Machine Kinematics
solved, then drawn
Concept

Tangential entry — where it appears

The condition that a driving pin enters a slot along the slot's own direction, which makes the driven part start from rest. Without it the driven part is asked to change speed instantaneously, so the condition is what separates an indexer that runs from one that hammers itself apart.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A 6-slot Geneva wheel. The driving pin enters a radial slot, carries the wheel through 60°, and leaves. The centre distance is not free: it must be crank ÷ sin(180°/6) = 60.00 so that the pin enters along the slot, with the crank and slot perpendicular. That is the mechanism's one design requirement and it is what makes the driven wheel start and stop from rest — measured here at 1.5e-3 against a peak of 1.000. What it does not fix is the acceleration, which peaks at 1.35 and is why film sprocket holes tear.

Stopping thirty times a second

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

cams · Cam 30 teeth and two pallets. An escape wheel of 30 teeth and a pair of pallets spanning 4 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 -0.0323° from it. Dragging the pallet through its whole engagement moves the wheel by 0.107° of recoil. Of the 6.0° the wheel turns each beat, -37.3% is drop and does nothing.

The escapement that could not alternate

Four drawings of intermittent mechanisms that appear everywhere and would not work: pallets spanning a whole number of teeth, a pawl whose pivot is on the wrong side of the tooth face, a Geneva at the wrong centre distance, and an intermittent gear with no locking arc. Each one is put through the library that draws the working version, and each returns a number.

wrong · Misconception 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

Leave teeth on part of a gear's circumference and take the rest off, and the output turns for part of the input's revolution and stops for the rest. It is the cheapest intermittent drive there is and it engages at full speed, so its output's velocity has a step and its acceleration is not a large number — it is not a number, and the way to report that is to watch a difference quotient refuse to converge.

timing · Intermittent 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

A Geneva's motion law is forced by its slot count and a cam indexer's is chosen, so the fair comparison gives the cam the Geneva's own index angle and step. On peak acceleration the cam wins only below a slot count that depends on the law — 5.19 for cycloidal, 6.23 for modified sine, 8.06 for simple harmonic — and above it the Geneva does. What no slot count removes is the step: the pin arrives with an acceleration of exactly tan(π/n).

timing · Intermittent The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at 0° of driver, with the driver's locking disc of radius 27.0 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, outside the 6 concave locking arcs cut into it, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 236.3°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot.

The disc decides the pin count

A Geneva's pin count is usually bounded by the slots: p indexes must not overlap, so fewer than 2n/(n−2) pins fit. The other half of the mechanism has its own inequality and nobody had measured it. The locking disc must be cut away wherever the wheel passes through it, that cut-away is wider than the index sweep at every slot count, and it is the binding condition everywhere — one pin only, from four slots upward.

timing · Intermittent

Named alongside it

The objects these essays reach for when they reach for this one.

the Geneva mechanismIntermittent motionDwellIndexingMotion lawAccelerationJerkDesign ruleMutilated gearCamCentre distanceClearance

All concepts