Machine Kinematics
solved, then drawn
Concept

Idler — where it appears

A body a strand runs over that drives nothing, carried either to guide the route or to take up length. An idler pressed in from outside a run wraps the other way, so its wrap counts negative in the turning number and the two bodies either side of it gain exactly what it takes.

Named by 7 essays across 2 fields — each of them below, with the objects they name alongside it.

What the wraps add up to, and what decides it. Four runs this field draws, with every wrap angle signed by the way the strand goes round its body. The right-hand column is their sum divided by a full turn, and it is a whole number every time — the turning number of a closed plane curve, arrived at by adding up a handful of angles that were computed one at a time from tangent lines. It is 1 for a loop that goes round its pulleys once and 0 for a crossed belt, whose two wraps are equal and opposite whatever the two radii are. The serpentine's idler contributes -25.3°, and the total is still exactly one turn: a tensioner lengthens the path without changing what the path is.

The wraps add up to a turn

Every wrap angle in a closed run is computed on its own, from a pair of tangent lines that knows nothing about the others. Signed by which way the strand goes round, they add to exactly one turn — or to exactly nothing, for a crossed belt — and the integer is decided by the route rather than by any of the geometry.

strands · Strand A tackle's ratio, differentiated rather than counted. Four parts of line between two blocks 260 mm apart, drawn as one strand over real sheaves. The ratio a tackle is sold with is the number of parts supporting the moving block, and it is the limit of the true velocity ratio rather than its value: the parts are not parallel, so each of them shortens by less than the lift. Differentiating the run's own length gives 3.9471 here, 1.32% short of 4, and the gap closes as the blocks separate. positioned by solving, not by drawing.

A ratio that is a derivative of a length

A tackle is sold by counting the parts of line holding the moving block: four parts, four to one. Differentiate the strand's own length instead and a four-part tackle gives 3.927 with its blocks 220 mm apart and 3.617 at 90 mm — and the integer it is named for is a limit it reaches nowhere.

strands · Strand What a Ravigneaux gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 5 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.

Four speeds from two numbers

A Ravigneaux gearset has five tooth counts and gives seven exact ratios. Two of the counts do not appear in any of them — the short planet is an idler and its size is free — and the remaining three enter only through two dimensionless numbers, so the whole shift ladder of a four-speed automatic is a function of ring-over-sun and ring-over-the-other-sun. A Simpson three-speed is a function of one number.

transmission · Transmission Four pulleys, one idler, and one equation. A closed strand over four fixed pulleys and an idler carried on an arm. The idler sits outside the loop the other four make and bulges the strand out to reach it, so its wrap is signed the other way — -25.34° against the four positive ones — and the five still add to exactly one turn. The arm angle is not a free choice: the strand has a length, and matching it is one scalar equation, whatever the number of pulleys. At this position the run is 1018.046 mm and the idler takes up 0.4387 mm of strand for every millimetre it moves along the bisector of its two spans. positioned by solving, not by drawing.

The tensioner is the unknown

A linkage closes when a vector comes back to where it started: two equations, two unknowns. A strand closes when a number does — its length — however many bodies it runs over. So a run with one free body is determined, a run with two is not, and the arm angle that takes up 1,020 mm of belt is the root of one scalar equation solved to 1·10⁻¹³ mm.

strands · Strand One tendon over two joints. A two-link chain with a strand anchored off to the left, running over an idler at each joint and terminating on the far link. Each idler is centred on its joint's axis, which is the one arrangement that makes the strand's length a linear function of the joint angles: the coupling measured here is 14.000000 and 10.000000 mm per radian, against radii of 14 and 10. Total strand 232.845 mm. positioned by solving, not by drawing.

One strand over many joints

Route a tendon over an idler centred on a joint's axis and the strand's length becomes an exactly linear function of the joint angle — 8·10⁻¹⁴ mm of departure over 203 degrees of travel. Move that idler 6 mm off the axis and the same drive's arm swings from 8.00 to 16.52 mm per radian.

strands · Strand Two taut paths, and no way between them. A strand from one point to another past a peg. Each side gives a path that is taut — straight where it can be, on the surface where it must be, leaving at a right angle — and each is the shortest path on its own side: 219.165 mm on one and 200.643 mm on the other, against 200 mm of open air the strand cannot use. Neither can turn into the other without passing through the peg, which is a different kind of non-uniqueness from the assembly branches of a linkage: those are separate roots of one equation, and these are separate classes of one minimisation. positioned by solving, not by drawing.

The taut path has more than one answer

A strand from one point to another past a peg has two taut paths — 200.643 mm on one side and 219.165 on the other, against 200 mm of open air it cannot use. Both are shortest. Neither can become the other without passing through the peg, and no computation recovers which one was threaded.

strands · Strand What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing.

Where a strand stops touching

A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.

strands · Strand

Named alongside it

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

Design ruleStrandTurning numberVelocity ratioWrap angleApproximationBelt driveConvex hullDerivativeHomotopy classTangencyTaut

All concepts