The mechanism that waits

A hundred and six essays on this site have been about mechanisms that move. Not one of them has been about a mechanism that stops — and stopping, for the mechanisms in this field, is not a failure or an edge case or a singularity. It is the working state, and it occupies most of the cycle.

Turn a four-bar’s crank by a degree and every point on its coupler goes somewhere. Turn an arm’s shoulder by a degree and its tool goes somewhere. Turn the driver of a Geneva drive by a degree and, two thirds of the time, nothing whatever happens: the driven wheel is held still by a locking arc and the pin is nowhere near a slot. Turn a clock’s escape wheel and it moves half a tooth and stops, and then waits for a pendulum.

What is actually different #

The difference is not that these mechanisms are complicated. A ratchet has two moving parts.

The difference is in what has to be computed. Everywhere else on this site a configuration is the solution of one system of equations, and that system is valid across the whole of the input’s range. A four-bar has two closure equations and they hold at every crank angle it can reach; a spatial loop has six and they hold everywhere the loop closes; an open chain has none at all and needs none. The mechanism’s position may stop existing — that is what a non-Grashof linkage’s dead range is — but while it exists it is given by one rule.

An intermittent mechanism has a sequence of rules and a set of conditions saying which one is in force. A Geneva’s driven wheel obeys a slot-and-pin relation while the pin is in a slot and obeys nothing at all while it is not. An escape wheel obeys a contact with one pallet’s locking face, then with the same pallet’s impulse face, then with the other pallet’s locking face, and in between it obeys nothing and simply spins.

So the state of one of these mechanisms is not a configuration. It is a pair: a configuration, and a discrete label saying which surfaces are in contact. Inside one value of the label the problem is the site’s ordinary one and is solved the site’s ordinary way — this field’s contacts are circle-against-circle and circle-against-line, which have closed forms, and every one of them is checked against a Newton solve of the same residual because a closed form is exactly where the habit of checking is cheapest to drop. What is new is the label, the inequalities that say when it is still valid, and the rule for what it becomes when it is not.

And the quantities the whole subject argues about live at the switches rather than inside the pieces. Drop, lock, draw, recoil, lost motion, index error: every one of them is a statement about what happens when one contact hands over to the next. None of them has an analogue anywhere else on this site, because nowhere else on this site is there a handover.

Two of the four columns are routinely got wrong #

The ledger above is computed, and the reason to insist on that is visible in two of its columns.

The first is the Geneva’s driven fraction. A six-slot Geneva indexes its wheel by 60° at a time, and the number that gets quoted for how much of the driver’s turn that takes is $1/n$ — a sixth. It is not. The pin is inside a slot while the driver sweeps $180° - 360°/n$ either side of nothing, so the fraction is

$$\frac{n-2}{2n}$$

which is a third at six slots, not a sixth. The two expressions agree at $n = 4$ and nowhere else, and four slots is the case a diagram is most likely to show. This site had the wrong one for three phases; nothing read it, which is why it survived, and correcting it doubled the stated moving fraction of every Geneva here.

The second is whether the output starts from rest. A Geneva’s does, exactly, and the reason is the one design requirement the mechanism has: the slot is radial and the pin enters along it, so the crank and the slot are perpendicular at entry and the driven wheel’s angular velocity there is zero. But “zero” measured by differencing a sampled sweep is not zero, it is the first sample’s forward difference, and that is a number of order the sample spacing. The ledger takes the entry rate in closed form instead — the numerator of the derivative collapses to $c^2\cos^2\beta - c^2\cos^2\beta$ — and gets $0$ and $1.9 \times 10^{-16}$, which are the two doubles that mean nothing at all.

Three of the six have no acceleration at all #

The last column is the one that separates the field into two halves, and it is not a matter of degree.

A Geneva’s driven wheel starts from rest with a finite acceleration and finishes at rest with a finite acceleration. A mutilated gear’s does not: its teeth engage at the full pitch-line velocity, so the driven wheel is stopped one instant and running the next. Its angular velocity has a step, and the acceleration of a function with a step in it is not a large number — it is not a number.

Reporting that honestly is a small problem in itself. Quoting a large acceleration would be quoting an artefact of whatever step the difference was taken at. What is reported instead is the measurement refusing: a second central difference of a smooth function settles as the step is halved, and across a step in the first derivative it doubles, every time.

The ratchet and the escapement are in the same half for the same reason, and it is worth being clear that this is not a defect of manufacture. No tolerance improves it and no polish removes it. A pawl dropping into a tooth arrives with whatever velocity it has; an escape wheel arrested by a locking face goes from turning to stopped. These are mechanisms whose velocity is discontinuous by construction, and a designer who wants an intermittent motion with bounded accelerations has exactly one route: a cam, which will produce any motion law asked for at the cost of a surface that must be cut.

The input does not determine the output #

There is a stronger statement to make about the escapement, and it is the one this field’s later essays are built on.

Give a four-bar its crank angle and its coupler is somewhere definite — possibly in one of two assembly branches, but definite once the branch is named. Give this escapement its pallet angle and the escape wheel may be in any of three places. Not because the equations have several roots, but because which equation is in force depends on where the mechanism has come from: a pallet at 1.1° may have a tooth on its locking face on the way out, or on its impulse face on the way back, or nothing on it at all while the other pallet is engaged.

That crossing is not pathology. It is what an escapement is for: the whole point of the mechanism is that the wheel advances by a fixed amount every time the pendulum does the same thing, and it cannot do that if the wheel’s position is a function of the pendulum’s.

The path is also drawn without any notion of time. Every point on it is a contact solve at a pallet angle, and the order they are visited in is the order the contacts hand over. What time would add is the rate, and the rate is a question about torque.

What decides the switch #

One more distinction, because it is easy to collapse and this whole field depends on it not being collapsed.

Elsewhere on this site, when a mechanism stops moving it is because its closure equations have stopped having a real solution. That is what happens at a non-Grashof four-bar’s limit position, at the boundary of a parallel platform’s workspace, at the edge of an arm’s reach. The mechanism runs out of configurations.

Nothing of the kind happens here. A Geneva’s driven wheel has a perfectly good configuration during its dwell — it is simply being held in it. An escape wheel is not out of positions; it is against a face. The stopping is imposed by a contact, which is an inequality rather than the failure of an equation, and the difference shows up the moment anything is asked about it: a mechanism at a limit position is at a singularity and its Jacobian has dropped rank, and a mechanism held by a contact is nowhere near one.

Where the wasted travel goes #

One number to end on, because it is larger than it has any right to be.

An escapement’s wheel advances exactly half a tooth pitch every beat — six degrees, on thirty teeth — and that half pitch divides into the impulse, during which the tooth is sliding along a pallet’s inclined flank and the train is doing the only useful work it does all cycle, and the drop, during which the tooth has left one pallet and not yet reached the other and nothing is touching anything at all.

Two thirds. On the deadbeat escapement this field is arranged around, 65.9% of the escape wheel’s motion is free flight in which the mechanism does nothing whatever, and there is no adjustment that recovers it. That is not a criticism of escapements; it is a statement about what the mechanism is buying with the waste, which is the ability to be stopped at all, reliably, thirty times a minute, for a century.

The dwell is not a loss of mobility #

It is tempting to describe a dwell as the mechanism briefly having nothing to do, and worth checking that against the count the first field of this site is built on.

A Geneva drive during its dwell is three links — the frame, the driver and the wheel — with a revolute at each of two shafts and one higher pair where the driver’s locking arc bears on the wheel’s concave face. Grübler’s planar count is

$$M = 3(n-1) - 2j_1 - j_2 = 3(3-1) - 2(2) - 1 = 1$$

and the mechanism has one degree of freedom, exactly as it does while the pin is in the slot. Nothing has been constrained away. The driver is free to turn and does; what is zero is not a freedom but the derivative of the transmission function, and it is zero over an interval rather than at a point.

That distinction is the difference between this field and everything the site has done with singularities. A toggle position is a point at which the output’s velocity vanishes and the mechanism’s Jacobian drops rank; a dwell is an interval on which it vanishes and nothing drops rank at all. The first is a property of a configuration and the second is a property of a surface — and which surface it takes is one thing, one curve, used by three different mechanisms in this field for three different reasons. That is the fourth essay here.

The counting also explains the fourth column of the ledger. A higher pair is a contact along a line or at a point, and a contact can only push. Every irreversibility in this field — the ratchet that turns one way, the escapement whose wheel never goes round backwards, the pawl that will not be pushed out — comes from that one fact about what a contact can do, and none of it is visible in a mobility count, which cannot tell a pair that pushes from a pair that pulls.

What a figure here is allowed to show #

The rule the rest of the site draws under is that a figure may not show a configuration the closure equations refuse, because there are no coordinates to draw it from. This field keeps the rule with one term exchanged: a figure may not show a configuration the contact refuses.

Every tooth resting on a face in this field’s figures is placed by solving where that face meets the tooth circle, and a pallet angle at which the contact does not exist stops the build rather than being quietly drawn at the nearest angle that does. The one thing that has to be drawn rather than solved is the drop, and the figures say so: wherever a drop appears, the wheel is shown in flight at the pallet angle of the release, because where the pallet has got to by the time the wheel lands is not a question this site answers.

The angles are also drawn at true scale. That is worth stating because it makes several of the figures here look like nothing is happening: the recoil this field spends four essays on is a tenth of a degree on a wheel six centimetres across, and a figure that magnified it to make it visible would have stopped being a measurement. Where a quantity is too small to see, it gets a plot instead of an arrow.

The design quantities are all on the boundary #

There is a structural difference between this field and every other one on the site that is worth naming early, because it says where a designer should look and the answer is not where the rest of the collection has trained them to look.

In a continuous mechanism, the quantities that decide a design live in the interior of the motion. A transmission angle is worst somewhere in the middle of a stroke. A coupler curve’s curvature is a property of every point of it. A cam’s peak acceleration happens at a particular fraction of the rise. In each case the mechanism is doing the same thing throughout and the design question is where along that motion it does it worst.

In an intermittent mechanism nothing interesting happens in the interior at all. During a dwell the output is still and there is no quantity to optimise; during an index the mechanism is an ordinary gear pair, or an ordinary slot-and-pin, doing what such things do. Every design quantity is at a switch. Drop is the gap at a release. Lock is the depth at an arrival. Lift is the interval between two contacts changing. The index angle is the difference between two switch positions.

That is why the ledger’s columns are what they are and why they are all transition quantities. And it explains a real hazard for anybody arriving from the rest of the site: the instruments that matter elsewhere — sweep the mechanism, find the worst configuration, quote it — return nothing here, because the worst configuration is not in the sweep. It is at an endpoint, where two rules meet.

The switches themselves are geometric coincidences: a tooth arriving at a face, a pin entering a slot, a corner passing a corner. Each is an equality between two positions rather than an inequality being violated, which is why a solver does not find them by failing. They have to be looked for, one at a time, as the roots of a contact condition — and once found they are exact, since they are the solution of an equation rather than the end of a sweep.

Which gives this field its own version of the site’s habit. Elsewhere, nothing is drawn that was not solved; here the addition is that nothing is quoted that was not found at a switch, and a quantity computed by sampling the interval between two switches is a quantity computed in the one place the mechanism has nothing to say.

What this field will not take #

The same boundary the site has kept since its first phase, and it bites twice here in ways worth naming in advance.

Nothing here evaluates a force. Two of the field’s central quantities look like statics and are not. Whether a pawl holds and whether an escapement’s lock draws itself deeper are both the question of which side of a pivot a contact normal passes, which is a length with a sign — the same quantity as a transmission angle, and it decides the sense of a moment without ever evaluating one.

And one ledger does not close. The escape wheel’s angular budget closes exactly on geometry; the pallet’s does not, because one of its terms is how far the pallet travels while the wheel is in free flight, and that is an integration under a torque. It is reported as missing rather than estimated, and that is an essay of its own.