Machine Kinematics
solved, then drawn
Concept

Reachability — where it appears

Whether a mechanism can be driven from one state to another by motions it allows, which for a one-way mechanism is stricter than sharing a piece of free space. Connectivity is symmetric because a path can be run backwards; reachability is an order, because a ratchet's or a cable tie's paths cannot.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

One piece, and still not reachable. The state of a linear ratchet — a cable tie — is how far in it is pulled, and its free space is the whole interval: every state is connected to every other, with no barrier anywhere. What it does not have is a way back. From the marked state the reachable set is everything forward and only as far back as the tooth the pawl has already dropped into, so 52.1% of ordered pairs are reachable and 4.2% are reachable both ways. Connectivity is symmetric; reachability is an order, and a component count answers the first question and cannot be asked the second.

One piece, and still not reachable

The site decides whether two configurations can be joined by asking whether they are in the same connected component, and that relation is symmetric because a path run backwards is a path. A one-way mechanism breaks the symmetry and leaves the connectivity alone: its free space is a single interval with no barrier anywhere in it, and about half of the ordered pairs of states cannot be joined by any admissible motion.

timing · Intermittent The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at -40° of driver, with the driver's locking disc of radius 13.5 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, short of the 6 flat chords cut across it, one between each pair of slots, 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 126.8°, 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.

Six things an indexer is not

More pins make a Geneva smoother, a finer ratchet is a more precise one, a locked wheel is a still wheel, a curved slot gives a smooth start, a cam indexer is always gentler than a Geneva, and a mechanism whose states are all in one piece can go from any of them to any other. Six things said about indexing mechanisms, each the natural reading of something true, each answered by a number the timing essays measured. They fail the same way: each judges the mechanism by its motion, and an indexer is decided at its switches — the instants a pin enters, a pawl drops or a lock takes over.

wrong · Misconception

Named alongside it

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

RatchetAssembly branchConfiguration spaceConnected componentContact stateDesign ruleFree spacethe Geneva mechanismIndexingIntermittent motionLockMisconception

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