The joint that is not constant velocity

Two shafts that do not line up, and a cross between them. It is the joint at each end of a car’s propshaft, at the corner of a socket-set extension, at the head of a hand drill’s flexible drive. Hooke described it in 1676 and Cardan wrote about the mechanism before that, and its usual name in engineering is whichever of the two the textbook prefers.

The thing almost everyone is told about it is that it transmits rotation between misaligned shafts. That is true. The thing almost nobody is told is that it does not transmit it evenly.

It is a four-bar #

Before anything about speed, it is worth seeing what kind of mechanism this is, because the answer is unexpected and it makes everything else easy.

Strip the joint down to its axes. There are four of them: the input shaft, the first pin of the cross, the second pin of the cross, and the output shaft. Each is a revolute joint. They form a closed loop — going input shaft, first pin, second pin, output shaft returns to the frame the two shafts are mounted in.

Four revolute joints in a closed loop is a spatial four-bar. And all four axes pass through one point, the centre of the cross, which makes it the special kind called a spherical four-bar: every point of every link moves on a sphere about that centre.

That gives it its geometry in three statements. The first pin is perpendicular to the input shaft. The second pin is perpendicular to the first. The output shaft is perpendicular to the second pin. Everything else about the joint’s behaviour follows from those, plus the angle β between the two shafts — and there is nothing else to specify.

Kutzbach counts a four-joint spatial loop at −2 degrees of freedom. This one is in tens of millions of vehicles. The screw system explains it: four axes through a common point have screws with no translational part at all, so the 6 × 4 matrix has rank 3 rather than 4, and the mobility is 4 − 3 = 1.

What the output actually does #

Set the input shaft angle to θ and solve the loop for the other three joint angles. The output shaft’s rotation φ comes out of that solve, and comparing it with θ gives the joint’s real behaviour.

The classical relation, which Cardan’s contemporaries had:

$$\tan\varphi = \frac{\tan\theta}{\cos\beta}$$

At θ = 0 and θ = 180° the two agree. At θ = 90° and θ = 270° they agree again. Everywhere else φ ≠ θ, and differentiating gives the instantaneous speed ratio:

$$\frac{d\varphi}{d\theta} = \frac{\cos\beta}{1 - \sin^2\!\beta\,\cos^2\!\theta}$$

which runs between cos β at its slowest and 1/cos β at its fastest, twice per revolution.

The two routes to that curve are worth separating carefully, because they are the site’s habit and because it is easy to make them the same route by accident.

The closed form is a statement about a spherical triangle. It never touches the solver.

The measurement takes the loop, drives the first joint to θ, closes the loop with Newton–Raphson, reads the fourth joint’s angle, and does it again a hundred-thousandth of a radian either side. The derivative comes from differencing two solved positions. Nothing in that path knows about Cardan.

Across ninety-six positions at each of three shaft angles, the worst disagreement between the two is about 10⁻¹⁰ — which is the finite-difference error, not a physical discrepancy, and it is the number the figure prints rather than a claim that they “agree”.

There is a sign in that comparison which is easy to get wrong and does not announce itself. Traversing the loop passes through the output shaft’s joint in the opposite sense to the one an observer watching the shaft turn would use, so the solved angle is the negative of the output rotation. Get it wrong and the two routes still agree on the magnitude of the ratio; the figure would show a joint running backwards and every printed number would be right.

Where the variation comes from #

The formula is easy to quote and hard to feel, so it is worth saying in words what the mechanism is doing.

The first pin of the cross is carried by the input yoke, so it sweeps a circle in the plane perpendicular to the input shaft. The second pin is carried by the output yoke and sweeps a circle perpendicular to the output shaft. Those two planes are at β to each other, and the two pins are rigidly at right angles in the cross.

When the first pin lies in the plane containing both shafts, it is at its furthest from the output shaft’s plane, and the cross has to be tilted most steeply to keep the second pin square to it. A quarter turn later the first pin is perpendicular to the shaft plane, the cross sits square, and nothing is tilted at all. The output yoke is therefore being pulled round by a lever whose effective length changes twice per revolution — long at the tilted positions, short at the square ones — and a lever of changing length turns at a changing rate.

That is why the period is half a revolution rather than a whole one. The geometry repeats after 180°, because tilting one way and tilting the other are the same amount of tilt.

It also explains the four crossings. At 0°, 90°, 180° and 270° the instantaneous ratio passes through 1 on its way between cos β and 1/cos β, and at those four instants the joint really is a perfect coupling. It is not at any other instant of the turn.

Grashof, on a sphere #

A planar four-bar is classified by Grashof’s condition: compare the sum of the shortest and longest links against the sum of the other two, and the answer says which links can rotate fully. There is a spherical analogue, and applying it to this mechanism explains a practical limit.

On a sphere the links are not lengths but angles — the angular separation between consecutive axes. For the universal joint those are 90°, 90°, 90° and β. Grashof’s spherical form compares those angles in the same way, and it says the joint’s shafts both rotate fully as long as β stays below 90°.

At β = 90° the mechanism reaches a limit and stops: the two shafts are perpendicular, the formula’s denominator 1 − sin²β cos²θ goes to zero at θ = 0, and the ratio goes to infinity. That is a genuine singularity, and it is exactly the kind the planar essays call a toggle — the mechanism arrives at a configuration where an infinitesimal input produces an unbounded output, and it seizes.

Nobody builds a universal joint at 90°, but the reason the practical limit is usually quoted as 30° or so is not a safety factor away from that singularity. It is the vibration, and the vibration climbs long before the geometry gives out.

Between those two positions the output runs through its whole range, and the quarter turn is where it is fastest — so the joint is worth seeing at that angle as well as at the two extremes.

Why the spread is what it is #

The extremes are cos β and 1/cos β, so the ratio between fastest and slowest is exactly 1/cos²β. That is a better thing to check than “the ratio varies by a lot”, and the check in the library is against that number rather than against a threshold.

It also gives the practical rule directly. At 5° the spread is 0.8%. At 15° it is 7%. At 30° it is 33%. At 45° it is a factor of two: the output shaft is doing half speed at one point in the turn and double at another, ninety degrees later.

Nothing about that is a defect in manufacture, and no amount of precision reduces it. It is the geometry of the joint. A perfect universal joint at 30° has an output that varies by a third.

What it costs downstream #

A varying speed means a varying acceleration, and a varying acceleration on a shaft with inertia means a varying torque. The second derivative of φ is where the trouble is: the output shaft is being angularly accelerated and decelerated twice per revolution by the joint alone, with no load change at all.

At driveline speeds that is a vibration at twice shaft frequency. It is audible, it fatigues the components either side, and its amplitude climbs steeply with β because the ratio’s excursion does. This is the reason propshaft angles are kept small even when packaging would allow more, and the reason a lifted vehicle with steeper driveline angles develops a vibration it did not have before.

It is also a case of a pattern this site keeps meeting from different directions. A linkage’s velocity ratio is a function of configuration and not a number, and quoting the mean loses precisely the excursion that matters. A universal joint has a mean ratio of exactly 1 — over a full turn the input and output make the same number of revolutions, always, at any shaft angle. Averaged, it is a perfect coupling. What the average hides is the whole engineering problem.

That is worth putting alongside the gear pair, whose ratio is constant instant by instant and not merely on average, and which is constant because the tooth profile was chosen to make it so. The universal joint’s geometry has no such freedom: the cross is what it is.

Two joints, and the condition on the second #

The classical fix does not remove the variation. It cancels it.

Put two universal joints in series with an intermediate shaft. The first joint’s output leads and lags; feed that into a second joint at the same shaft angle, and its own leading and lagging can be arranged to be equal and opposite, so the far output tracks the near input exactly.

“Can be arranged” is doing the work, and the arrangement has two parts, both of which get got wrong in practice.

The two shaft angles must be equal. If the first joint works at 20° and the second at 15°, the cancellation is partial and a residual variation remains. This is why a propshaft installation is specified by its angles at both ends and not just by where the parts fit.

The intermediate shaft’s two yokes must be in the same plane. The first joint’s output is fastest where the second’s must be slowest, and that phasing is fixed by the orientation of the yokes on the shaft between them. Fit the intermediate shaft a quarter turn out and the second joint does not cancel the first — it adds to it, and the far output varies about twice as much as it would with one joint. There is nothing in the assembly to prevent it, and the parts fit either way.

This is the shape of a problem the site meets elsewhere: a mechanism with two configurations that are geometrically identical and functionally opposite, with no feature to distinguish them. A toggle and the worst transmission angle look the same in a still frame and are opposite situations; a propshaft phased correctly and phased wrongly look the same on a bench.

The constant-velocity alternatives #

If the variation cannot be removed from a single Hooke joint, and cancelling it with a pair requires conditions that are awkward to guarantee, the remaining option is a different mechanism.

A constant-velocity joint is one whose transmitted ratio really is 1 at every instant and any shaft angle. The geometric requirement is a clean one: the contact points between the two halves must always lie in the plane that bisects the angle between the shafts. A Hooke joint’s cross does not satisfy that — its pins are perpendicular to the shafts rather than sitting on the bisector — and the Rzeppa joint, which is what the front wheels of most cars have, arranges its ball tracks so that the balls are forced onto the bisecting plane at every angle.

The Rzeppa is not a four-bar and is beyond what the loop machinery here handles, so this site does not draw one. But the reason it exists is exactly the measurement in the figure above, and the fact that the fix required inventing a different mechanism rather than refining this one is the useful thing to take away.

A double Cardan joint — two Hooke joints back to back in one housing, with a centring mechanism that keeps the two shaft angles equal automatically — is the intermediate answer, and it is the pair-cancellation trick with the two conditions enforced by hardware rather than by the fitter.

What a still picture cannot show #

There is a reason the joint’s behaviour is so widely unknown, and it is the reason this site exists.

A universal joint drawn once, at any instant, is a perfectly correct picture of a coupling. Nothing about the drawing says the output is running fast. The pins are square, the yokes are where they should be, and the geometry is exactly what the catalogue shows. The variation is a property of the sequence of positions, and a single frame does not contain a sequence.

That is the same failure mode as a four-bar drawn rather than solved, where every individual frame is a plausible picture of a linkage and the animation is of a machine that would tear itself apart. Here the frames are all genuine — the joint really does pass through them — and what is lost is the timing. A figure that shows only positions can be entirely truthful and still leave the reader with the wrong belief.

The site’s answer is the one it applies everywhere: the claim about timing gets its own figure with its own measurement, and the measurement is taken by a route the claim does not control. Hence the dots in the ratio plot, which come from differencing solved positions and would disagree with the line if the line were wrong.

There is a second thing the still picture cannot show, and it is why the essay says “at this shaft angle” throughout. β is not a property of the joint. It is a property of the installation, and the same part behaves differently depending on how it is fitted — which puts the variation outside the manufacturer’s control and inside the fitter’s. That is unusual for a machine element and it is worth stating plainly: this is a component whose most important characteristic is decided after it leaves the factory.

The honest summary #

The universal joint transmits rotation between misaligned shafts, and it transmits the number of revolutions exactly. What it does not transmit is the speed, and the excursion is 1/cos²β between fastest and slowest — a third at 30°, a factor of two at 45°.

That is not a manufacturing tolerance, an approximation, or something that improves with better bearings. It is what four axes through a point do, and the solve says so at every position rather than the formula being taken on trust. Cardan knew; the fact that the joint is still routinely described as though it were a coupling is the part worth changing.

The comparison worth ending on is with the two mechanisms this site treats as the standard for a fixed ratio. A gear train’s ratio is a count of teeth and is exact by construction — the same integer every instant of every turn. A gear mesh holds it instant by instant because the involute was chosen to make the common normal stand still. The universal joint’s ratio is exact once per revolution and wrong in between, and there is no profile to choose: the cross is the mechanism. When a machine needs a constant ratio through a variable angle, the answer has never been to refine this joint. It has been to build a different one.

The condition on the second joint is the practical content and it is worth stating as the general rule it is. Two universal joints cancel each other’s variation only if the second is arranged to undo what the first did — equal shaft angles, and the two yokes on the intermediate shaft in the same plane. Get either wrong and the variations add instead of cancelling, which is worse than a single joint rather than better. That is a genuinely unforgiving arrangement: a correction that depends on an exact relation between two parts, with no margin and with failure in the direction of making things worse. It also explains why the assembly is drawn with a phasing mark and why a propshaft reassembled a quarter turn out is a fault a driver feels immediately. A cancellation is a coincidence held by assembly, which is the same species of arrangement as every overconstrained mechanism on this site — and it fails the same way, all at once, for a small error in the wrong place.