Field

Motion that stops

Every other mechanism here moves whenever its input does. A ratchet, an indexer and an escapement are still for most of a turn and moving for the rest, and what decides which is not an equation running out of answers but a tooth arriving at a face. Whether one holds is the sign of a lever arm; what it wastes is two thirds of its own travel.

$Every way of stopping, on the same four questions. Six mechanisms that all turn a continuous input into an output that moves and then waits. Index is how far the output steps. Moving is the fraction of the input's turn the output is actually going for; the rest is dwell. From rest says whether the output starts and stops at zero velocity, and acceleration whether its acceleration is a number at all. Every entry is computed from the mechanism's own library, which matters for two of them: a Geneva's moving fraction is (n − 2)/2n and not 1/n, and its entry rate is zero in closed form rather than to the accuracy of a sampled sweep. The three rows whose acceleration is not a number are not badly made — they are mechanisms whose output velocity has a step, and no tolerance improves that.$

The mechanism that waits

The class of mechanisms whose output stops, why their transmission function is piecewise, and the four columns on which six of them can be compared.

A 12-tooth ratchet, holding. A ratchet wheel with a 8° tooth face and its pawl. The long line is the face extended; the short one through the contact is the normal the tooth pushes along, and both belong to the wheel and turn with it. The pawl holds, and the reason is which side of those two lines its pivot is on: 1.080 wheel radii from the face's line and 0.495 from the normal's, and on opposite sides of them. Nothing about how hard anything pushes enters the question: a normal is a direction and a moment arm is a length with a sign. Dragging turns the wheel the way it may go, and the pawl's tip is solved onto the wheel's surface at every frame: it is lifted up a tooth's back and dropped into the next notch, once per click.

A joint that works one way

The ratchet and pawl, the pair of cross products that decide whether it holds, and the map of pivot positions whose boundaries are the tooth face and the contact normal.

Lost motion on a 24-tooth ratchet. A pawl can only drop into a tooth, so an input that has moved by less than one tooth pitch has moved the output by nothing at all. On 24 teeth the pitch is 15.0°, and that is the worst case with one pawl. Two pawls offset by half a pitch halve it and three thirds it, because whichever pawl is over a root drops first. None of this is a manufacturing question: the numbers are the same on a perfectly made ratchet, which is what separates them from the lost motion an as-built measurement would.

The resolution is the pitch

Lost motion by design rather than by tolerance, why extra pawls divide it exactly, and the geometric limit on how fine a ratchet's teeth can be.

$The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell a machine has. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake that is easy to make. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of.$

The arc that is concentric with the pivot

The single geometric device behind every exact dwell, measured on five curves, with the sampling floor quoted beside the answer.

A quantity that is not there, refusing to converge. A second central difference of a function with a bounded second derivative settles as the step is halved; across a step in the first derivative it doubles, every time. The rising line is a mutilated gear at the instant its teeth engage, where the driven wheel goes from stopped to full pitch-line speed: its successive estimates grow by a factor of 2.000000, which is the signature and not an accident of the step. The flat line is a six-slot Geneva at the same point in its cycle, whose pin enters along the slot and whose acceleration is a number. This is the only way to report an acceleration that does not exist: not by quoting a large one, but by showing the measurement refuse.

The gear with its teeth cut away

Intermittent gearing, the velocity discontinuity at engagement, and how a quantity that does not exist is measured by a second difference that doubles with every halving of the step.

30 teeth and two pallets. An escape wheel of 30 teeth and a pair of pallets spanning 4 and a half tooth pitches. The heavier line at each pallet is the locking face, here an arc about the arbor; the lighter one is the impulse face the tooth slides along once it is let go. The wheel is drawn where the contact puts it, not where it looks well: at this pallet angle the tooth in play sits on the lock face and the wheel is -0.0320° from it. Dragging the pallet through its whole engagement moves the wheel by 0.106° of recoil. Of the 6.0° the wheel turns each beat, 65.9% is drop and does nothing.

Where the tooth lets go

An escapement's anatomy built from four numbers, the beat budget that closes exactly on geometry, and the second budget that does not close at all.

How far back the wheel is pushed. Recoil against the part of the pendulum's swing that happens after the tooth has landed. The arc cut concentric with the pallet arbor is the flat line on zero, and it is zero as a matter of arithmetic rather than of smallness: rotation about the arbor carries that arc into itself, so the tooth's resting place does not move and every sampled value is the same double. Every other face rises. The flat face with no draw at all is the interesting one — it starts at zero and curves, because it agrees with the arc to first order and not to second, which is precisely the amplitude sensitivity a deadbeat exists to remove.

The wheel that goes backwards

Recoil measured against the supplementary arc for four locking faces, the exact zero an arc gives, and the quadratic error a straight cut leaves behind.

One angle, read twice. Draw against the angle the locking face is tilted by. It is not a separate design quantity from recoil: the pallet's moment per unit of wheel torque is minus the rate at which the wheel is driven backwards, which is virtual work and holds to nine figures. So an escapement cannot be made to hold its own lock without also being made to push its train back — and at δ = 0 the arc has neither, which is a lock that any disturbance opens. The flat face starts above zero because the tooth rests below the corner, where a straight face is no longer tangent to anything.

The angle that holds the lock

Why an escapement's lock must be tilted to hold itself, the closed form for the moment it produces, and the identity that makes draw and recoil the same number.

The fork and the roller, which is a Geneva pair. A lever escapement's fork with the balance's impulse pin inside it. The pin is on a roller of 0.150 against a fork of 1.00, and the slot points at the pin — which is the entire kinematic relation and is the same one a Geneva drive's pin and slot obey. The same eight lines answer both, and the check is that they reproduce the cams field's independently written Geneva solution to fourteen figures. The pin is in the fork for 44.4° of the balance's swing out of 540°, so the balance is left alone for 91.8% of the beat. Drag the balance.

Detached, and safe while detached

The detached lever escapement, the fork and roller as a Geneva pair, and the sine rule that decides how much of each beat the balance is left alone for.

One test, three mechanisms

The lever-arm sign that decides a ratchet, an escapement's draw and a linkage's transmission angle, and where exactly the boundary between geometry and statics runs.

A period, in the space the mechanism lives in. One whole period of an escapement, plotted as wheel angle against pallet angle. Give a four-bar its crank angle and its coupler is somewhere definite; give this its pallet angle and the wheel may be in any of 3 places, because what settles it is which face of which pallet a tooth is against. The path crosses itself and no amount of solving removes the crossing. The vertical jumps are the drops, drawn at the pallet angle of the release because during a drop nothing is touching anything and where the pallet is by then is a question about torque. Over the period the wheel advances 12.000000°, which is one tooth exactly.

Where the input stops deciding

The configuration of an intermittent mechanism as a pair — a contact label and a solve — and the self-crossing path a whole period traces in the space of both.

One piece, and still not reachable

Reachability against connectivity on a linear ratchet, and why a connectivity result leaned on by the earlier essays on intermittent motion needs a second half.

One pin and two, on the same wheel. A 6-slot Geneva wheel at one instant of its index, driven by a crank carrying one pin and by the same crank carrying two. The wheel, the slots and the centre distance are identical; only the pin count differs. Each pin drives the wheel through one slot pitch while the driver sweeps 120°, so 1 pin gives 1 index a driver turn and leaves 67% of it at rest, 2 pins give 2 indexes a driver turn and leave 33% of it at rest. Nothing else about the mechanism changes, which is why the whole question is arithmetic.

Two pins and no dwell at all

The multi-pin Geneva: the driven share p(n − 2)/2n, the dwell test and the pin-overlap test shown to be one inequality over fifty-six pairs, the three integer solutions of n(p − 2) = 2p searched to four hundred by four hundred, and the trade between a dwell that falls in a straight line and a peak acceleration that falls as a square.

A ratchet with no teeth

The roller clutch's lost motion measured as geometry with no friction in it: the gap between a flat and a bore narrowing as a secant, the closed form 2c(R − r)/(R − 2r)·sin ψ and its convergence to the exact travel, the inverse-sine dependence on the wedge angle, and the ratchet comparison in which extra pawls divide the lost motion and extra rollers do not.

A Geneva's acceleration against three cam laws, at the same index angle. The output's acceleration through one index, divided by the step over the square of the input angle it takes — the unit in which a cam law's acceleration coefficient is quoted — so every law is one fixed curve whatever the step. The dark curve is a Geneva of 6 slots: its step is 360°/6 and its input turns 120° while indexing, and a cam indexer is given exactly those two numbers. The Geneva's coefficient is 5.653 against cycloidal 6.283, modified sine 5.528 and simple harmonic 4.935, so at 6 slots modified sine and simple harmonic have the lower peak. The Geneva's curve starts and ends away from nought, at tan(π/6) = 0.577 in absolute terms: its acceleration steps the instant the pin enters. Dragging the slot count moves only the Geneva.

When the index law becomes a choice

The Geneva's displacement law in closed form, set against cycloidal, modified-sine, simple harmonic and constant-acceleration cams at the same index angle and step; the real-valued slot counts at which each law's peak acceleration crosses the Geneva's; the entry step tan(π/n) and its share of the peak; the limit in which a Geneva becomes simple harmonic motion; and what a cam gives up in dwell to match a Geneva's peak.

The disc decides the pin count

The locking disc's cut-away measured by sweeping the wheel into the driver's own frame: 119° at three slots against a 60° index, 236° at six against 120°, and a pin limit of one from four slots upward where the slot arithmetic allows two or three.

A flat holds the wheel to second order

The Geneva's locking disc with the wheel held on flat chords rather than concave arcs: the largest disc a flat allows (crank × sin π/n, set by the slot mouths), the cut-away the disc needs swept in the driver's frame for both locks across three to twelve slots, the pin counts each allows against the slot arithmetic, and the locked wheel's rock against the running clearance, measured by turning the wheel's material into the disc — first order for an arc, c/(a sin ψ), and second order for a flat, √(2c/a).

A V holds the wheel to first order

A Geneva wheel locked by V notches between its slots: the wheel, the driver's disc and its cut-away as the V steepens; why a lock's hold is the lever of its contact normal about the wheel's centre, nought for the flat; the rock against running clearance for V angles from nought to 20°, measured by turning the wheel's material and against the exact law; the cut-away against the V angle at four slot counts with the two-pin line; and the largest two-pin V at every slot count from four to twelve, its disc and its rock against the flat's and the arc's.

A curved slot bends at its mouth with the crank's circle, or against it. A six-slot Geneva wheel whose slots are curved: mouth curvature 0.5 of the crank's. The driver's pin runs round the dashed circle and enters each slot along its mouth, which is radial so that the wheel starts from rest. At entry the pin's acceleration is square to the slot, and the part of it the slot's own bend takes up does not turn the wheel. Solved over the index, the wheel starts with an acceleration of 0.289 and stops with a deceleration of 0.866, in units of the crank's speed squared, against 0.577 at both ends for a straight slot; the two always add to 1.155. Its peak is 1.649 against the straight slot's 1.350.

A curved slot moves the step

A Geneva wheel whose slots are curved: a six-slot wheel with slots bent at the mouth by a slider; the wheel's acceleration and deceleration steps against the mouth's curvature at four slot counts, collapsing onto 1 ∓ κ̂; the acceleration through the index for a straight slot, a half-bent one and one bent to the crank's circle and straightened; the peak against the bend; and 246 slots straight at the mouth and bent inside, none of which peaks lower than the straight slot.

A pin steered off its circle carries the wheel on any law. A 4-slot Geneva wheel and its driver. The pin rides in a radial guide on the driver and a stationary cam sets its radius, so its path through the index (solid) is not the crank's circle (dashed). Here it is cut for the modified trapezoid law over the circle's own window of 90° of driver: the wheel starts with an acceleration of nought and stops with nought, against tan(π/4) = 1.000 at both ends for the crank pin, and peaks at 3.112 against the crank pin's 5.407, in units of the driver's speed squared. The pin's radius moves between 0.667 and 0.708 on a crank of 0.707.

A steered pin keeps only the window

A Geneva wheel driven by a pin that a stationary cam steers off the crank's circle: the wheel and the steered path for four laws; the wheel's acceleration through the index for the crank pin and three steered laws; the pin's radius through the index against the crank; the steered peaks over the crank pin's from three to sixteen slots, with the crossovers at 5.19, 6.23 and 8.30; the peak against the share of the dwell the slot holds the wheel; and a ledger of steps, peaks, cam travel, pressure angle and slot depth.

Three steered pins hand a six-slot wheel from slot to slot. A 6-slot Geneva wheel with no locking disc and a driver carrying 3 pins, each in a radial guide whose radius a stationary cam sets. Each pin is in some slot for 120° of every driver turn: it runs in along a resting slot, indexes the wheel on a modified sine law over 40°, and runs out along the slot it moved. The solid curve is the path every pin follows while it is in a slot; the dashed circle is the crank's. At the moment shown pin 1 is running in, pin 2 is clear of the wheel, pin 3 is clear of the wheel; the wheel is held by one slot.

A wheel held by its slots needs three pins

A Geneva wheel with no locking disc held by several steered pins in turn: the wheel and three pins at one moment of a cycle; each pin's window against the slot count, with the angle each of p pins must cover; the timelines of four disc-free drives over a driver turn; the steering cam's pressure angle against the index's share of the cycle; the wheel's rock in its slots through the dwell against a V lock's; and a ledger of the four drives against the crank pin.

All essays