Three mechanisms, one subtraction

Three mechanisms, from three different trades.

A differential screw — Hunter’s screw — is a rod with two threads of nearly equal pitch cut on it in the same hand, one turning in the frame and one in the nut. Turn it once and the frame’s thread advances the rod by $p_1$ while the nut’s thread retracts along it by $p_2$, so the nut moves by $p_1 - p_2$. With pitches of 1.0 and 0.9 mm that is a tenth of a millimetre per turn, from two threads that are both coarse and easy to cut. It is the adjusting mechanism in fine instruments and in the micrometer heads on optical benches.

A differential pulley — the Weston chain block hanging in every workshop — is one compound sheave with two slightly different diameters welded together, an endless chain, and a movable load block. Pull one turn of the sheave and the chain is taken up on the large diameter and paid out on the small one, so the load rises by half their difference. With diameters of 20 and 19 the velocity ratio is 40 to 1.

A compound epicyclic is the drive in the last essay: two rings, a compound planet, and a reduction of $1/(1 - z_1z_4/z_2z_3) = 20.53$ from a difference of two products, 2176 and 2070.

They are the same mechanism.

The shared structure #

Each mechanism has two quantities that are nearly equal — two pitches, two radii, two products of tooth counts — and each produces an output proportional to their difference:

$$\text{output} \;\propto\; a - b, \qquad a \approx b .$$

The gain is large because $a-b$ is small, and the gain is large for that reason and no other. That is what makes them one family rather than three coincidences: there is no other way to get a hundred to one out of two parts, and all three found the same one.

The immediate consequence is the one this essay is about. If $a$ is in error by a relative amount $\varepsilon$ — the thread cut a little coarse, the sheave turned a thousandth over, a ring with one tooth too many — then the output is in error by

$$\frac{\Delta(a-b)}{a-b} \;=\; \varepsilon \cdot \frac{a}{a-b},$$

and $\kappa = |a/(a-b)|$ is the amplification. It is exactly a condition number in the numerical-analysis sense: the factor by which a relative error in the input is multiplied on its way to the output, for the operation subtract two nearly equal numbers, which is the classic ill-conditioned operation and the reason a numerical analyst distrusts a difference.

Measured, not quoted #

A formula that says a mechanism is sensitive is a claim about the mechanism, and this site’s habit is to get it a second way. So each mechanism is perturbed: one of its two quantities is increased by one part in ten million, the output is recomputed by the mechanism’s own machinery, and the ratio of relative changes is measured.

The residuals are the finite-difference step, as they should be. The two routes share no algebra: one is $|a/(a-b)|$ written down, the other is the library computing an advance, a lift and a null space at two nearby parameter sets and dividing.

The correction the measurement forced #

The first version of that table had the epicyclic’s conditioning number at 20.528, and the perturbation came back with 19.528. Exactly one apart, which is the signature of a definition rather than a bug, and it was.

The reduction is $R = B/(B - A)$ with $A = z_1z_4 = 2070$ and $B = z_2z_3 = 2176$. Two candidate expressions present themselves:

$$\frac{B}{B-A} \;=\; 20.528 \qquad\text{and}\qquad \frac{A}{B-A} \;=\; -19.528,$$

and they differ by 1 because $B/(B-A) - A/(B-A) = 1$ identically. The first is the gain — it is the reduction — and the second is the conditioning. Writing the table with the first meant claiming that a mechanism’s mechanical advantage is its sensitivity, which is nearly true and is not true, and would have made the whole family look like an identity instead of a relationship.

The perturbation caught it because a perturbation cannot be argued with. It also, incidentally, explains the relationship the table now shows: the gain and the conditioning are one apart for the epicyclic, a factor of two apart for the pulley (velocity ratio $2D/(D-d)$ against $\kappa = D/(D-d)$), and equal for this particular screw only because $p_1$ happens to be 1. Those factors are unit conventions — what counts as “the output” — and the conditioning is the invariant.

That is worth stating as a rule. The gain of a difference mechanism depends on which quantity is called its output; its conditioning does not. A chain hoist quoted as 40 : 1 and a robot gearbox quoted as 20 : 1 are equally fragile, and comparing their quoted ratios is comparing two different things.

What the fragility looks like in each #

The abstraction earns its keep when it predicts what each mechanism does when it is made badly, and the three failure modes read as though they were about completely different machines.

The screw. A relative error of $10^{-3}$ in the coarse pitch — a thread cut one micron over per millimetre — moves the advance per turn by one per cent. So the instrument’s calibration is ten times more sensitive to thread error than the threads are themselves, which is why a differential screw head is calibrated by measurement rather than by arithmetic, and why the pitches are chosen to be simple numbers so that the difference is exact by construction.

The pulley. With $D = 20$ and $d = 19$, one part in a thousand on $D$ is 2% on the lift per turn. That does not matter, because nobody measures how far a chain hoist has lifted; what matters is that the same sensitivity applies to the difference being positive at all. A hoist whose two diameters were meant to differ by one and differ by 0.1 has a velocity ratio of 400 and a chain that has to be pulled ten times as far.

The epicyclic. The applied field measured this one directly: one tooth on one ring takes the reduction from 20.53 to 28.63, a 39% change from a part 1.4% different. And $\kappa = 19.5$ predicts it: a tooth is $1/69$ of the ring, which is $1.4\%$, and $19.5 \times 1.4\% = 28\%$ — the right order, with the rest being that a single tooth is not an infinitesimal.

That last comparison is the honest one to end on. The conditioning number is a derivative, and a tooth is not small. For the epicyclic, the parameter is an integer and the smallest possible change is already outside the range where a linearisation is accurate. Which means the conditioning number is a good guide to how a family of designs behaves and a poor predictor of what a specific extra tooth does — and the site has made that distinction before, where a first-order tolerance stack was compared with the real thing.

The pulley, worked through #

The chain hoist is the one of the three whose arithmetic is least often written down, and doing it makes the family resemblance obvious.

The compound sheave has two diameters, $D$ and $d$, welded together and turning as one. An endless chain goes down from the large diameter, round the load block’s pulley, and back up onto the small one. Turn the sheave once: the large diameter takes up $\pi D$ of chain and the small one pays out $\pi d$, so the loop holding the load shortens by $\pi(D - d)$. The load hangs on two falls of that loop, so it rises by half of that per turn:

$$\text{lift per turn} \;=\; \frac{\pi(D-d)}{2}.$$

The hand chain, meanwhile, has been pulled $\pi D$. So the velocity ratio — chain pulled per unit of lift — is

$$\frac{\pi D}{\pi (D-d)/2} \;=\; \frac{2D}{D-d},$$

which is 40 for $D = 20$ and $d = 19$.

Every term is a length and two of them are nearly equal, which is the family. And the conditioning is $D/(D-d) = 20$: make the large diameter 0.5% over and the lift per turn is 10% out, which on a hoist means the operator pulls ten per cent more chain than the chart says and notices nothing at all.

The reason nobody notices is worth a sentence, because it is the difference between this and the other two. A hoist’s output is distance lifted, and nobody measures it. A micrometer head’s output is distance too, and its whole purpose is that somebody measures it. Identical mechanisms, identical conditioning, and one of them is a precision instrument and the other is a lump of steel on a hook.

The screw that is not a screw #

One caution, of the kind the fleet’s consolidation notes keep insisting on: this essay’s “differential screw” and the site’s screw field are two different objects sharing a word.

A screw pair here is a helical joint: a thread, a pitch in millimetres per turn, a mechanism that converts rotation to translation. A screw in the sense of the spatial field is an element of a six-dimensional space of instantaneous motions, with a pitch that is a ratio of a translation rate to a rotation rate and no thread anywhere. The second contains the first as a special case, in that a helical joint’s motion is a screw of that pitch, and using either vocabulary in the other’s argument produces sentences that parse and mean nothing.

The two differential mechanisms here are both screw pairs and neither has anything to do with reciprocal screw systems. The compound epicyclic has neither. What all three have is subtraction.

What a differential gear is doing in the list #

There is a fourth member of the family that has been in this field since its first essay, and it is worth naming because it makes the structure clearer rather than adding to it.

A differential gear computes a sum: $\omega_L + \omega_R = 2\omega_C$. Reverse the roles and it computes a difference — drive the cage and one wheel, and the other wheel’s speed is $2\omega_C - \omega_L$, which is a subtraction with no amplification at all, because the two quantities are not nearly equal.

That is the useful contrast. A differential is a difference mechanism with a conditioning number of order 1, and it is therefore robust in a way none of the three above is. What makes the others fragile is not that they subtract; it is that they subtract two quantities that have been deliberately made almost equal, in order to make the result small.

Put another way: subtraction is only ill-conditioned when the operands are close. A designer who wants a large gain from a difference mechanism must bring the operands close, and therefore must accept the conditioning. There is no third option, and that is the whole content of the family.

It also says where to look for a mechanism that gives a large ratio without the fragility, and the answer is that there is not one of this kind. The escape is to stop using a difference: two stages of ordinary gearing give a hundred to one with a conditioning number of 1 in each stage, at the cost of two shafts, two bearings and a case. That trade — compactness against conditioning — is the real content of the choice between a harmonic drive and a two-stage reducer, and it is a kinematic statement, not a manufacturing one.

What the site has done with this shape before #

The difference structure is not new to the site; what is new is the family being named.

In the timing field the drop of an escapement turned out to be a remainder — impulse plus drop plus lock adds to exactly half a tooth pitch, so the drop is whatever the other two leave, and every degree added to the impulse is a degree taken from the drop to one part in $10^{12}$. That is a difference mechanism made of angles, and its conditioning is why a finer escape wheel does not give a better one: the tolerance stack barely moves while the nominal drop falls twentyfold, so the margin collapses from 26 to 1.4.

In the practice field it appeared as a direction: a mechanism’s sensitivity to its dimensions is not a scalar, and the interesting question is which combination of parameters the output is a difference of.

And in approximate synthesis it appeared as the reason a linkage that satisfies its precision points exactly can be unbuildable — the design’s quality is a small difference between two large quantities, and the difference is where the whole answer lives.

What this essay adds is that when the difference is deliberate — when a designer has chosen to make two quantities nearly equal in order to get a large gain — the conditioning number is not an incidental property to be measured afterwards. It is the gain, up to a constant, and it was chosen at the same moment.

Conditioning belongs to the operating point #

The differential gear is set against the three as the well-conditioned member because it computes a sum rather than a difference, and that contrast is nearly right and worth sharpening — because a sum can be ill-conditioned too, and the same differential proves it.

The cage speed is $(\omega_L + \omega_R)/2$, and a sum of two quantities is ill-conditioned when they are nearly opposite rather than nearly equal. So drive one wheel forward and the other backward at nearly the same rate — a car turning on the spot, one wheel on ice while the other is driven, a limited-slip unit being tested on rollers — and the cage speed is a small difference of two large numbers with all the amplification that implies.

So the differential gear is not a well-conditioned mechanism. It is a mechanism that is well conditioned where it usually operates, with both wheels turning the same way at similar rates, and badly conditioned in a regime it can perfectly well be put into. The conditioning is a property of the operating point and not of the gearset, and the same is true of every mechanism in this essay read the other way round.

That reframes the family properly. A differential screw, a chain block and a compound epicyclic are not mechanisms with a bad conditioning number; they are mechanisms deliberately parked at the ill-conditioned point, permanently, because that is where the ratio is. The fragility is not a defect that came along with the gain — it is the same fact as the gain, and a designer who wanted less of one would be asking for less of the other.

Read that way the family has a clean membership test that does not mention subtraction at all. A mechanism belongs to it when its useful operating point is near a place where its own map is singular. The differential gear is in the family only when driven that way; the three named mechanisms are in it always, by construction; and a plain gear pair is never in it, because its map has no singularity to sit near.

It also says where to look for a member nobody has named. Any mechanism whose ratio has a pole — a linkage near a toggle, a cam near a cusp, a platform near a singularity — offers arbitrarily large gain to anything willing to sit close to the pole, and pays the reciprocal in sensitivity. The three here are simply the cases where somebody built the mechanism at the pole on purpose and gave it a name.

Why the family is worth naming #

The practical dividend is a design heuristic that is exact rather than a rule of thumb.

If a mechanism’s output is a difference of two nearly equal quantities, its ratio and its fragility are the same design decision. Not related, not correlated — the same decision, taken once. A designer asking for twice the reduction from a difference mechanism is asking for twice the sensitivity to every dimension in it, and there is no arrangement of parts that separates the two, because the gain is $1/(a-b)$ and the conditioning is $a/(a-b)$ and they differ by a constant.

That is why a hundred-to-one harmonic drive is a precision component and a five-to-one gear pair is a commodity, and it is why the tolerances on a difference mechanism are the interesting part of its drawing rather than a note in the corner. It is also why the one difference mechanism in this essay that escapes — the harmonic drive, whose design family moves both tooth counts together and keeps the difference fixed at 2 — is the one that is actually manufactured in quantity.