Two errors, one ellipse

Constant velocity is a mirror found that a double Cardan joint keeps its output exactly in step with its input for one reason: a reflection that swaps the two ends of the joint also carries every configuration of it onto itself. It found too that the argument is all or nothing. A joint that is almost symmetric has no symmetry, and the argument has nothing to say about it.

Every real joint is almost symmetric, which is the position a parallelogram a micron wrong is in at its change point: the exact argument has lost its footing and the machine has not noticed yet. Its two working angles are equal only as well as the vehicle’s ride height and the propshaft’s installation allow, and the yokes on its middle shaft are in line only as well as the splines were cut and assembled. The sweeps in that essay showed the lag spread growing in proportion to either error, from nought, and stopped there: nothing in the symmetry argument predicted the slope.

It left a suggestion about where the slope would come from. An error splits into the part the mirror carries onto itself and the part the mirror reverses, and only the second can move a configuration off its own image. This essay follows the suggestion to the end. Both slopes come out in closed form, the errors that cost nothing are identified, and the way the two costly errors combine turns out to be exact geometry rather than statistics.

An ellipse of equal lag #

The joint throughout is the W arrangement from the mirror essay: an input shaft and an output shaft meeting the middle shaft at working angles near 15°, bent the same way, so that the input and output lines meet. Two things can be wrong with it. The middle shaft’s two yokes can be turned out of line by a phasing error δ. And the two working angles can differ, by ε.

The measured spreads, from thirty-five joints solved from their pins with no formula for the lag, lie on a family of ellipses:

$$\text{spread} \approx \tan a\,\sqrt{(\sin a\,\delta)^2 + \varepsilon^2},$$

with a the mean working angle. The worst of the thirty-five is 0.06% off it, at a phasing error of 3° with the angles a degree apart, where the next order is beginning to show. Along the δ axis alone the spread is 0.069° per degree; along the ε axis alone it is 0.268° per degree. The ellipse is 3.86 times as wide as it is tall, which is 1/sin 15°.

The rest of the essay takes that formula apart. There are three pieces to it: why only these two errors appear, where each slope comes from, and why they combine as a square root.

What the mirror does to each error #

The mirror of the W joint is the reflection in the plane that bisects the input and output lines. It carries the input shaft onto the output shaft, the first cross onto the second, and the middle shaft onto itself end for end. An error in the joint is a change to its geometry, and the mirror carries each change onto some other change. It can do one of two things with it.

Raise both working angles together by the same amount, and the mirror carries the changed joint onto itself. That change is even, and the joint keeps its symmetry exactly. The all-or-none argument then applies to it as it did to the perfect joint, and the joint keeps step at any size of change, not only to first order.

Make the first angle larger than the second by ε, and the mirror, which swaps the two crosses, makes the second larger than the first: it carries ε to −ε. Turn the second yoke of the middle shaft through δ relative to the first, and the mirror, which reverses the middle shaft end for end, turns it through −δ. Both of those are odd.

An odd error is the kind that can move a configuration off its own image, and at first order its effect on the lag has to change sign with the error. Every error a real joint can have is a sum of an even part and an odd part. So the lag at first order is set entirely by the odd part, and the even part adds nothing until the second order at the earliest.

The figure checks both halves of that. Raising both angles together by up to 6° leaves the spread below 10⁻¹⁵ rad, which is rounding. The joint has kept its mirror and is exact. Splitting a difference ε evenly, 15° + ε/2 and 15° − ε/2, is a pure odd error, and the spread follows the line. Putting all of the difference on one side, 15° + ε and 15°, is the same odd error plus an even rise of ε/2, and the spread follows a line whose slope is tan 15.25° instead of tan 15° at a 0.5° error: the even half has only moved the mean angle the formula is evaluated at. At one degree the one-sided joint’s spread is 3.5% above the even split’s. At six degrees, the far end of the plot, it is 21% above, because the even half has become three degrees of mean angle.

That is the whole content of the suggestion the mirror essay made. It is also a practical statement. A double Cardan joint does not care about the size of its two working angles as long as they match, and the one thing a propshaft’s installer has to get right is their difference.

The slope for a phasing error #

The phasing error’s slope can be had by an argument short enough to do in the head. A single cross at working angle a turns its output at a rate that swings between cos a and 1/cos a of its input’s, twice a turn; that is Cardan’s law as the joint that is not constant velocity drew it. In the double joint, the second cross’s input is the middle shaft. A phasing error δ is a shift of where the second cross thinks its input angle is, so to first order it changes the output’s angle by δ times the second cross’s rate, and changes the lag by δ times that rate less one. Over a turn, that rate less one ranges from cos a − 1 to 1/cos a − 1, so the lag spreads by

$$\frac{d\,\text{spread}}{d\delta} = \frac{1}{\cos a} - \cos a = \sin a\,\tan a.$$

At 15° that is 0.06935: a degree of phasing error spreads the lag by 0.069°.

The pins agree with the line to 0.02% up to 2° of phasing error and to 0.18% at 6°. The slope is small because it is second order in the working angle: sin a · tan a is about $a^2$. A shallow joint barely notices its phasing. At 5° the slope is 0.0076, so the yokes could be turned a whole degree out of line and the lag would spread by less than a hundredth of a degree.

The rule that the middle shaft’s yokes must be in one plane is right, and the mirror essay found that a quarter-turn out of phase is the worst setting there is. But the rule’s cost near the right setting is gentle, and it is gentlest exactly on the shallow joints that most propshafts are.

The slope for unequal angles #

The angle difference needs one more step. Write a cross’s law as $\tan\varphi = \cos\beta\,\tan\theta$, with β its working angle. Changing β by a small amount changes the output’s angle by

$$\frac{\partial\varphi}{\partial\beta} = -\frac{\sin\beta\,t}{1 + \cos^2\!\beta\;t^2}, \qquad t = \tan\theta,$$

and over a turn, as t runs through every value, the fraction $t/(1 + \cos^2\!\beta\,t^2)$ reaches its largest value, $1/(2\cos\beta)$, where $t = 1/\cos\beta$. So the output’s angle wobbles by $\pm \sin\beta/(2\cos\beta)$ per radian of error, and the lag spreads by

$$\frac{d\,\text{spread}}{d\varepsilon} = \tan a.$$

At 15° that is 0.268 per degree, and the even-split dots in the angle figure lie on it. This slope is first order in the working angle, where the phasing slope was second. So the ratio between the two, the price of a degree of phasing error in degrees of angle difference, is sin a.

Five angles, two arrangements #

The two closed forms were each derived for a W joint at one angle. A table checks them over the range a real propshaft uses, and in the other arrangement too.

At 5°, 10°, 15°, 20° and 30°, in both the W and the Z arrangement, both measured slopes are within 0.0002% of sin a · tan a and tan a, and the ratio is sin a. The Z arrangement, whose symmetry turned out to be a point reflection rather than a mirror, has exactly the same slopes as the W. That is what the split predicts, and it is worth noticing because the two arrangements move so differently: in the Z arrangement the output hub never turns relative to the input’s and only its offset goes round, the relative motion of an Oldham coupling. A point reflection also swaps the two crosses and also reverses the middle shaft, so it also carries δ to −δ and ε to −ε, and the second cross sees the same shifts either way.

The last column puts the slopes in proportion. One cross at 15° spreads the lag by 1.99° over a turn, and that is the error a double joint exists to cancel. A degree of angle difference brings back 0.268° of it, about a seventh, and a degree of phasing error brings back 0.069°, about a twenty-eighth.

Why they add as a square root #

Neither slope says what two errors do together. The simplest guess is that the spreads add, since each is a peak-to-peak swing of the lag and two swings can line up. The ellipse says they do not add. They combine as the root of the sum of their squares, for every combination, not on average.

The reason is in the shape of each error’s mark on the lag over a turn. A phasing error’s mark is a constant plus a cosine of twice the input angle; the constant does not spread anything, and the cosine has a half-height of 0.605 thousandths of a degree per degree of error. An angle difference’s mark is a sine of twice the input angle, with a half-height of 2.338. Where the cosine peaks the sine passes through nought, and where the sine peaks the cosine is at its middle. A cosine of amplitude p plus a sine of amplitude q is a single wave of amplitude $\sqrt{p^2 + q^2}$, and so the two spreads combine as the hypotenuse of a right triangle whose sides are the two spreads alone. The ratio of the two half-heights is 0.2588, which is sin 15° again.

Worst case and the square root found that a root sum of squares on a linkage’s tolerances is bought entirely with an assumption that the errors are independent. When they can line up, their worst case is their straight sum, and the root-sum-square figure is only a statement about how often the worst case happens. Here the square root is not bought at all. The two errors cannot line up, because the joint’s own geometry puts their effects an eighth of a turn apart. For these two errors the worst case is the root sum of squares, and a designer can use it with no assumption about how the errors were made.

A tolerance stack-up of the usual kind is taken at one position of the machine, as where a stack-up stops working takes it at each of 180. At one position every error’s effect is a single number, and whether two of them reinforce depends only on their signs. A spread over a whole turn is a different quantity, the difference between a curve’s highest and lowest points, and two errors whose curves peak at different angles cannot both be at their worst at once. Here they peak an eighth of a turn apart by construction. It would be a mistake to generalise from this. A third error with a mark of its own, a cosine of twice the input angle shifted by some other amount, would line up partly with each of these, and the three would no longer combine as a clean square root. The two errors of a double Cardan joint happen to be exactly orthogonal because one moves the second cross’s input and the other its law, and those two changes land a quarter of a period apart.

What a lag budget allows #

A specification works the other way round from the slopes. It fixes how much lag spread the driveline can tolerate and asks how much of each error that permits. That is the question a length is a range asked of a linkage’s bars, and the answer has the same form as the four lengths do not matter equally: not one tolerance for everything, but a ranking of which error the output is sensitive to.

For a spread of a tenth of a degree the limits are $0.1°/(\sin a \tan a)$ for phasing and $0.1°/\tan a$ for the angles. At 15° a joint may be 1.44° out of phase or have its angles 0.373° apart. At 5° it may be 13.1° out of phase and still meet the budget, and its angles may be 1.14° apart. At 30° the phasing limit falls to 0.35° and the angle limit to 0.17°. Three joints solved from their pins at the angle limit have spreads of 0.1000° at 5°, 15° and 30°. The limits are for each error alone. Together they share the budget along the ellipse, so a joint at 15° with its phasing 1° out has a little over two thirds of its angle allowance left.

Drivetrain practice asks installers to match the working angles within about a degree and to line up the yokes by the marks on the shaft. The mirror argument said why both instructions exist: each restores a symmetry. The slopes say what each is worth once it is nearly met. The measurements say the second of those instructions buys far less than the first on a shallow joint, and the gap narrows as the joint steepens. At 30° a degree of phasing costs half what a degree of angle difference does.

What the first order leaves out #

The next order. Every slope here is the first term of an expansion, and the figures show where the next term begins: at 6° of phasing error the pins are 0.18% above the line, and a one-sided angle error drifts off by the even half’s effect on the mean angle. None of the second-order coefficients is derived.

Clearance and elasticity. The joint is solved with rigid pins in perfect bearings. A real cross has play in its needle bearings and a real shaft twists under torque, and neither is modelled. The mirror argument does not care about those if they are symmetric, but a worn bearing at one end is an odd error of its own.

What the lag does to the vehicle. A spread of a tenth of a degree is a statement about kinematics. Whether it is felt depends on the shaft’s speed, its inertia and what it drives, since the lag’s second derivative is an angular acceleration applied twice a turn. That is a dynamics question.

Still open: a Rzeppa cage with clearance #

The mirror argument explained the Rzeppa joint without solving it: the cage holds every ball in the plane that bisects the two shafts, and a ball in that plane is a configuration that is its own mirror image. A real cage has clearance, and each ball can sit a little off the bisecting plane.

The distinct argument there would be the Rzeppa joint’s version of this essay. A ball displaced off the plane is an odd error, since the mirror sends a ball displaced one way to one displaced the other. So the lag should again be first order in the displacement, with a slope derived from the joint’s own geometry. What the double Cardan joint cannot show is the question of many balls. Six balls, each with its own small displacement, each able to take the load or not, is not two errors but six. It is not obvious whether the lag is set by the worst ball, by the average, or by something like the root sum of squares found here. The measurement would be the joint’s lag against a given cage clearance, with the balls placed at every corner of their clearances.