Machine Kinematics
solved, then drawn
Concept

Taut — where it appears

The state in which a strand's length constraint is active, so that the strand lies on the shortest path its route permits. It is a geometric condition rather than a statement about tension: a taut strand meets every surface it lies on at a right angle, and a slack one imposes nothing at all.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

One routine, six strand systems. Every row is the same function: a list of bodies, each with a sense, handed to a routine that returns the tangent runs between them, the arcs on them, and the total. Nothing in it knows what a belt is, what a tackle is or what a tendon is. The right-hand column is what each row was checked against — a textbook formula, an integer, a convex hull's perimeter, a second route to the same length — and it is the reason the middle column can stay the same all the way down. positioned by solving, not by drawing.

A member with no length of its own

Every link of a pin-jointed machine holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a rope and a tendon do neither: a strand's path is decided by the bodies it touches, and its constraint is an inequality that does nothing at all until it is taut.

strands · Strand Four tangents, and the two signs that choose between them. Two circles admit four common tangents, and a strand takes whichever one its two wrap senses name. Same sense at both ends — both centres on the same side of the strand — gives the two outer tangents, each 169.0444 mm long. Opposite senses give the two that cross between the circles, each 154.9193 mm. There is no search and no case analysis anywhere in this: the run's length is √(D² − Δ²) with Δ the signed radius difference, and changing one sense changes Δ from -18 to 70. positioned by solving, not by drawing.

Where a strand leaves a body

A taut strand meets the surface it lies on at a right angle, and every book draws it that way. It is not a rule about strands: it is what being shortest looks like, and a family of paths that were never told about tangency has its minimum exactly there — 200.64346 mm against the construction's 200.64346.

strands · Strand Where a point held by three strands may be. Three anchors, three strands of 130, 130, 120 mm, and a point tied to all three. A rigid link of those lengths would leave nothing to decide — three distance equations in two unknowns have no solution at all — and three strands leave a region, because each of them says no further than rather than exactly. The region is the intersection of the three discs; its area here is 2721.0 mm² and it has 3 corners. Inside it nothing is taut and the point has both its freedoms; on an arc one strand is taut and it has one; at a corner two are taut and it has none. positioned by solving, not by drawing.

The strand that is slack

A rigid link removes a freedom wherever the mechanism stands. A strand removes one only where it is taut — so a point held by three of them has two freedoms in the middle of its region, one on an arc, none at a corner, and no single mobility count describes it at all.

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.

StrandConvex hullTangencyWrap angleApproximationBelt driveConfiguration spaceConstraintDesign ruleIdlerInextensibleShortest path

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