears — is the angle split down the middle.

Four variants of the bevel family

Within the family, what changes from one variant to the next is how the tooth is traced on the cone and where the shafts cross — or don't. Each choice moves two things: the smoothness of the mesh and the difficulty of the assembly.

diagram

The straight bevel is the simplest: the teeth are straight and point at the common apex, like the ribs of an umbrella. They mesh all at once: the whole tooth flank takes contact almost simultaneously, just like a spur gear. It's the easiest to model and to understand, and the noisiest: each tooth gives a small impact as it enters, and at speed that's hum and vibration.

The spiral bevel carries the teeth curved in an arc across the cone, so that contact starts at one end of the tooth and progresses toward the other as it turns. It's the same move as the helical versus the spur in parallel gears: the mesh is gradual, there's always more than one tooth in contact, and that makes it much quieter and smoother. The price is a geometry that's very hard to model and an axial thrust the assembly has to take. This last point is worth clarifying, because it's a common misunderstanding: every bevel gear thrusts axially — the cone's own taper generates a force that tends to push the pinion and gear apart, so the straight bevel too loads the bearings axially. What the spiral adds is an extra axial component whose direction depends on the hand of the spiral and the direction of rotation: depending on the combination, it can push outward (separation) or toward the apex (attraction), and this second case is the dangerous one, because it drives the cones into each other and tends to seize.

The crown gear (crown gear) is a curious limiting case: one of the two cones has opened up so much that it's gone flat, a disc with teeth on its face, and it meshes against a pinion with straight teeth instead of against another bevel. It has a notable but bounded practical mounting advantage: the axial position of the pinion stops being critical, because the pinion slides along its own tooth without changing the mesh. Watch out, though — that forgiveness is on one axis only: the axial position of the disc is still as critical as in any bevel. And all of it at the cost of tolerating torque and speed worse.

The hypoid breaks the common-apex rule on purpose: it offsets the shafts so they cross in space without intersecting, skewed, at different heights. In doing so, the pitch surfaces are no longer pure cones rolling without sliding, but hyperboloids, and a net sliding appears between the flanks along the tooth. That sliding is exactly what pays for the extra overlap and smoothness — and what in automotive use demanded special oils — but in a printed part the sliding is wear and heat, and the offset complicates the assembly. In FDM it rarely pays off; keep it as a concept and reserve the spiral when you're after smoothness.

The tooth face prints uphill

Here's the first trap specific to FDM, and it comes straight from the tooth not being prismatic. On a cylindrical gear you lay the part down with the axis vertical and the teeth come out perfect: each flank is a vertical wall, the beads stack up clean, and the envelope profile comes out as you drew it. The bevel gear gives you no such escape. Because the tooth grows from the apex toward the rim, its surface is inclined relative to the bed whatever orientation you pick, and that incline is exactly the one FDM handles worst.

If you print the bevel with the axis vertical — usually the most convenient — the tooth flanks come out like a sloping overhang: each layer juts out a little relative to the one below, and the curved tooth profile comes out stepped, faceted in steps the height of a layer. That faceting isn't cosmetic. The flank of a tooth is the surface that rolls against the flank of the other, and its quality directly governs the smoothness of the mesh: a stepped flank turns clean rolling into a succession of little bumps layer by layer. The finer the step — the lower the layer — the smoother, but you'll never reach the continuous flank of a cylindrical wheel laid down.

The alternative is to print with support under the overhanging flanks, or to reorient so the most loaded face is better supported, accepting that support leaves a mark right on the functional surface. There's no free orientation: on a bevel you always pay faceting on some flank. The decision is which flank you sacrifice and how far you drop the layer so the step doesn't show in the mesh. The same orientation logic that governs any moving part you have in Layer orientation for motion; on a bevel it weighs double, because orientation decides not just the tooth's strength but the noise and wear of the whole drive.

Backlash isn't the only gap: seat the cones at their depth

The second trap is in the assembly, and it's the one that ruins the most bevel pairs after they've been printed well. On a parallel gear, the only gap you manage is backlash: the play between flanks, which you adjust by moving the shafts closer or farther apart in their plane. On a bevel there's one more degree of freedom, and it's treacherous: the axial position of each cone.

Remember that correct mesh only exists when the two apexes coincide at the crossing point. That means each cone has to seat at an exact axial distance along its shaft — the mounting distance. And here's the key that ties the two gaps together: on a bevel pair, moving the axial depth is the way to adjust the backlash, just as on a parallel you adjust it by moving the center distance. If you push a cone too far in, the two overlap more than they should, the teeth mesh too deep, and the drive seizes: the bottom of one tooth's gap bottoms out against the top of the other and there's no clearance left to breathe through. If you leave it too far out, the cones barely touch at the tip of the teeth, contact reduces to a corner, the backlash shoots up, and the mesh comes out loose, noisy, and prone to skipping under load. The margin between the two extremes is narrow — tenths of a millimeter — and it's not controlled by the tooth geometry but by how the part seats on its shaft.

And there a problem arises that's worth facing head-on: tenths of a millimeter is exactly the order of magnitude of the typical dimensional error of an FDM part. A printed seat fattens up, a hole comes out tight, and those tenths eat into your mounting distance just as they eat into any fit. That's why a fixed printed stop — a shoulder modeled into the part itself — inherits the print error and rarely lands on depth the first time. What really works on an FDM bevel is an adjustable stop: a replaceable spacer of known length, a shim washer, a nut that advances on a thread. Something you can fine-tune tenth by tenth to seat each cone at its distance and, with it, fix the design backlash. The way to calibrate that gap is the usual one, only applied to the axial direction as well as the play between flanks: you have it in Tolerances for moving parts.

When you really need it

The bevel exists for one thing only: passing rotation between shafts that cross at an angle, almost always 90°, where two parallel wheels physically won't fit. If you can solve your drive with parallel shafts, do it: a cylindrical pair is easier to model, much easier to print, and much more tolerant in the assembly. The bevel is the tool for when the geometry forces you to turn the corner — motion comes in horizontal and has to leave vertical, or two shafts converge at an angle and you need one to drive the other.

Once inside the family, the choice is almost always between straight and spiral. Start with the straight bevel: it's the one that prints best and the easiest to get right, and for most low-speed mechanisms its noise is perfectly tolerable. Move to the spiral only when the smoothness and quiet pay off against the modeling difficulty and the extra axial thrust it puts into the bearings. Reserve the crown for when you want a forgiving assembly on the pinion's axis and the torque is modest, and leave the hypoid to the books: the offset between shafts that justifies it rarely shows up in a printed part, and the sliding that comes with it is wear that plastic doesn't thank you for.

And always keep an eye on the three failure modes that will visit you, in this order. First, the mesh mispositioned by an axial mounting depth that's loose or overshot — the hardest to diagnose, because the part itself is fine. Second, the faceted flanks from a print orientation that steps the tooth face, paid for in noise and accelerated wear. And third, with everything mounted right, the tooth breaking under torque: the fatigue crack starts at the root, near the inner end, where the tooth is thinnest and the stress concentration highest. Beware the temptation to add teeth to the pinion to spread the load, because more teeth at the same diameter means thinner teeth, and a thinner tooth is more fragile, not stronger: you win load-sharing with a good contact ratio and a pinion with enough teeth to avoid undercut, not by cramming teeth until they weaken. Get the orientation right, fix the axial depth with a truly adjustable stop, and size the torque so no tooth works at the limit, and the bevel will turn your drive through the corner without complaint.