Machine Kinematics
solved, then drawn
Series

Strand — the series

21 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · strands
  2. 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.

    part 2 · strands
  3. 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.

    part 2 · strands
  4. 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.

    part 3 · strands
  5. 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.

    part 3 · strands
  6. 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.

    part 4 · strands
  7. The shape is what the demanded arm implies. A strand leaving a convex body runs along a tangent, so the only thing about the body the strand can feel is the perpendicular distance from the axis to that tangent — the shape's support function h(ψ). Ask for a rate and you have asked for h, because turning the body by dψ pays out h dψ; the shape then comes back from h with no solve at all, as h(ψ)û + h′(ψ)û⊥. This one was asked for h₀(1 + 0.3 cos 2ψ) with h₀ = 34 mm. The strand drawn here leaves the axis at a perpendicular distance of 24.1286 mm, which is what the demand asks for at this angle. positioned by solving, not by drawing.

    The drum that is not round

    The only thing about a body a strand can feel is the perpendicular distance from the axis to the tangent it leaves along. Ask for a rate and you have asked for that distance at every angle — and the shape comes back from it with no solve at all, unless the demand exceeds 1/(n²−1), at which point there is no shape.

    part 4 · strands
  8. The radius a winch works at is not a property of the winch. A drum of core radius 30 mm filling with line 6 mm thick, one layer at a time. Line speed per turn is 2πr, and r is the radius of the layer being wound rather than anything about the machine: it runs from 207.35 mm on the first layer to 395.84 mm on the 6th, a factor of 1.909. A winch quoted at one speed is quoted at one layer, and which one is rarely said. positioned by solving, not by drawing.

    The radius a winch works at

    Line speed per turn is 2πr, and r is not a property of the winch. It is the radius of whichever layer is being wound, so a six-layer drum runs from 207.35 mm per turn to 395.84 — a factor of 1.909 with nothing about the machine changed, and a length that is quadratic in the turns rather than proportional to them.

    part 5 · strands
  9. 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.

    part 5 · strands
  10. 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.

    part 6 · strands
  11. 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.

    part 6 · strands
  12. 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.

    part 7 · strands
  13. 4 turns, and none of them in a plane. A strand wrapped 4 times round a drum of radius 50 whose barrel has room for 8, with each turn lying beside the last — so the pitch is the strand's own diameter, 10 here, and the path is a helix rather than a circle. The helix angle is 1.823°, the length per turn is 314.318 against the planar model's 314.159, and the whole run is 1257.27 long where a plane would have said 1256.64. The plan view is a planar wrap exactly: projected onto the plane perpendicular to the axis, the helix is a circle of the drum's own radius to 2.8e-14 traversed 4.000000 times.

    The wrap that walks along the axis

    Every figure in this field is drawn in a plane, and a strand that goes round twice cannot be: the second turn has to lie beside the first. The plan view of the helix that results is a planar wrap exactly — so every wrap angle survives and the length does not, by five hundred parts per million on a real rope.

    part 8 · strands
  14. An inner cable in its sheath, pulled and pushed. A sheath routed through two bends — the first turning 90° on a radius of 70, the second turning back 90° on 55 — drawn with its bore exaggerated to a clearance of 7 so the inner's two positions can be seen. Pulled, the inner is taut and takes the shortest path the tube allows, hugging the inside of each bend; pushed, it is pressed against the outside. Both paths are strands over pulleys of radius R ∓ c at the bend centres. At this clearance the pulled inner is 20.53 shorter than the centreline and the pushed one 23.63 longer, against c times the total turning, 21.99. At a real clearance of 0.25 the pulled inner is short by 0.7834 against 0.7854. Dragging changes the first bend's angle.

    A strand in a tube

    A Bowden cable's inner runs inside a sheath with a little clearance, and pulled it takes the shortest path the tube allows — a strand over pulleys of radius R − c at every bend. So its lost motion is the clearance times the total angle the sheath turns through: no bend radius in it, no route shape, and the S-bend that turns nowhere net loses as much as the U that turns back. Steering a handlebar changes it by exactly c times the change in angle.

    part 9 · strands
  15. The taut inner, and the walls it is actually held by. A sheath of two 8° bends on a radius of 40, its bore drawn at a clearance of 4, with the shortest path from ferrule to ferrule inside it. Nothing about a bend goes into that path: it is the shortest route the tube allows, found by tightening a funnel against every cross-section in turn, and the places it reaches the wall are where it is held. At this clearance the two bends hold 53% and 53% of their own turn, and the inner is short of the centreline by 0.854 against the closed form 1.117 — 76% of it. Widen the bore and the path lifts off.

    Which walls a strand is held by

    A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.

    part 10 · strands
  16. An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

    A drum is a size, a wrap is a shape

    A strand's whole behaviour is decided by where it leaves each body and how far round it goes, and both are angles. So a belt drive's velocity ratio, its wrap angles and its tackle's mechanical advantage transfer between drives of any size — and the one thing that does not is how much strand there is.

    part 13 · strands
  17. One belt, five steps. Two stepped pulleys 400 mm apart, each with five steps whose radii on the two shafts always sum to 160 mm, giving five speeds a factor of 1.6 apart. The belt is on step 4: 98.46 mm driving 61.54 mm. Solid: an open belt, 1306.066 mm on this step against 1302.655 mm on the middle one. Dashed: a crossed belt, 1367.552327 mm — the same on every step, because a crossed belt's length depends on the two radii only through their sum. Faint circles are the other steps. Dragging moves the belt to another step.

    One belt for every step

    A lathe changes speed by moving one flat belt between the steps of two stepped pulleys, so every pair of steps has to take the same belt. A crossed belt's length depends on the two radii only through their sum, so steps whose radii sum to the same value take the same crossed belt exactly — to the last bit of the arithmetic, over 266 random pairs. An open belt's length depends on the difference too: at constant sum it grows by d²/C, then d⁴/12C³, so an open belt made for the middle step of a five-speed pair is 12.3 mm short on the outer steps, and those steps must give up 3.74 mm of radius sum to take it.

    part 14 · strands
  18. One open belt on two drums. Two identical drums on parallel shafts 400 mm apart, one reversed against the other, seen from the side; the belt runs between them at a station 60 mm from the middle of its 300 mm travel. The drums are cut so that the speed ratio grows by the same proportion for every millimetre the belt is moved — from 1/2.56 to 2.56 over the travel — and so that one open belt fits at every station. At this station the driving radius is 94.45 mm, the driven 64.85 mm, a ratio of 1.456. The profiles are not cones, but the difference is a millimetre or two on radii of eighty and cannot be seen at this scale; the figures below measure it.

    A cone drive is not two cones

    Take a stepped pulley to its limit and the belt runs on two drums, shifted continuously along them. Drawn, the drums are straight cones of opposite taper, and a crossed belt fits them everywhere because their radius sum is constant. But straight cones grade the speed unevenly — 0.58% per millimetre of shift in the middle, 0.72% at the ends — and an evenly graded drive needs drums whose radii are logistic curves. And an open belt, the common kind, fits neither: on a lathe-sized drive it is 12.3 mm short at the ends, and two identical drums that take it and keep the grading must each be shaved in proportion to their radius, 2.69 mm at the large end and 1.05 mm at the small.

    part 15 · strands
  19. The drums a ratio law asks for. The two drum radii along the travel for the law speed even, the power λ of the ratio changing evenly (its logarithm at λ = 0) with λ = 1, over the range 1/2.56 to 2.56 in 300 mm, with a crossed belt so that the radii sum to 160 mm everywhere. Faint: the even proportion's logistic drums; dotted: the driven drum turned end for end, which lies on the driving drum only if the two are one shape. For this law they are not: the two shapes differ by up to 30.7 mm. The driving drum departs from the straight cone between its end radii by up to 16.18 mm, and its steepest taper is 30.8° from the drum's axis, against the straight cone's 13.2°.

    A drum for every law

    A cone drive's drums can be cut for any ratio law along the travel, not only the even proportion a machine tool wants. A crossed belt fits every law whose ratio stays positive, and one open belt fits any of them after a shave that depends only on the ratio at each station, so one table of shaves serves them all. What the law decides is everything else. Only the even proportion keeps the two drums one shape; the speed changing evenly or a fan's power changing evenly each need two different drums, 30 and 55 millimetres apart, and pack the change of speed into the slow end at three and fifty times the even law's rate.

    part 16 · strands
  20. Softening a law is drawing a taut string through its tolerance. The speed even law's ln q along the share of the travel (dashed), the tube of speeds within 11% of it with both ends of the range kept exact (shaded), and the law with the least peak grading anywhere in the tube (solid): it climbs from the slow end at the least peak slope the tube allows until it rejoins the demanded law, at 26% of the travel, and from there is the demanded law exactly; it never lags by more than the tolerance. The dotted line is the even proportion. Exact, the law needs 884 mm of travel for a belt that needs 300 mm on the even proportion; softened by 11%, 547 mm.

    A tolerance shortens the drum

    A cone drive's drums must be long enough that a belt never straddles too large a change of ratio across its own width, so a demanded speed law sets the drums' length through its steepest point. Allowed to deliver speeds within a tolerance of the demand, a law's least steep version is a taut string through a tube: the demanded law with its steep end capped, exact everywhere else. Softening the speed law by 11% shortens its drums from 884 mm to 547, where the even proportion needs 300, and leaves it exact over three quarters of the travel. What softening never does, at any tolerance a designer would accept, is turn the law into the even proportion: that needs 52% for the speed law and 165% for a fan's.

    part 17 · strands
  21. On even drums the law lives in the cam's groove. The station the belt must stand at on drums cut for the even proportion, 300 mm long, against the share of the operator's handwheel travel, for the speed law (solid) and for the even proportion, whose groove is straight (dotted). Speed ratios are marked at the right; on the even drums the logarithm of the ratio is proportional to the station, so the groove is the demanded law redrawn. The speed law's groove is steepest where the handwheel starts, 2.95 times the even proportion's slope, and least steep at 0.45 times it; on a barrel of 40 mm pitch radius with a 30° pressure-angle limit it needs 6.1 turns of the handwheel, against 2.07 for the even proportion.

    A cam moves the law off the drums

    A cone drive cut for a steep speed law needs long drums, because a belt cannot straddle a steep change of ratio across its own width. Cut the drums for the even proportion instead and let a cam between the handwheel and the belt fork place the belt, and every law runs on the same 300 mm. The steepness does not go away. The cam's groove is steeper than the even groove by exactly the factor the law's own drum was longer, 2.95 for the speed law and 44.7 for a fan's, and it is paid in handwheel turns instead of drum: six turns for the speed law against 884 mm of drum, and a fan law that needed thirteen metres becomes buildable.

    part 18 · strands

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