Machine Kinematics
solved, then drawn
Concept

Pin in slot — where it appears

A joint that permits sliding along one line and free rotation about the pin, so it removes one freedom rather than two. It is the joint a Geneva mechanism's crank pin makes with its slot, and the freedom it leaves is what lets the pin enter and leave rather than staying engaged.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

A 3-stage stack at 35°. Every bar is the same length and every stage is at the same angle, which is a consequence of the solve rather than an assumption in the drawing: the pins at the crossings and the roller on the ground leave one freedom, and it has been used. The platform is at 1547 mm and the ram — the heavy line from the ground pin — is 428 mm long. Moving the ram one millimetre from here raises the platform 5.0 mm; at the bottom of the travel the same millimetre is worth 17.6 mm.

A roller is not a slider

A scissor lift has one degree of freedom, at every height and for any number of stages. Grübler's criterion agrees — if the rollers under it are counted as pins in slots. Count them as slider blocks, which is how every textbook draws a slider, and the same formula declares a machine holding a car in the air to be a structure with minus one.

applied · Mobility The wheel is the coupler — double wishbone. The suspension solved at 0 mm of bump, with the whole travel ghosted behind it. The two arms are the cranks and the upright between them is the coupler; the wheel is bolted to that coupler, so camber is the coupler's rotation and nothing else. Camber here is 0.00° and the contact patch has moved 0.0 mm across the road. The cross is the instantaneous centre of the upright, found from the solved velocity field; the roll centre is where the line from it to the contact patch crosses the car's centreline, and it is at 73 mm here.

The wheel is the coupler

A double wishbone is a four-bar standing on end whose coupler carries a wheel, so camber is coupler rotation and scrub is a coupler point's path. Both are computable, and the second one comes out with the opposite sign from the model every suspension book uses — by more than the whole scrub.

applied · Fourbar 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

A lever escapement touches its balance for a twelfth of each beat and leaves it alone for the rest. What connects the two is a pin entering a radial slot — the same pair a film projector's Geneva drive is made of, solved by the same eight lines — and the sine rule then fixes the balance's lift angle at 44.4°, which is the number a Swiss lever is specified at.

timing · Intermittent 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's pin arrives with an acceleration of tan(π/n) of the crank's speed squared, and no choice of entry direction removes it. A choice of slot shape can: bend the slot's mouth with the crank's own circle and the slot absorbs the pin's acceleration instead of handing it to the wheel. But the pin leaves by the mouth it came in by, and there the same bend adds to the step. Solved at four slot counts and seven curvatures, the entry and exit steps are tan(π/n)(1 ∓ κ̂) and always sum to 2 tan(π/n). A curved slot moves the step from one end of the index to the other, every bend raises the peak, and no slot tried beat the straight one.

timing · Intermittent 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's step at entry and exit belongs to its pin's circle, not to its slots. Carry the pin in a radial guide on the driver and set its radius with a stationary cam, and the wheel can be given any law at all — the pin is simply where the driver's ray meets the slot's line. Cut for a modified trapezoid, a six-slot wheel starts and stops with no step and peaks 14% below the crank pin, while the cam moves the pin by 1.2% of the crank. What the circle cannot hand over is its window: no ray before the crank pin's entry meets a resting slot, so the index can never be longer, and every degree spent holding the wheel in its slot comes out of it as the square.

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 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 steered pin can hold a Geneva's wheel by running along a resting slot, but every pin is in some slot for exactly the crank pin's window, 180° less 360°/n of every turn, because it can leave a slot only by the mouth it will reach. Two windows of less than half a turn can never cover a turn, so two pins cannot hold a wheel without its disc at any slot count. Three can, on six slots and more, and four on four. At the Geneva's own share of index to dwell the index is no harder than the crank pin's, and the slots hold the wheel tighter than any lock in the field. The disc's work moves to the steering cam: on a run the cam's pressure angle is the run's own angle, and the six-slot drive's runs are 40°.

timing · Intermittent

Named alongside it

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

the Geneva mechanismIndexingLockMotion lawConstraintDwellPressure anglePrismaticTangential entryAccelerationBalanceBanking

All concepts