Universal joint (Cardan/Hooke): transmitting rotation between angled shafts

When two shafts have to turn together but you can't line them up — because one enters the housing on one side and the other exits at a corner, or because you have a fixed motor and a moving output — a rigid coupling won't do: it either doesn't fit, or it flexes until it breaks. The universal joint solves that by bending the drive. It's the cross that connects two yokes and lets torque cross an angle, the same part that has spent a century carrying engine torque to the rear axles of automobiles. It works, it transmits real torque, and you can print it fully assembled in FDM (print-in-place). But it has a kinematic catch that almost nobody warns you about until you've built it and notice the output isn't turning the way it should, and it has four small spots that decide whether the part lasts or wears into uselessness. That's what this article is about.

The cross: how torque turns a corner

The geometry is deceptively simple. You have two yokes, each fixed to its shaft, with its arms spread into a U. Between them sits a cross: a cross-shaped part with four trunnions, two in each arm, and the two arms are perpendicular. One yoke pivots on one pair of trunnions; the other yoke pivots on the perpendicular pair. So the input yoke drives the cross, the cross drives the output yoke, and because each yoke is only required to turn about its pair of trunnions, the assembly can bend: the two shafts can form an angle of several degrees and still keep turning together.

The key is why you need four pivots and not two. If you joined the two yokes by a single axle, they could bend in that plane but not in the perpendicular one, and as the joint turned it would jam. The two arms of the cross, crossed at 90 degrees, give the cross the two rocking degrees of freedom it needs to follow the rotation while the angle between shafts stays fixed. Every full turn of the input shaft forces each trunnion to swing back and forth inside its yoke, tracing a small oscillation; that rocking is the price of turning the corner, and it's also the root of the problem coming up next.

It's not constant velocity: why it oscillates

Before you use a Cardan joint for anything serious, understand this: a simple universal joint does not transmit rotation uniformly. If you turn the input shaft at constant speed, the output shaft does not turn at constant speed. It speeds up and slows down twice per turn, and the amplitude of that oscillation grows with the angle between the shafts. At a few degrees it's negligible; at twenty or thirty degrees it's perfectly noticeable, and the output delivers a pulsing rotation.

The reason is purely geometric. When the shafts are bent, the plane in which the input yoke's trunnions oscillate is tilted relative to the output's. Over one turn, the projection of that motion onto the output shaft isn't linear: for half a turn the trunnion runs ahead of the output and for the other half it lags behind, so the output sometimes leads and sometimes trails a perfect rotation. The instantaneous output velocity oscillates about the input velocity with a period of half a turn — two cycles per revolution — and the larger the angle, the more pronounced the oscillation. It's not a manufacturing defect or slop: it's the kinematics of the Hooke joint, and you would see it identically in a machined joint held to micron tolerances.

You can even put a number on it. With the input shaft at constant speed, the output velocity varies between cos β and 1/cos β, where β is the angle between the shafts. That gives you a concrete bound on how much the output pulses: at 10 degrees the velocity deviates by at most ±1.5%, at 20 degrees it's around ±6%, and at 30 degrees it exceeds ±15%. Below about 10 degrees almost nobody notices; past 20 degrees it's already audible vibration. The threshold at which the pulsing stops being negligible depends on your mechanism, but that progression — growth roughly as the square of the angle — tells you why a small angle buys you a lot.

The classic solution is to use two joints in series — the double Cardan — mounted so the second joint's oscillation cancels the first's. For the cancellation to be exact, three conditions have to hold: the two joints have to form the same angle, the yokes of the intermediate shaft have to be in phase (the two yokes of that center shaft aligned in the same plane), and the three shafts — input, intermediate, and output — have to be coplanar, meaning the assembly lies in a Z or a W and isn't twisted in space. If you respect that, input and output turn at the same instantaneous speed and the oscillation is confined to and compensated on the intermediate shaft. This is what lets a real driveshaft with two crosses deliver smooth rotation even though each joint on its own pulses.

The four trunnions: printing the cross without leaving it unsupported

The physically delicate part of printing a Cardan is the cross: four small pivots that have to turn and that, badly oriented, end up unsupported. You have two paths.

The first is to print in place (print-in-place): print the cross already nested inside its yokes, with a design clearance at each trunnion so they don't fuse together and turn as soon as you pull the part off the bed. It's elegant and asks for no assembly, but it forces an orientation that's rarely good for all four trunnions at once: the cross has two perpendicular arms, so if you leave one pair of trunnions as clean vertical cylinders, the other pair ends up horizontal and overhung. A horizontal trunnion printed without support comes out ovalized and sagged on top, and an out-of-round pivot binds and stutters with uneven play.

There are two tricks that rescue the print-in-place without pins. The first is to orient the cross diagonally, at 45 degrees: that way all four trunnions emerge symmetric, none fully hanging horizontal, each with a similar overhang and within what the machine holds on its own. The second is to give the trunnion a teardrop or rhombic profile instead of a cylindrical one, so the downward-facing side isn't a pure horizontal that collapses, but an edge the printer can lay down without support. With either one — or both together — a print-in-place comes out functional for smooth rotation; the trunnion no longer depends on a critical overhang.

The second path, when torque is involved, is to print the cross and the yokes as separate parts and join them with separate pins — printed or, better, metal. You lose the charm of the print-in-place, but you gain control: you orient each part so its critical bearing surfaces are well-formed vertical cylinders, and the trunnions stop depending on an overhang. If the joint is going to transmit meaningful torque, this is almost always the sensible path.

Whichever route you take, the golden rule when orienting is that no trunnion that has to turn should be printed as a critical overhang, and the cross shouldn't end up hanging in the air over the bed waiting for a support to hold its arms — because that support leaves a rough surface right in the contact zone that has to slide. The vertical cylinder is the roundest shape an FDM printer produces; reserve that orientation for the surfaces that roll.

Clearance, torque, and where it breaks

A Cardan has four pivots in series between input and output, and that sets the quality of the transmission: the play adds up. The backlash — the angular play you feel if you hold the output and rock the input back and forth without the output responding — isn't that of one pivot, it's that of all four accumulated. The linear play you give each trunnion so it turns freely reappears at the output as an angle — that play divided by the trunnion radius — and the four contributions add up. That's why the clearance on a Cardan's trunnions is tuned finer than that of a lone pivot: you need them to turn without binding, but anything beyond that shows up at the output, all four contributions summed. The method for fixing that number — printing a coupon and measuring what clearance slides without play on your machine — is the same as in Tolerances for moving parts, only here the play budget is stricter because it accumulates.

The most effective remedy against accumulated play is metal pins in the trunnions. A smooth metal rod of whatever diameter you have on hand — a hinge pin, a cut nail, a piece of straight steel — in each arm of the cross gives you a real cylindrical bearing, round and to a repeatable dimension, instead of a printed cylinder that comes out out-of-round and rough. Play and friction drop, strength rises, exactly at the point where the main failure mode lives. Housing those pins with the right fit is itself an embedded-hardware problem; you'll find it in Embedded hardware: magnets, bearings, and inserts.

The failure that truly breaks a printed Cardan is the fracture of a trunnion under torque. At any instant, torque doesn't pass through all four trunnions at once, but through the pair on the arm that's loaded in that position; over the turn, the load alternates from one arm of the cross to the other. And those trunnions are thin by geometry: all the torque that has to be carried passes through them in shear and bending. A trunnion printed entirely in plastic, further weakened if it came out as an ovalized overhang, gives way there before anywhere else: it's the smallest section under the highest load. That's why, if the joint moves real torque, the metal pin isn't a luxury but the difference between a part that lasts and one that snaps an arm off the cross at the first overload.

Three other failure modes remain. Accumulated play in the four pivots, which grows with use as the plastic-on-plastic contact wears. Vibration from non-uniform velocity, if you bridged a large angle with a single joint, which on top of that fatigues the trunnions by loading them in pulses. And a failure specific to print-in-place at speed: frictional heat. Two plastic surfaces sliding against each other under continuous rotation heat up, and PLA softens at low temperature; the trunnion wears fast or seizes outright. That's what limits an all-plastic printed Cardan to slow rotation, and one more reason for the metal pin when the part is going to turn continuously.

When it's the right part

Use a universal joint when you have to transmit rotation between two shafts you can't align: a steering column dropping to the rack past the dashboard, a drive that enters a mechanism at an angle, a crank that drives something offset past a corner. Where a rigid coupling would demand that the two shafts be collinear and a flexible joint wouldn't transmit enough torque, the Cardan turns the corner and keeps the torque.

The angle can even vary while the part is working — the Cardan tolerates it without seizing — but don't treat it as a free benefit: a changing angle makes the amplitude of the pulsing change too, and it breaks the "same angle in both joints" condition the double Cardan depends on to cancel. If the angle moves, the pulsing moves with it and there's no longer an assembly that compensates it cleanly. Treat it as a complication, not a feature.

And keep in mind what it isn't: it's not a constant-velocity coupling. If you need the output to follow the input degree by degree without pulsing — metrology, synchronizing two shafts, anything that turns fast — a single joint at an angle won't give you that; you need the double Cardan in phase or a constant-velocity joint, which is another part with another geometry. The Cardan joint is for transmitting torque through an angle, not for delivering instantaneous angular precision. If you get that right, along with the orientation of the trunnions and a metal pin wherever there's torque, you'll have one of the most useful drives you can print.

Before you fix the clearances of the four trunnions, calibrate your machine and work out the clearance per side: Tolerances for moving parts takes you from function to number, and here that number decides how much play the drive carries.