Machine Kinematics
solved, then drawn
Concept

Misconception — where it appears

A claim about a mechanism that is widely repeated and measurably wrong. This site collects them as a field, because the way each of them fails says something about which quantities were never computed.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

Six gearsets, three conditions. Every one of these can be drawn, and five of the six are drawn in some textbook or other. The columns are the three conditions a planetary has to satisfy: that a whole planet fits between the sun and the ring, that the sun and ring teeth add to a multiple of the planet count, and that the planets clear each other. The last column is how far out of mesh the worst planet station is, in teeth — a quantity that is zero or is not, and that no drawing shows, because a drawing of a planetary at this scale draws circles.

The gearset that could not be assembled

A planetary drawing shows a sun, a ring and three or four planets between them, and if the circles are the right sizes at the right stations it looks right. The condition that decides whether the second planet can actually be dropped in is arithmetic — the sun and ring teeth must add to a multiple of the planet count — and it appears in no drawing, at no scale, in any style.

wrong · Misconception What each of them can reach. Nine hundred control histories of four legs each, from the same starting configuration, with the resulting position plotted. The wheel's cloud is two-dimensional and fills the region; the trolley's is one-dimensional and lies exactly on its rail — the same number of coordinates, the same number of constraints, the same count of freedoms, and a reachable set of a different dimension. Nothing here is a matter of degree.

Not unreachable, only expensive

The sentence is false of every wheeled mechanism in this field and true of exactly one — the trolley bolted to a rail. A rolling constraint forbids a direction and reaches everywhere; the mistake is reading a statement about instants as a statement about intervals, and it is made in both directions.

wrong · Misconception Which of these two tracks was made by the front wheel. The rear wheel of a bicycle is towed, so it points at the front wheel at every instant: the tangent to the rear track, extended forward by the wheelbase, lands on the front track. Done that way round the tangents land a mean of 0.13 mm off; done the other way round they land 461 mm off, a factor of 3625. It is a measurement rather than an eye for tracks, and it needs neither the wheelbase nor the direction of travel to be known in advance.

Which way did the bicycle go

Two tyre tracks in mud, and a question with a definite answer. The rear wheel is towed, so its tangent extended forward by the wheelbase must land on the front wheel's track — and it does, to 0.13 mm one way round and 447 mm the other. The test needs neither the wheelbase nor the direction of travel, and it returns both.

rolling · Rolling Forwards it settles, backwards it runs away. A trailer starting a hundredth of a radian out of line, with the steering held straight. Driving forwards the angle decays as e^(−s/d); reversing, the same equation runs the other way and it doubles every 4.16 m. Nothing about forces is involved and nothing about the driver: it is the sign of one exponent, and the length scale is the trailer's own length. The curve flattens at the top because the sine that generates it saturates — the runaway is exponential only while the angle is small.

The angle that doubles

A trailer a hundredth of a radian out of line decays back into line as e^(−s/d) driving forwards and grows as e^(+s/d) reversing — doubling every 4.16 m for a six-metre trailer. And a jackknifed rig is not a rig that has lost anything: its growth vector is 2·3·4·5 at a hitch angle of zero, of ninety degrees and of a hundred and eighty.

rolling · Rolling What a sweep at the wrong resolution reports. One machine — a crank passing a stud a tenth of a unit across — swept at six sample counts. At eight samples the smallest gap found is 0.0809 and the machine reads as clear; at twelve it is 0.0067 and still clear; from sixteen on it is negative and the crank is inside the stud by 0.0099. Nothing about the twelve-sample answer looks wrong: the curve it draws is smooth, its minimum is interior, its margin is small and positive. A sweep cannot report what it did not look at, and the repair is not more samples but a bound on what happens between them.

Six things a body is not

A verdict read as a measurement, a hull read as a part, a sweep read as a proof, a drawing read as a configuration, a geometry read as a force, and a plane read as a place. Six claims, each of them what a careful person would say, each answered with a number.

wrong · Misconception The best split for a given variator rating. The most span a single-planetary power split can have when its variator may carry no more than the share on the axis, for variators spanning 2, 3.333 and 6. Up to a share of one the best is a split with no circulation, reaching the variator's own span exactly at one; beyond it, a split whose variator carries more than the engine, gaining R − 1 of span per unit of share. The dashed curves are the third branch, with the variator's power running backwards, which is below the best everywhere. The ring marks the previous essay's split at a unit rating: 1.700, which is 2 − 1/R. The same variator in a split with no circulation, its share never passing 0.9, spans 2.703, and it approaches 3.333 as that share approaches one.

Six things a power split is not

A multiplier of its variator's span, a machine whose neutral the belt decides, a source of any ratio near that neutral, a travel that slides towards its pole for free, a design whose attachment is the choice that matters, and a machine a second planetary always improves. Six claims about continuously variable transmissions built as power splits, each of them what a careful person would say, each answered with a measured number.

wrong · Misconception A roller offset by 5.88, at 42.3° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, drawn with the follower fixed above and the cam turned anticlockwise to 42.3°. The follower's line of travel is 5.88 to the right of the cam's centre. The contact normal leans 21.81° from that line, which is the pressure angle, and measured off the drawn pitch curve it is 21.81°. At this offset the peak on the rise is 21.81° and on the return 21.81°; centred, they are 27.37° and 15.33°.

Six things a cam drive is not

A follower that is better off offset, a flat face with no size limit, a swinging arm that improves any follower, a groove that keeps its roller on one wall, a smooth law that is gentle everywhere, and a camshaft that delivers the law it was cut to. Six things people say about cam drives, each the natural reading of something true, each answered by a number the cam essays measured. They fail the same way: each reads the cam at first order, in positions and slopes, and the drive is decided at second order, in accelerations and curvatures.

wrong · Misconception 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 The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out.

Six things a balancing spring is not

More springs share the load, a spring that must push wants a compression spring, a stiffer spring balances better, a refitted stiffness repairs a free length, least squares finds the balance, and the best spring is one of no free length. Six things said about springs that balance a mechanism, each the natural reading of something true, each answered by a number the essays on the desk lamp and the rocking slide measured. They fail the same way. Each treats the spring as the thing to be sized, and a balance is decided first by where the spring is hung and how its error is judged.

wrong · Misconception a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1809 here, between coupler · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour.

Where the boundary moved

An earlier essay drew a line around what the essays here compute and listed one thing on the far side as a gap rather than a boundary: interference between links, which needed no new physics, only a body and a test. Here is what that turned out to cost and what it turned out to open.

practice · Practice

Named alongside it

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

Design ruleRolling constraintControllabilityEpicyclicFree spaceGrowth vectorInterferenceJackknifeLayer assignmentLink bodyNonholonomicOff-tracking

All concepts