Machine Kinematics
solved, then drawn
Field

Members that pull

Every link on this site so far has been rigid: it holds two points at a distance, in both directions, and a configuration is the root of an equation. A belt, a chain, a rope and a tendon are none of that. A strand has no shape of its own — where it runs is decided by the bodies it touches — and it constrains one way only, so it does nothing at all until it is taut, and a mechanism made with one has a different mobility in different places.
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

The strand as a mechanism member — one closed form for the tangent runs and wrap arcs, the six systems it produces, and the two ways it is not a rigid link.

 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

The common-tangent solve that decides where a strand leaves each body, the four tangents two circles admit, and the minimisation that makes the right angle a consequence rather than a construction.

 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

The turning number of a taut strand, why a crossed belt's two wraps are always equal, and how one integer catches the mistake this geometry invites.

 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

The velocity ratio of a block and tackle computed by differentiating the run's length, checked against the sum of the parts' cosines, and the parity rule that decides which block carries the dead end.

 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

The unilateral constraint a strand imposes, the reachable region it leaves instead of a configuration, and why no number of strands can hold anything still.

 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

Closing a five-body run by solving for an idler's arm angle, the interval over which the idler is touching at all, and the closed form that says a tensioner near the edge of contact takes up nothing.

 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 support function as the arm a strand feels, a profile recovered from a demanded rate in closed form, and the exact ceiling on how much variation a demand may ask for.

 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

A drum filling with line in layers, the ratio that drifts as it fills, and the exact agreement between the layered reading and the spiral one at every layer boundary.

 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

What changes when a strand is made of finite links: the inscribed polygon, the quantised centre distance, and the fraction of a tooth that no adjustment can remove.

 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

Tendon routing as a length constraint, the one arrangement that makes the coupling constant, and the winding number a wrap angle cannot carry.

 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

Routing as a discrete input: the homotopy classes of a taut strand, how they differ from a mechanism's assembly branches, and what happens at the configuration where an obstacle stops obstructing.

 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

The boundary at which a body enters or leaves a taut run, the quadratic law that governs it, and why authority vanishes exactly where contact does.

 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

Sorting the strand field's quantities by what a scaling does to them, and what a measurement of a drive's motion recovers about the drive.

 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

A multi-turn wrap as a helix, with its plan view shown to be a planar wrap to ten decimal places, the length correction measured against the one ratio it depends on, and the fleet-angle rule derived.

 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

The inner of a Bowden cable modelled as a taut strand hugging the inside of every bend of its sheath, and pushed as a strand pressed against the outside; the lost motion measured on six routes against clearance times total turning, with the departure shrinking in proportion to the clearance; independence from bend radius over a ninefold range; net against total turning; and the change in a brake cable's take-up as a handlebar steers, matched to c·Δφ.

 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

The taut inner solved as a shortest path inside the bore by the funnel algorithm rather than as a wrap of one pulley per bend: the share of each bend still in contact falling from 95% to nothing, the threshold matching asin(c/L) to a quarter of a degree, and the lost motion saturating at the route's own chord deficit.

 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

Two five-step pulleys with one belt, open and crossed, on each step; the belt length each step needs at constant radius sum; the open belt's excess against d²/C and the next term d⁴/12C³; the radii an open belt needs on each step, with the shortfall in radius sum against d²/πC; the wrap on the smaller pulley step by step; and 266 random stepped-pulley pairs, open and crossed.

 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

A continuous cone drive: two identical drums, one reversed, with an open belt at one station of its travel; how evenly straight cones and logistic drums grade the speed ratio along the travel; the drum profiles as departures from a straight cone, for a crossed belt and for an open one; the shave that fits one open belt and keeps the grading, found by bisection against a proportional law; and an open belt's length error on straight cones as drawn, and with one constant shave.

 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

Cone-drive drums for a family of demanded ratio laws, in which some power of the speed ratio changes evenly along the travel: the two drums each law asks for against the straight cone; the proportional change of speed per millimetre along the travel for four laws; a table of each law's departure from the cone, drum mismatch, steepest taper and peak grading; the speed-even and roll-surface-speed laws using one pair of drums exchanged; and the open-belt shave for every station of four laws falling on one curve in the ratio.

 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

Cone-drive drums for demanded speed laws softened within a tolerance on the delivered speed: the speed law, its tube of tolerance and the least-peak law through it; the travel each law needs against the tolerance, with the tolerance at which it can become the even proportion; the grading along the travel, exact and softened; the drums for the speed law exact, softened and the even proportion, each as long as the same belt needs; and a ledger of travels at four tolerances.

 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

Cone-drive drums cut for the even proportion with a barrel cam on the handwheel placing the belt: the groove each law asks for on the even drums; the cam's steepest slope against the length the law would need cut into its own drums, for every law and tolerance; the belt's axial walk on the cam's drums and on law-cut drums; the two pairs of drums to scale; and a ledger of drum length against cam turns.

All essays