One test, three mechanisms

Three questions from three fields of this site, separated by six phases and a hundred essays.

Will this pawl hold, or will the first tooth lever it out?

Will this escapement’s lock hold itself, or will the smallest knock open it?

Can this four-bar’s coupler actually drive its rocker here, or is it pushing along a line that does nothing?

They are the same question, and the fact that they are is more interesting than any one of them, because it says something about where the boundary of this site actually runs.

The instrument #

A contact — a tooth on a face, a pin in a hole, a coupler pinned to a rocker — can transmit along exactly one direction. For a pin that is the line joining the two centres; for a surface contact it is the normal at the point of contact. Which of the two senses along that line applies is settled by which body is pushing on which, and pushing is the only thing a contact does.

So the direction is known before anything else is, and it is known from the geometry rather than from anything about loads. Given the direction $\hat{\mathbf{n}}$ and the vector $\mathbf{a}$ from a pivot to the contact, the quantity

$$\mathbf{a} \times \hat{\mathbf{n}}$$

is a length with a sign: the perpendicular distance from the pivot to the line of the push, signed by which side of it the pivot is on. It is the moment the push would exert per unit of force, and it can be computed without ever knowing what the force is.

Every one of the three questions is that number’s sign.

Three answers, three uses #

The pawl. $\mathbf{a}$ runs from the pawl’s pivot to the contact, $\hat{\mathbf{n}}$ is the direction the tooth pushes the pawl. The sign says whether the pawl is driven towards the root of the tooth — engaging — or towards its tip, and off. On the ratchet this field draws, the arm is 0.495 wheel radii on the holding side, and moving the pivot to the other side of the face’s line gives 0.175 on the other, which is a mechanism the first tooth throws off.

The pallet. $\mathbf{a}$ runs from the pallet arbor to the contact, $\hat{\mathbf{n}}$ is the direction the tooth pushes the pallet. The sign says whether the wheel’s own torque carries the pallet deeper into its lock — draw — or lifts it out. At a locking face tilted by $\delta$ the magnitude is $\rho \sin\delta$, which for this escapement’s $\rho = 0.8588$ and $\delta = 1.5°$ is 0.022481, against 0.022482 measured. A concentric face gives $6.5 \times 10^{-17}$, which is zero, and a lock with no tendency to stay shut.

The four-bar. $\mathbf{a}$ runs from the output pivot to the coupler’s pin, $\hat{\mathbf{n}}$ runs along the coupler. The magnitude — $|O_4B| \sin\mu$, where $\mu$ is the transmission angle — is how much moment the coupler delivers per unit of push, and the sign is which way it turns the rocker. At a crank angle of 0.6 rad this linkage’s transmission angle is 58.9° and its mechanical advantage 17.7; at 2.0 rad the angle is 87.5° and the advantage 3.0.

What separates the three is not the calculation. It is what the mechanism does with the answer.

A four-bar uses the moment arm: its value, not merely its sign, is the velocity ratio and the mechanical advantage, and a designer wants it large. An escapement uses the sign to hold and pays for the magnitude in recoil, so a designer wants it small but not zero. A pawl uses only the sign, and the magnitude is irrelevant to it — a pawl 0.495 radii on the right side and a pawl 0.05 radii on the right side both hold, and the second one holds no worse.

Where virtual work applies, and where it does not #

The three are not equally amenable to the identity that makes the escapement’s case so tidy, and the difference is worth pinning down.

For the escapement, the pallet is driven by the pendulum and the wheel by the train, and the two are joined by a contact. Over any small motion the work cancels, so

$$\frac{M}{\tau} = -\frac{\mathrm{d}\theta}{\mathrm{d}\varphi}$$

and the draw and the recoil rate are the same number in two units. Measured independently — the left side by a cross product, the right by differencing the contact solve — they agree to nine figures.

For the four-bar the same identity is the familiar one: mechanical advantage is the reciprocal of the velocity ratio, which is why the advantage runs to 17.7 exactly where the output is slowest.

For the pawl it does not apply in the same way, and reading it as though it did gives a wrong answer. A pawl carries no load except the contact and its own pivot reaction, so its moment about the pivot is not balanced by anything: it is not in equilibrium, it is being driven. What decides whether the ratchet holds is the pawl’s own moment, which drives it towards the root or away from it — not a virtual-work statement about the coupled pair, which is satisfied whichever way the mechanism goes.

That distinction is easy to miss and expensive to miss. The coupled system always does work in the direction it moves; the question is which direction, and answering it needs the free body that is being driven rather than the pair that is being coupled.

The wheel gains a little while the pawl seats #

The pawl does have a transmission ratio, and it is worth having, because it is a third kind of lost motion nobody counts.

When the tooth’s push drives the pawl towards the root, the tip slides down the face, and the wheel has to move to keep the face under the tip. Solving that contact — a circle against a line, in closed form — gives $\mathrm{d}\theta/\mathrm{d}\gamma = -0.5428$, and the seating travel is the distance from where the pawl bears to the root, which is 3.89° of pawl rotation.

So the wheel gains 2.11° while the pawl settles, on a twelve-tooth wheel whose pitch is 30°. That is seven per cent of a pitch, added to the design lost motion and the clearance lost motion and different in kind from both: it is not the input travelling without the output moving, it is the output moving after the input has stopped.

It also stops at a geometric limit rather than at a balance. The tip reaches the root and the seating is over, whatever is pushing; nothing about the load decides where a ratchet ends up.

The degeneracies line up too #

Each of the three has a configuration where the moment arm goes to zero, and the three go by different names for the same thing.

For the four-bar it is the toggle: the coupler in line with the rocker, transmission angle 0° or 180°, the moment arm gone, and the linkage unable to drive its output however hard it is pushed.

For the escapement it is a locking face exactly concentric with the arbor: the push through the pivot, no draw, and a lock in indifferent equilibrium.

For the pawl it is a pivot on one of the two lines: either the face extended, where the push cannot slide the tip anywhere, or the normal extended, where it has no moment at all.

And the three are wanted at different distances from their degeneracy. A four-bar is designed to stay well away — the usual rule is a transmission angle no worse than 40° from a right angle. A toggle clamp is designed to sit exactly on it, because the infinite advantage is the point. An escapement is designed to sit just past it: a degree and a half of draw, which is as close to indifferent as a mechanism can be while still holding.

Three mechanisms, three distances from the same zero, chosen for three different reasons.

There is a fourth position on that axis and this field has it: the locking arc of a Geneva drive or a cam’s dwell, which sits on the degeneracy deliberately and for a whole interval rather than at a point. A dwell is a toggle that has been made to last, and the way to make a toggle last is to replace the point where the moment arm vanishes with a curve along which it vanishes — which is exactly the concentric arc, and which is why that essay and this one are about the same object seen from two sides.

A fourth, and why it is not in the list #

There is an obvious candidate for a fourth member and it does not qualify, which is worth working through because the reason is the whole discipline.

A friction clutch holds one way and slips the other, exactly as a ratchet does. Its verdict looks like the same kind of thing: a wedge angle, a contact, a decision about whether the mechanism grips. And it is not the same kind of thing at all, because the answer is $\tan\theta < \mu$ — a comparison between an angle the drawing sets and a coefficient the drawing does not.

Change the material and the same geometry gives a different verdict. Nothing like that can happen to a ratchet: the pawl in the first figure holds on steel, on brass, on ice, and with the surfaces oiled. A self-locking wedge is a mechanism whose behaviour is a property of the pair of materials; a pawl is a mechanism whose behaviour is a property of a drawing, and only the second kind is on this site.

The same test excludes two more things the field brushes against. Whether a cam follower jams in its guide is a pressure angle against a friction coefficient, and the pressure angle is here while the jamming verdict is not. Whether a gear tooth will scuff is a sliding velocity against a lubricant, and the sliding velocity is here while the scuffing is not.

The rule that comes out of it: a geometric criterion is one whose verdict survives every substitution that leaves the shape alone. Three of this essay’s four survive; the fourth does not, and it is the one everybody’s intuition wants to add.

What the three have in common at the far end #

One more resemblance, and it is at the opposite end of each mechanism’s argument from where this essay started.

In all three cases the sign of the cross product is a robust thing and its magnitude is a fragile one. A pawl 0.495 radii clear holds, and so does one 0.05 radii clear, and so does one 5 radii clear — the verdict is a partition of a plane by two lines and nothing in between matters. An escapement with 1.5° of draw holds; with 0.15° it also holds, and recoils a tenth as much, and is a tenth as safe against a knock. A four-bar with a transmission angle of 87° drives well and one at 5° drives badly and both drive.

So each of the three mechanisms has a binary property decided by geometry and a graded one decided by the same geometry, and design is nearly always about the graded one. What this essay’s cross product settles cleanly is the binary question, which is the one that gets drawn wrongly; the graded question is what the rest of this field’s essays measure.

Four mechanisms on one axis #

The three questions are one cross product with three different uses, and adding the locking arc gives a fourth — at which point the four can be arranged on a single axis, which is the most compact statement of what the whole comparison is for.

Normalise the moment arm by the distance from the pivot to the contact, and what is left is the sine of the angle between the contact normal and the line to the pivot. It runs from $-1$ to $+1$, its sign is the binary question all three mechanisms ask, and its magnitude is the graded quantity each of them uses differently. Four positions on it, and every one of the four is somewhere a mechanism deliberately sits.

At the far end, near one. A four-bar wants its transmission angle near a right angle, which is this quantity near its maximum. A large moment arm is a large mechanical advantage and a mechanism far from any degeneracy, and everything the linkage fields say about transmission angle is a statement about staying up here.

Small and clearly one-signed. A pallet’s draw is one or two degrees on a clock and ten to fifteen on a watch, so the quantity is a few hundredths to a quarter. Small enough that the recoil it buys is affordable, large enough that the lock holds itself against a knock. The whole design argument is about how far up from zero to sit.

Exactly zero. A locking arc, a cam’s dwell, a deadbeat’s concentric face: the normal passes through the pivot, the moment arm vanishes, and nothing turns. This is the arc theorem, and it is the only one of the four positions where the value rather than the sign is being specified exactly.

And one-signed with the magnitude irrelevant. A pawl needs its pivot on the correct side and does not much care how far; the sign settles whether it holds and the magnitude only affects how it seats. This is the position where the binary property is everything.

Read as one axis, the four say something the three-way comparison does not. The same scalar is being specified in four different ways — maximised, chosen small, set to zero, and asked only for its sign — and which of those a mechanism does is what the mechanism is for. A four-bar transmits, so it wants magnitude. An escapement holds and releases, so it wants a small positive value. A dwell holds only, so it wants zero. A pawl decides, so it wants a sign.

That is a tidier account of the family than a list of three resemblances, and it predicts where a fifth member would sit. Anything wanting a large magnitude of the opposite sign would be a mechanism arranged to be driven backwards deliberately and hard, which is a back-driving device rather than a holding one — and this site has no example of it, which is the honest end of the list.

Where the boundary actually is #

This site has said since its foundation that it computes kinematics and not dynamics, and this essay is the sharpest available statement of what that means in practice.

What is geometric: the direction a contact can transmit along. The sense of the moment that direction produces about any pivot. The magnitude of the moment arm — a length. Ratios of moment arms, which are velocity ratios and mechanical advantages and transmission ratios and recoil rates, all of them dimensionless and all of them computable from a drawing. Whether a mechanism holds, jams, draws or toggles.

What is not: any force, any torque, any pressure, any stress. How fast anything moves. How much anything deflects. Whether a part is strong enough. What the mechanism does over time under a load — which is why the pallet’s angular budget does not close and the wheel’s does.

The line between those two lists is exactly the line between a ratio and a magnitude. A mechanism’s whole transmission behaviour is ratios, and ratios are lengths divided by lengths; the moment the question needs an absolute value of anything with mass or stiffness in it, it has left.

It is worth noticing how much of engineering practice sits on the geometric side. A four-bar is selected on its transmission angle, an escapement is specified by its lock, lift, drop and draw, a ratchet is drawn by putting a pivot on one side of a line — and none of those decisions needs a force. What the forces decide is how large the parts have to be, which is a separate discipline with its own textbooks, and which begins after the mechanism has been chosen. That ordering is not an accident of how the subject is taught. It is a consequence of the fact that ratios are settled by shape, so the question “does this arrangement work” can be answered before the question “how big must it be” is asked.

Which is also the answer to the objection this site’s premise most often attracts — that a kinematic account of a machine leaves out the part that matters. It leaves out the sizing. It does not leave out whether the thing works.