Machine Kinematics
solved, then drawn
Concept

Tensioner — where it appears

An idler pressed against a run to make its length match the strand fitted to it, usually carried on a spring-loaded arm. What it takes up per millimetre of its own travel is twice the sine of half its wrap, so its authority falls to nothing exactly where its contact does.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 A chain is a strand with a smallest piece. 53 teeth driving 11 at a pitch of 12.7 mm. The taut strand is drawn as this field computes it; inside each pitch circle is the polygon the pins actually sit on. The two agree on the straight spans and not on the wraps, and the gap is what the applied field's chordal action is: the effective radius swings between R cos(π/n) and R within every tooth, which is 4.05% on the small sprocket and 0.176% on the large one. A strand has no such number, because a strand has no pitch. positioned by solving, not by drawing.

A chain is not a strand

A chain has a smallest piece, and two things follow that no continuous model can have. Its pins sit on a polygon, so the radius that matters swings by 4.05% within every tooth of an eleven-tooth sprocket — and its loop must contain a whole number of pitches, so the centre distance that closes it comes in steps of 6.4834 mm.

strands · Strand The number on the tackle is a limit. The velocity ratio of a 4-part tackle as its blocks separate, measured by differentiating the strand's own length. It is 3.6174 at 90 mm and 3.99992 at 6898 mm, and it reaches 4 nowhere. The shortfall is geometric and has nothing to do with friction: each part of line makes an angle with the lift, and a part at angle β shortens by cos β of the movement. A tackle used at close quarters — which is when a tackle is useful — is the case furthest from its own rating.

Six things a strand is not

A tackle that is not four to one, a wrap that is not a half turn, a tensioner whose travel takes up nothing, a winch speed that is not a property of the winch, a cable rig that holds nothing still, and a shaped pulley that cannot be asked for what it is usually asked for. Six claims, each with the number that kills it.

wrong · Misconception 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 ruleStrandWrap angleApproximationIdlerBelt driveBlock and tackleCentre distanceChain driveChordal actionConvex hullDerivative

All concepts