The gear with its teeth cut away

The cheapest way to make an output turn for part of a revolution and stop for the rest is to take a pair of gears and remove most of one gear’s teeth. Leave them on an arc of a hundred and twenty degrees, put a locking arc on the rest of the circumference to hold the driven wheel while they are away, and the mechanism is finished. It costs one gear blank and one extra operation.

It is also, on the one column that separates this field into halves, the worst mechanism in it.

What happens at the first tooth #

While the teeth are engaged the mechanism is an ordinary gear pair. The velocity ratio is the tooth-count ratio, it is constant, and it is constant for the reason every involute pair’s is: the common normal at the contact passes through a fixed point on the line of centres. Nothing about the engaged phase is unusual.

The trouble is the first tooth. Before it arrives the driven wheel is stationary, held by the locking arc. After it arrives the driven wheel is turning at $\omega_1 r_1 / r_2$. There is no interval in between and no mechanism arranging a transition: the tooth is out of mesh and then it is in mesh, and the driven wheel’s angular velocity jumps from zero to its full value.

An angular velocity with a jump in it has no derivative at the jump. Not a large derivative — no derivative. And that is awkward to report honestly, because every instinct is to quote a big number, and every big number available is a number about the measurement rather than about the mechanism.

How to measure a quantity that is not there #

The site’s finite-difference checks are written to agree with something. This one is written to disagree, in a stated way.

Take the second central difference of the driven wheel’s angle at the moment of engagement, at step $h$, then at $h/2$, then at $h/4$. For a function with a bounded second derivative these settle: successive estimates converge, and their ratio goes to one. Across a step in the first derivative they do not settle. The second difference of a step of size $s$ at spacing $h$ is of order $s/h$, so halving $h$ doubles it, and it goes on doubling for as long as arithmetic lasts.

Two point zero zero zero zero zero zero. That is the signature of a step in the first derivative, it is not a property of the step size, and it is the honest form of the sentence “this mechanism has no acceleration here”. A ratchet dropping into a tooth and an escape wheel arrested by a locking face are in the same class for the same reason, and none of the three is improved by manufacture.

The rate of divergence says which derivative is missing #

The refusal to converge is reported above as a ratio of 2.000000, and that number is doing more work than it looks. It is not merely evidence that the limit does not exist; it says exactly what kind of discontinuity is there, and a different value would have meant a different mechanism.

Take a second central difference of the driven angle at step $h$. If the angle itself were to jump, the numerator would carry a term of the jump’s size and the quotient would grow like $1/h^2$ — halving the step would multiply the reading by four. If the angle and its first derivative were both continuous and only the acceleration stepped, the quotient would converge to a finite value and halving the step would change nothing. Between those, a jump in the first derivative puts a term proportional to $h$ in the numerator, the quotient grows like $1/h$, and halving the step doubles the reading.

Two, then, and not four and not one. The measurement is a classification, not merely a failure: the driven wheel’s position is continuous, its velocity is not, and its acceleration is absent rather than large. That is the whole kinematic content of engaging at full speed, and it has been read off a sequence of difference quotients that were never told what to look for.

It is worth appreciating how unusual that makes the check. Every other finite-difference test on this site is written to agree with something — a closed form, a second route, a conservation law — and reports a residual. This one is written to disagree, and its output is the rate at which the disagreement grows. A check that succeeds by diverging at a specific rate is a stranger object than a check that succeeds by returning zero, and it is the right instrument here because the quantity being asked about does not exist.

The same reading would catch a mistake that would otherwise be invisible. Suppose the mechanism’s geometry were subtly wrong — the teeth arranged so that the first one engages with a small but non-zero approach velocity match, say, or the locking arc mis-cut so that the wheel is already moving when the tooth arrives. Then the velocity step would be smaller and the ratio would still be 2, because the classification does not care about the size of the jump. But a geometry that accidentally removed the step, by smoothing the entry, would return a converging sequence, and the check would report a finite acceleration where a mutilated gear must have none. The instrument distinguishes the mechanism this essay is about from anything that is not it.

That is also the cleanest statement of what separates this mechanism from the Geneva. Both are intermittent, both have exact indexes, both dwell. Run the same difference sequence on a Geneva and it converges, to 5.41 at four slots — a large number and a number. Run it here and it doubles for ever. The ledger’s last column is the difference between a quantity that is big and a quantity that is not there, and the two are told apart by an exponent rather than by a magnitude.

What the Geneva pays to avoid it #

The Geneva mechanism exists to solve exactly this, and the way it solves it is worth stating precisely because it is a single condition with a very long shadow.

The driving pin must enter the slot along the slot. That makes the crank and the slot perpendicular at the moment of entry, and the driven wheel’s angular velocity there is then zero — which can be had in closed form rather than from a sweep. Differentiating the wheel angle with respect to the crank angle at entry, the numerator collapses:

$$x\dot{y} - y\dot{x} = c^2\cos^2\beta - c^2\cos^2\beta = 0$$

and the sampled mechanism returns $0$ and $1.9 \times 10^{-16}$ at four, six and eight slots. That is not a small entry velocity. It is the absence of one.

What that costs is the whole of the mechanism’s freedom of proportion. Tangential entry fixes the centre distance; the centre distance fixes the wheel radius; the slot count fixes the index and the dwell. A Geneva designer chooses one integer and receives a mechanism. A mutilated-gear designer chooses the tooth counts, the sector angle, the centre distance and the locking arc independently, and can have any index ratio at all — which is why the mechanism is still made.

The acceleration a Geneva does have #

Avoiding an infinite acceleration is not the same as having a small one, and the Geneva’s is neither small nor a design variable.

Measured over the engagement in normalised units, the peak angular acceleration of the driven wheel is 5.41 at four slots, 1.35 at six and 0.70 at eight. It falls steeply with the slot count, and the reason is geometric: more slots means a smaller index, a shallower slot angle and a gentler carry. It also means a shorter dwell, because the driven fraction $(n-2)/2n$ rises with $n$ — a quarter of the turn at four slots, a third at six, three eighths at eight.

So the Geneva’s two useful properties pull against each other. A long dwell wants few slots and gentle accelerations want many, and there is no third parameter to relieve the tension because there is no third parameter at all.

This is the fact behind the mechanism’s most famous application and its most famous failure. A cinema projector’s Geneva pulls the film down thirty times a second, and the acceleration it does so with is what tears sprocket holes — which is a genuine complaint about a mechanism whose acceleration is genuinely finite and genuinely too big. The answer, when there was money for it, was never a different Geneva. It was a cam.

The practical fix, and what it really is #

Nobody builds a mutilated gear that engages cold at full speed on a machine that matters. The standard remedy is a lead-in: a specially shaped first tooth, longer and relieved, sometimes with a spring-loaded plunger, whose job is to accelerate the driven wheel up to speed over a small angle before the ordinary teeth take over.

That is worth looking at squarely, because of what it is. A tooth whose profile is designed to produce a stated motion over a stated interval is a cam. The remedy for a mutilated gear’s velocity step is to put a small cam in front of it, and the reason the remedy works is the reason cams work: a surface can be cut to any programme, including one that starts from rest.

There is a second and older remedy, and it is the same observation from the other side. Instead of shaping the first tooth, shape the gap: cut the driven wheel’s first tooth space wide and give the driver a pin rather than a tooth for its first engagement, so that the pin enters the space along its centreline. That is a Geneva pair grafted onto the front of a gear train, and it works for the Geneva’s reason — the driving point arrives along the direction the driven part is about to move in, so the entry velocity is zero.

Which is the honest summary of the whole comparison. There are exactly two ways to bring an output up from rest smoothly — arrange the geometry so that the driving point arrives along the direction of motion, as a Geneva does, or cut a surface that says what the motion is to be, as a cam does. The mutilated gear does neither, and every fix for it is one of the two in miniature.

The arc that does the holding #

Nothing so far has said what holds the driven wheel still while the teeth are away, and the answer is the same curve the previous essay is about: an arc of a circle about the driver’s own axis, bearing on a matching concave face on the driven wheel.

Without it, the mechanism does not have a dwell at all. It has an interval during which the driven wheel is free, which is a different object: the wheel keeps whatever motion it had, arrives at the next engagement wherever it happens to have arrived, and the mesh has to catch it there. That is not an intermittent drive, it is a drive with a gap in it.

The locking arc is also where a drawing of a mutilated gear is most often incomplete, and the incompleteness is invisible: a diagram showing the gear blank with its teeth on one sector and a plain circular hub elsewhere is showing a mechanism whose output is free for two thirds of every revolution. Reading it as a dwell is reading in a component that is not drawn.

Any ratio at all, which is why it survives #

The reason the mechanism is still made, in spite of everything above, is that it is the only member of the ledger whose index is a free choice.

A Geneva’s index is $360°/n$ and $n$ is a whole number, so the available indexes are 120°, 90°, 72°, 60°, 51.4°, 45° and so on down — a fixed and rather sparse list. An escapement’s is half a tooth pitch. A ratchet’s is a tooth pitch. A mutilated gear’s is the sector angle times the tooth-count ratio, and both of those are free: a sector of 120° on a pair geared 3 : 1 gives an output step of 40°, and a sector of 97° on a pair geared 1 : 1.6 gives 155.2°. Nothing has to divide anything.

There is a second reason, and it is about packaging. A Geneva needs its centre distance to be crank ÷ sin β, which for a long dwell is large: a four-slot Geneva’s centres are $\sqrt{2}$ crank radii apart and its wheel is as big again. A mutilated gear’s centres are the ordinary sum of pitch radii. On a machine where the two shafts are where they are, the intermittent gear may be the only mechanism that fits between them, and no argument about acceleration moves a shaft.

The output step is exact, and that is the point #

For all its faults in the last column, the mutilated gear is exact in the first — and it is worth being clear about why, because the reason is different from the Geneva’s.

A Geneva’s index is exact because the pin carries the wheel through exactly one index and the locking arc holds it there. A mutilated gear’s is exact because the number of teeth that pass through the mesh is an integer. The driven wheel turns by (teeth engaged) ÷ (teeth on the driven wheel) of a revolution, and both of those are counts. There is no accumulation over a million cycles, no dependence on where the wheel started, and nothing a tolerance does to it: cut the sector to carry eleven teeth and it carries eleven.

That puts the mechanism in the same company as the gear train generally, and for the same reason. A ratio built out of tooth counts is a rational number known exactly, and every kinematic property that follows from it is exact too. The velocity step at engagement is a defect of the transition, and the transitions are the only part of an intermittent mechanism that is not integer arithmetic.

What is on the other side of the boundary #

An angular velocity with a step in it implies an impulsive angular acceleration, and an impulsive angular acceleration implies an impulsive torque, and that is where this site stops.

What can be said here is exactly what has been said: the kinematics of the mechanism contain a discontinuity, its location is known to the tooth, and its size is $\omega_1 r_1/r_2$ — a velocity, which is a kinematic quantity and is drawn. What cannot be said is what that costs, because what it costs depends on the inertia of the driven train and on the compliance of everything between, and neither of those is a length.

It is also worth noticing that the discontinuity is in the idealised mechanism, which is a slightly unusual position for this site to be in. Everywhere else the ideal mechanism is the well-behaved one and reality roughens it; here the ideal mechanism has an infinity in it and every real one does not, because real teeth have tip relief, real shafts twist, and the contact that arrives at full speed arrives at a surface that gives. The kinematic model is the pessimistic one, and it is pessimistic about the right thing — the location of the problem, which is the first tooth, and its size, which is the pitch-line velocity.

It is worth noticing how much of the design argument survives the boundary anyway. The choice between a mutilated gear, a Geneva and a cam is decided almost entirely by three kinematic facts — whether the output starts from rest, whether its acceleration is bounded, and how much of the turn it is moving for — and the ledger this field is built around contains all three. The dynamics decide how much the difference matters. They do not decide which mechanism is which.