Machine Kinematics
solved, then drawn
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The thread: Somebody had a problem, part 2 of 2

Every mechanism in the canon was invented to solve something specific — a piston that had to move straight, an indicator that had to trace, a film that had to stop thirty times a second. The problem explains the shape.

194 essays in 2 parts: part 1 · part 2

The same multiplier, applied to the error. A tong whose units are cut to an angle 0.01 radians away from the drawing. If one unit is out, the span is out by that unit's share and nothing more; if every unit is out the same way — which is what a machine setting or a worn tool produces — the error is multiplied by the unit count, exactly, to 7.6e-14. The third column is what would happen if the errors were independent and equally likely either way: the accumulation goes as the square root of the count instead, and the difference between the two columns at thirty-two units is a factor of 5.66. Which column applies is a question about how the parts were made, not about the mechanism. As built

The error that is repeated

Thirty-two units cut on one setting of one machine are thirty-two copies of one error, not thirty-two draws from a distribution — so a tong's span is out by thirty-two times a unit's, not by the square root of thirty-two times it. The two estimates differ by a factor of 5.66, and the second one is the comforting one.

 A machine measuring its own shape. A four-bar with an encoder at each end of its one freedom. Every pose gives one scalar equation — Freudenstein's, which is linear in the three invariants — so 40 poses make a three-column least squares with a condition number of 8.758 and no instrument outside the machine anywhere in it. With perfect encoders the invariants come back to 8.64e-15; with encoders good to a milliradian they come back to 5.40e-3, an amplification of about 6.26. What comes back is a shape and only a shape: the same three numbers describe this machine and one a quarter of the size, and nothing an encoder can read separates them. Numbers that were measured

A machine that measures itself

Put an encoder at each end of a one-freedom loop and every pose gives one scalar equation. The equation is Freudenstein's, it is linear in three unknowns, forty poses make a three-column least squares at a condition number of 8.76 — and no instrument outside the machine is involved anywhere.

 The same cycloidal rise behind a bent joint. A cycloidal rise over 120° of cam angle, with the camshaft driven at constant speed through a Hooke's joint bent at 30°. Across: the fraction of the rise; up: the follower's acceleration in time, divided by its peak on a steady shaft. The joint turns the camshaft unevenly with a period of half a turn, so where the rise falls in that period — the clocking, the cam angle at which the rise begins, measured from where the shaft turns fastest — decides what the follower feels. At the worst clocking, 100°, the peak is 1.352 times the steady one; at the best, 30°, it is 0.828 — lower than on a steady shaft. Every curve still starts and ends at nought. Dragging changes the bend. Prescribed motion

A cam behind a bent joint

A cam's motion law is written in cam angle and delivered in time, and the two are the same only on a shaft that turns steadily. Drive the camshaft through a Hooke's joint and the follower's acceleration gains a second term — the law's velocity times the shaft's angular acceleration — that a steady shaft does not have. Where the rise falls in the joint's half-turn then decides everything: at a 30° bend the same cycloidal rise peaks 35% above its steady value or 17% below it. And past a bend of 34.5° a well-clocked cycloidal cam gives its follower less peak acceleration than the constant-acceleration law, the one designed to minimise it.

 Where each flank slides worst, and a shift that makes them equal. A 17-tooth pinion driving a 51-tooth wheel, module 1 and 20°, with the pinion's cutter moved out by 0.3574 and the wheel's in by the same, so the centre distance does not change. The two curves are each flank's specific sliding along the path of contact — the sliding speed over that flank's own rolling speed — clipped at ±6. Each is worst at its own root: the pinion's at the start of contact, where it reads 1.962 in size, and the wheel's at the end, 1.961. Unshifted the pinion's is 11.09 and the wheel's 1.38. At a shift of 0.3574 the two are equal, at 1.961. The contact path — the part of the line of action drawn — slides along the line as the shift changes. Dragging changes the shift. Teeth

The shift that evens the wear

A pinion's flank slides hardest at its root, and on a small pinion much harder than anything on its wheel: a 17-tooth pinion on a 51-tooth wheel reads 11.1 at its root against the wheel's 1.38. Moving the pinion's cutter out and the wheel's in by the same amount leaves the centre distance alone and slides the contact path along the line of action, and at a shift of 0.357 the two roots slide equally, at 1.96. That shift is not the undercut cure — the same pinion needs 0.006 for that — and it grows with the ratio, so a pinion that needs no shift at all to be cut cleanly still needs a third of a module to wear evenly with its wheel.

 The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at 0° of driver, with the driver's locking disc of radius 18.8 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, outside the 6 V notches cut into it, each face tilted 14° off the chord, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 159.8°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot. Motion that stops

A V holds the wheel to first order

A Geneva wheel locked on concave arcs holds to first order and carries one pin; locked on flat chords it carries two and holds only to second order. Cut the lock as a shallow V instead — two faces tilted a few degrees off the chord — and it keeps both. Its disc is the crank times sin(β + γ), small enough for a second pin up to 34° at four slots and 5.6° at twelve. It rocks as acos(cos γ − c/a) − γ, which is linear in the clearance below 2a sin²γ, and at a running clearance of 0.1 on a crank of 30 it holds five to eight times tighter than the flat. The trade was between two shapes, not a law.

 The same drive with a smaller eccentric: every pin touches. Twelve pins on a ring of 60, pin radius 5, and the disc exactly as those pins generate it, at 10° of the eccentric. Left, the eccentric at exactly R/N = 5, whose roots are round sockets: 9 pins touch (solid) and 3 stand clear (dashed), the pins near the roots. Right, the eccentric at 0.8 of that, 4.00, whose roots are smooth: 12 of twelve touch. Over a pin pitch the socketed disc touches 9 or 10, and every smaller eccentric measured touches all twelve. The shape is the unknown

A disc whose roots are not sockets

Put a cycloidal drive's eccentric at exactly the pin circle's radius over the pin count and every root of its disc is a round socket; take any smaller eccentric and the roots are smooth. The sockets looked like a benefit — a pin in one closes its gap both ways at up to the full pitch offset. Measured from a half of that eccentric to all of it, they are the opposite. The socketed disc touches nine or ten of its twelve pins, every smaller one touches all twelve, and its first contact keeps the smallest share of the pitch offset of the seven. What a smaller eccentric really costs is the pitch offset itself: the free turn per unit of clearance is 0.0365 divided by the eccentric's share.

 Six balls, two grooves each, and a cage. A Rzeppa joint bent 20°, input shaft along z and output shaft turned about x. Each ball lies where the plane of its inner-race groove, through the input's axis, meets the plane of its outer-race groove, through the output's axis; at equal input and output angles every ball is in the plane bisecting the two shafts, and the cage holds them there. Left, from the side: the two shafts and the bisecting plane edge-on, the balls on it. Right, along the cage's normal: the six balls round their circle, the plane of the bend drawn across it. At an input angle of 0° a cage clearance of c lets the output lag or lead by up to 0.350 c radians, c in ball-circle radii. What a joint is

A Rzeppa cage adds its clearances

A Rzeppa joint keeps constant velocity because its cage holds every ball in the plane that bisects the two shafts. Give the cage's windows a clearance c and the output can lag or lead the input. Six balls might be expected to limit that lag through the worst of them, or as a root sum of squares. They do neither. Opposite balls are one constraint, the cage's two tilts spend two of the three, and what is left adds the three pairs' allowances like clearances in series: 0.350 c at a 20° bend, twice what the worst ball alone would allow. The lag falls to nought each time a ball crosses the plane of the bend, grows with the bend itself, and a four-ball joint cannot hold it at all.

 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. Members that pull

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.

 Two wheels on one frame, on a rough road. A rigid frame on two wheels of 50 mm radius, 100 mm apart — 0.5 of the road's wavelength — on the road of 10 mm amplitude and 200 mm wavelength. The dashed curve is the path every wheel's centre follows on this road, whatever the frame does; the frame only fixes where on it the second wheel is. Over each wavelength of travel each wheel counts 204.847 and 204.847 mm, the length of one period of that path, against the 200 mm travelled; the frame pitches by up to 11.53°. Dragging changes the wheelbase. Wheels, and where they may not go

A second wheel cannot correct the first

A wheel on a rough road counts the length of its centre's path, not the distance travelled. Put two on one rigid frame and average them, and the error does not shrink at any wheelbase: both centres run on the same curve, fixed by the road and the wheel's size, and each wavelength of travel carries each of them once round it. A half-wavelength wheelbase smooths the count's ripple and rocks the frame instead. What does work is two wheels of different sizes. A bridging wheel's error falls as the inverse square of its radius, so two radii measure their own error: wheels of 0.6 and 1.2 m that overcount by 0.339% and 0.099% extrapolate to 0.018%, knowing nothing about the road.

 The best split for a given variator rating. The most span a single-planetary power split can have when its variator may carry no more than the share on the axis, for variators spanning 2, 3.333 and 6. Up to a share of one the best is a split with no circulation, reaching the variator's own span exactly at one; beyond it, a split whose variator carries more than the engine, gaining R − 1 of span per unit of share. The dashed curves are the third branch, with the variator's power running backwards, which is below the best everywhere. The ring marks the previous essay's split at a unit rating: 1.700, which is 2 − 1/R. The same variator in a split with no circulation, its share never passing 0.9, spans 2.703, and it approaches 3.333 as that share approaches one. Drawn wrongly

Six things a power split is not

A multiplier of its variator's span, a machine whose neutral the belt decides, a source of any ratio near that neutral, a travel that slides towards its pole for free, a design whose attachment is the choice that matters, and a machine a second planetary always improves. Six claims about continuously variable transmissions built as power splits, each of them what a careful person would say, each answered with a measured number.

 A pin with friction lets the cam's force miss the pivot, by the friction circle. A rope pulled to the right, gripped by a cam whose pivot pin has a radius of 0.16 (drawn larger) and friction 0.5, so the pin's force on the cam may lie anywhere tangent to or inside its friction circle, radius 0.072. The dashed lines from the contact touch that circle; the cam's force on the rope can lie anywhere between them, where a frictionless pin would allow only the line through the centre. The shaded wedge is the contact's friction cone, 21.80° either side of the normal for μ = 0.4. At a lean of 23° the cleat holds a forward pull (it holds up to 24.14°, against 21.80° with a frictionless pin) and lets go when pulled back (it locks backwards below 2.56°). Contacts that only push

A stiff pin buys grip with release

A cam cleat on a frictionless pin holds a rope while the line from the contact to the pivot leans no more than the contact's friction angle. Give the pin friction and the cam's force may miss the pivot by the pin's friction circle, so the cleat holds to a lean steeper by exactly the angle that circle subtends: tan γ ≤ μ + (ρ/h)√(1 + μ²). But the same friction makes a cleat leaning less than arctan(ρ/h) lock against a backward pull as well, and refuse to let go. Both edges of the working window rise, the lower at a slope of one and the upper at √(1 + μ²): the grip gained is almost all paid for in release.

 A tilting head balanced with one parallelogram and two springs. A desk lamp with a lower arm 0.4 m long, an upper arm of 0.35 m and a 0.5 kg head that tilts on the upper arm's end, its weight 0.08 m from that wrist. A parallelogram in the lower arm keeps a bracket at the elbow upright. Four springs of no free length: the lower arm's from a post on the base, the upper arm's from the elbow's bracket, and two for the head — one from the top of the elbow's bracket to a point on the head, which sees the head's angle from the vertical, and one across the wrist from the upper arm, which cancels the term the first brings in the angle between upper arm and head. At this pose, lower arm 0°, upper arm 80°, head 130° from the upward vertical, the lamp's energy differs from the first pose's by 5e-15 of the head's moment, and the torque at the wrist is 9e-10 N·m. Dragging moves all three joints. Machines you have met

Two springs stand in for a parallelogram

A lamp's head tilts on the end of its arm, and its weight's moment is a cosine of the head's angle from the vertical, so a spring can balance it only if the spring sees that angle. A second parallelogram carries the vertical out to the wrist and one spring does it. So do two springs with one parallelogram, or six with none: each spring that reaches across a joint brings a term in the joint's angle, and another spring across that joint takes it away. A spring across the wrist alone balances the head at one arm angle and leaves sin(Δ/2) of its moment over a range Δ.

 Six cables hold a three-by-three grid only when they run round a cycle. A grid of three by three square cells whose bars are rigid and whose six braces are cables, each on one diagonal of a different cell. A cable resists stretching and not shortening, so it forbids its cell's shear in one sense only. Right, the graph that decides it: one node per column and one per row, and one arrow per cable, from the column to the row for a rising diagonal and back for a falling one. This grid is rigid: its arrows run round a single cycle through all six nodes, so every node can reach every other, and a self-stress pulls on all six cables. Of the 5,376 ways to place six cables in six of the nine cells, twelve are rigid. Many of one thing

A cable is half a brace

A square grid braced with rigid diagonals is rigid when a graph on its columns and rows is connected, so a three-by-three grid needs five braces and eighty-one sets of five will do. Brace it with cables instead, which pull and do not push, and each cable becomes one direction of an edge. The grid is rigid when that directed graph can get from every column and row to every other, which takes six cables, and only twelve of the 5,376 ways of placing six do it.

 A spring hung below the body puts the block on alternate walls. The five prescribed poses, the crank that drives them and the compromise slide, with one spring of no free length from a fixed point on the frame at (3.41, 1.16) to a point (-0.60, -1.20) of the body's frame, 0.92 below its lower edge on an arm, stiffness 2.51 per unit length against a unit working load along the guide. The spring's pull is at most 11.7 times the load. With it the block bears on the walls + − + − + at the five poses, exactly the compromise's, and crosses its slot 4 times on the way. At pose 2 the spring pulls with 10.1 and the block needs, and gets, the − wall. The problem backwards

A spring balanced against the load

The best slide for five poses reaches all five with play in its slot only if its block bears on alternate walls, and neither the body's weight nor a working load puts it there. A spring can, and whether one does is a linear program. No point of the body can carry it: for every such point there are weights on the poses under which any spring's pull cancels. A point on an arm below the body can, and a spring from it pulling at most 11.7 times the load gives + − + − + exactly. The spring alone presses one wall throughout; it works by balancing the load, and only for loads within nine per cent of the one it was chosen for.

 The same links and pins, and between 12 and 15 link lengths in the stack-up. Every closed loop in a mechanism is one equation a tolerance analysis has to satisfy, and the equation involves every link the loop passes through. All 16 chains here have the same number of independent loops — 3, which is pins minus links plus one and is fixed by the two totals — but not the same shortest set of them. The bars are the total length of a minimum cycle basis, and they run from 12 to 15. So the smallest number of link dimensions that any stack-up on this mechanism can involve is decided by the graph, before a single dimension has been chosen, and two topologies a count cannot tell apart differ by 3 of them. As built

Where the shortest loops are

A tolerance stack-up goes round a loop, and every link the loop passes through is a dimension in it. Two eight-link chains with the same links, the same pins and the same number of loops can need twelve link lengths in their shortest independent set or fifteen — decided by the graph, before any dimension is chosen.

 Where a length comes from. A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. Each carries an error of 0.010, of which 60% is common to both because they were bored in one setup — the whole pattern shifts by that much and the distance between the holes does not change. What survives is the independent part, 0.0063 at each hole, combining to 0.0089 on the length. A drawing that tolerances the length at ±0.010 is describing a part nobody makes, and it is out by a factor of 0.894 — optimistic below a shared fraction of one half and pessimistic above it. Numbers that were measured

Where a length comes from

A coupler 3.5 units long is a part with two holes in it, and neither hole's position is the length. What reaches the length is only the part of each hole's error the two do not share — so a drawing that tolerances the length is describing a part nobody makes, and is out by a factor of √2 in one direction or by everything in the other.

 How evenly the ram is driven through the working stroke. The ram's speed through the working stroke of an offset slider-crank with a time ratio of 1.2 of its own, crank 1, rod 2.719, offset 0.847, divided by its mean over the stroke, against the fraction of the stroke's duration. The number after each name is the ratio of the fastest to the slowest ram speed while the ram covers the central 80% of its stroke. Driven directly at constant speed the slide gives 2.02. The drag link that gives the highest time ratio, 2.71, gives 2.60. The most even design that still reaches 2, a drag link of 4.75, 4, 2.25 on a ground of 1, gives 1.19 at a ratio of 2.03 — the same pattern the shaper's rocker showed, measured on a ram. Dragging moves that design's phase. Linkages

The first stage does not know what it drives

A drag link that evens a shaper's rocker evens a ram driven by an offset slider-crank just as well — within 3.8% of the best that ram can have — once it is turned by the angle between the two machines' stroke starts, to within a degree. The two outputs cover their working strokes alike to 1.5% when they share an own time ratio, so what the first stage has to know about the second is that ratio and where its stroke begins. The exception is a centred slide with a short rod, whose skewed stroke moves the best phase by 12° more than the rule says and makes the train more even than any rocker's.

 Where contact runs on the line of action. A 17-tooth pinion driving a 51-tooth wheel, module one, drawn as its line of action from the pinion's base tangency (left end of each line) to the wheel's (right end), for four designs; the thick stretch is where the teeth are in contact. Specific sliding at a root grows without bound as contact nears that gear's own base tangency. Standard: operating pressure angle 20.00°, contact ratio 1.636, worst root 11.086; ±0.357 shift: operating pressure angle 20.00°, contact ratio 1.558, worst root 1.961; four knobs, shifts ≤ 1: operating pressure angle 26.18°, contact ratio 1.200, worst root 0.834; four knobs: operating pressure angle 27.94°, contact ratio 1.200, worst root 0.753. The four-knob designs push contact away from both ends at once, at a larger centre distance and pressure angle, until the contact ratio reaches its floor of 1.2. Teeth

Four knobs for the root

A shift of the cutter can balance the sliding at a gear pair's two roots, but on a 17-tooth pinion driving a 51-tooth wheel both then slide at 1.96. Free both shifts and both addenda, let the centre distance go where zero backlash puts it, and keep a contact ratio of 1.2, tips a quarter of a module thick and a tenth of a module of clearance: the worse root comes down to 0.83 with shifts no larger than a module, and 0.75 without that limit — two and a half times less. Every optimum has its two roots exactly equal, runs at an operating pressure angle of 24° to 29°, and sits on its contact-ratio floor, which is what the gain is paid with.

 A cycloidal rise on a chain-driven camshaft. A cycloidal rise over 120° of cam angle, the camshaft driven by a roller chain moving at steady speed over an 11-tooth sprocket. The chain leaves the sprocket at a radius that varies as each pin seats, so the shaft speeds up and slows once per tooth, and its angular acceleration jumps from one sign to the other at every seating. The follower's acceleration (solid) therefore steps at every tooth — 4 or so inside this rise — by up to 0.414 of the steady peak, and the cycloidal law's continuous acceleration, the reason it is chosen, is lost. At the worst clocking against the teeth the peak is 1.165 times the steady shaft's (dashed). Dragging changes the tooth count. Prescribed motion

A cam on a chain steps at every tooth

A roller chain leaves its sprocket at a radius that shrinks and recovers as each pin seats, so a chain-driven camshaft speeds up through every tooth and its angular acceleration jumps back at every seating. That puts a step into the follower's acceleration once per tooth, whatever the law — a cycloidal rise, chosen for its continuous acceleration, gets four steps inside a 120° rise on an eleven-tooth sprocket, each 0.41 of its peak. The harm falls with the tooth count, as one over it, not up as its square, and the simple harmonic law, whose peak sits where its velocity is nought, is hurt least of all.

 A wheel on a real road rides the long waves and bridges the short ones. Three metres of a class C road — the ISO 8608 average — built from 80 sines of random phase from 5 cm to 90 m wavelength, heights in millimetres and drawn far taller than they are. The second line is the lowest point of a wheel of 0.6 m radius rolled along it: where it lies on the road the wheel touches, and where it lifts off the wheel bridges a trough. On this stretch it touches 12% of the way. A wheel counts its centre's path, so the texture it bridges no longer adds to its count, and the waves it follows still do. Wheels, and where they may not go

A real road sets its own power

On a road of one sine, a wheel's overcount falls as the inverse square of its radius, and two wheels extrapolate it away. A real road is a spectrum. Over the band road standards describe, every wheel smaller than the road's tightest trough — about a metre and a half on an average road — counts the road's whole length, and its size makes no difference at all. With the texture below that band, bigger wheels do bridge more, but the overcount falls as the two-thirds power of the radius, a power the road's spectrum sets: 2(3 − w)/(5 − w) for a spectrum falling as n^−w. The inverse-square recipe removes a sixth of the error.

 Twelve cables, any one of which can go slack. A three-by-three grid of square cells braced with twelve cables: every row and every column of cells holds one cell with a rising cable, one with a falling cable and one crossed with both (RFX / FXR / XRF, reading R rising, F falling, X crossed). Right, its directed graph, one arrow per cable, and every column and row node has exactly two arrows in and two out. The dashed cable is the one gone slack, the rising cable of the cell in column 1, row 1. With it removed the graph is still strongly connected and a self-stress still pulls on all eleven remaining cables, so the grid is still rigid. The same is true whichever of the twelve goes slack. Many of one thing

Twelve cables survive a slack one

A three-by-three grid braced with cables is rigid on six and never survives the loss of one while each cell holds a single cable. Let a cell hold two crossed and every one of the 262,144 fillings can be tried: none of eleven cables survives a slack one, thirty-six of twelve do, and the rule for which is a count at each row and column, two cables in and two out. That count is the whole rule on every grid narrower than four cells. At four by four it stops being enough, and a grid can meet it everywhere and still hang on a single cable.

 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°. Members that pull

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.

 A roller offset by 5.88, at 42.3° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, drawn with the follower fixed above and the cam turned anticlockwise to 42.3°. The follower's line of travel is 5.88 to the right of the cam's centre. The contact normal leans 21.81° from that line, which is the pressure angle, and measured off the drawn pitch curve it is 21.81°. At this offset the peak on the rise is 21.81° and on the return 21.81°; centred, they are 27.37° and 15.33°. Drawn wrongly

Six things a cam drive is not

A follower that is better off offset, a flat face with no size limit, a swinging arm that improves any follower, a groove that keeps its roller on one wall, a smooth law that is gentle everywhere, and a camshaft that delivers the law it was cut to. Six things people say about cam drives, each the natural reading of something true, each answered by a number the cam essays measured. They fail the same way: each reads the cam at first order, in positions and slopes, and the drive is decided at second order, in accelerations and curvatures.

 The flank bends towards the disc, and the pin's radius is measured against that bend. One root of a twelve-pin cycloidal drive, pin radius 5 on a ring of 60, with the eccentric at 0.97 of R/N. The thin line is the curve the pin's centre traces in the disc's frame, its tip pointing at the disc's centre; the dashed line is that curve offset towards the disc by the pin's radius, which is the disc's edge wherever nothing trims it. Either side of the tip the flank bends the other way, towards the disc, most tightly 22.6° of the eccentric from the root, with a radius of 5.36 (the circle, centre marked). The pin's radius, 5, is measured along the same normal to the offset point. The offset point stops short of the centre of curvature, so the offset does not fold and the root is untrimmed. The two meet at 0.9740 of R/N. The shape is the unknown

The flank that trims the root

A cycloidal disc's roots are trimmed, and its pins start standing clear, above an eccentric that was known only for one drive: 0.974 of the pin circle's radius over the pin count, for twelve pins of radius 5 on a ring of 60. The trimming is a bend. Either side of each root the flank of the pin's path curves towards the disc, and where that bend is tighter than the pin the offset folds. The bend has a closed form, so the threshold does too, √(1 − ρ²(N + 1)³/27(N − 1)) for pins of radius ρ, and it falls well below one for fat pins or many of them. On every drive measured, the classical eccentric loses contacts and lever together.

 With a seventh leg the dead yaw is not a dip at all. The paired Gough platform held level at home, turned through sixty degrees of yaw centred on its dead yaw: the smallest singular value of its six leg screws, which falls to 8.9e-9 at the dead yaw, and of the seven screws with a seventh leg from the base circle to the platform circle at base 60°, platform 255°. With it the smallest singular value is 0.0288 at the dead yaw and no less than 0.0287 anywhere in the sweep, against the six legs' 0.0141 ten degrees off the dead yaw and 0.0320 thirty degrees off. The best placement of the 576 tried is the first the slider offers. Several legs, one platform

A seventh leg opens the dead yaw

Six legs can never carry a paired Gough platform across its dead yaw without passing a singular pose, because their determinant changes sign there and every path crosses nought. Seven legs make no square matrix and have no determinant to change sign. They fail only where the twist the six leave free also does no work on the seventh, a second condition that paths miss. Placed well, a seventh leg holds the platform's smallest singular value near 0.029 through the dead yaw, above the six legs' own value ten degrees off it, on every one of a hundred random paths across. What it cannot do is lift the platform further than the six's next-weakest direction allows.

 Where the head spring hangs decides whether the arm's spring must push. The tilting-head desk lamp with one parallelogram: lower arm 0.4 m, upper arm 0.35 m, a 0.5 kg head with its weight 0.08 m beyond the wrist, and four springs of no free length. The spring from the top of the elbow's bracket to the head is hung 0.03 m along the head (the slider). Solved for an exact balance, its stiffness is 131 N/m, the wrist spring's 687, the lower arm's 561, and the upper arm's −434, negative: a spring that would have to push (drawn red). The upper arm's spring pulls exactly when the head attachment point is beyond 0.0615 m, 76.9% of the way to the head's centre of mass. Machines you have met

A spring that must push is hung in the wrong place

Balancing a lamp with a tilting head exactly, with springs of no free length, sometimes asks one spring for a negative stiffness, a spring that would have to push. It looked like a property of the lamp: a light upper arm under a heavy head might admit no design in which every spring pulls. It is a property of one attachment point. The upper arm's spring pulls exactly when the head spring meets the head beyond a point that is always short of the head's centre of mass, 77% of the way on this lamp, and at most 99% even with a head fifty times the arm's mass. Hang the head spring at its centre of mass and every spring pulls on every lamp.

 A compound split sees its gearset through one cross-ratio. The four shafts of the Ravigneaux 69/74/100, planets 13 and 13 on its lever, each at the position that makes every shaft's speed a straight-line function of it: sun2 at 0.000 (engine), carrier at 0.483 (variator), ring at 0.840 (variator), sun1 at 1.000 (output). The best compound split puts the variator between the two shafts shaded together and the engine and output on the other two. What the split can do depends only on the cross-ratio of the four positions, here 5.613, and a split with that cross-ratio beats every split on a single planetary up to a variator share of 0.542. A share of a third needs a cross-ratio of 2.52 and a half needs 4.67. More than one input

One cross-ratio decides the compound split

A compound power split, its variator between two shafts of a four-shaft gearset, can beat every split on a single planetary, but four real gearsets let it do so only up to a variator share of about a third. Whether teeth can do better turns on one number: the cross-ratio of the gearset's four shaft positions on its lever. A third needs 2.52 and a half 4.67. Every Simpson inside a ring of 100 teeth stops at 3.87, a share of 0.454. A Ravigneaux with large suns and 13-tooth planets reaches 5.61, a share of 0.542, and solved on its lever it passes a half. With planets of 17 teeth, neither family does.

 A slide spends one ball of every pair. A 6-ball Rzeppa joint bent 20°, at the symmetric position, with its cage slid along its own normal by 0 of the window clearance c and then tilted to allow the largest lag. Each column is one ball's window, from −c to +c, and the dot is where the ball sits in it at that lag: −1.00, 1.00, −1.00, 1.00, −1.00, 1.00 c. The lag is 0.3500 c. Opposite balls lie on one line through the joint's centre, so a lag or a tilt moves them opposite ways and a slide moves them the same way: whatever the slide gives one ball of a pair it takes from the other, and the lag falls as 1 − s/c. What a joint is

A floating cage frees three balls and not six

A Rzeppa cage that may slide along its own normal as well as tilt has a third freedom, and counting constraints against freedoms says a six-ball joint should then lose its hold on the lag altogether. It loses nothing: a slide moves both balls of an opposite pair the same way, so it spends one ball's clearance for every one it frees, and the six-ball lag stays at 0.350 c. A three-ball joint, whose balls have no opposites, loses its bound outright, because three ball centres always lie in some plane. Offset grooves then add a second limit, set by the worst ball alone, which takes over from the cage only past an offset of about 0.35 times the tracks' play over the windows'.

 A cleat's lever, and the squeeze the load has left it. A cam cleat at a lean of 10°, its pivot pin with friction 0.5 and a radius a tenth of the pivot's height (drawn larger, with its friction circle), and a release lever on the cam. Along a load path the rope's pull rises to one, falls to nought and rises again, and the squeeze follows it between the band's edges, 4.50 and 7.62 times the pull. Here the pull is 1, the squeeze 4.50, and the lever must turn the cam with a moment of 0.412 times the pivot's height to free the rope. Contacts that only push

A cleat set hard lets go easiest

A cam cleat on a frictionless pin is freed by a lever with no effort at all. With pin friction the lever must beat h(T − N tan γ) + ρ√(T² + N²), and since a stiff pin lets the squeeze N sit anywhere in a band, the effort is set by where in the band it sits: nought at the top, twice the pin's friction moment at the bottom. A rising load drags the cam in to the bottom, so a cleat holding the load it was set by is the hardest to free — 0.412 times the pull times the pivot's height at a lean of 10° — while one jammed hard by hand, or one whose load has eased, frees itself. Below the backward-lock lean nothing jumps in the effort. What changes is the state: the slack cleat keeps the squeeze its largest pull left.

 One body point, one stiffness, and an attachment point for each load. The five prescribed poses with the lightest spring, from a point within a unit of the body's outline, that puts the rocking block on + − + − + against the body's weight: from the body point (-0.60, -1.20), on an arm 0.92 below the body, to an attachment point at (2.93, 1.97), stiffness 2.513, pulling at most 8.97. Designed against no load, the weight, the working load or both, the program picks the same body point and the same stiffness every time and moves only the attachment point: no load (3.04, 2.10), the body's weight (2.93, 1.97), the working load (3.41, 1.16), weight and working load (3.29, 1.03) (all four marked). The problem backwards

The arm belongs to the poses

A spring from the frame can put a rocking slide's block on the alternate walls its five poses need, but only from points on an arm below the body, never from the body itself. That looked like a fact about the working load the spring was balancing, and the body's own weight, which never changes, might have moved the reachable points onto the body. It moves them nowhere. Under no load, the weight, the working load and both, the program finds the same 155 points of the body's frame, the same lightest body point and the same stiffness, and only the spring's attachment point moves. The arm belongs to the five poses. And the earlier spring's band of nine per cent on the working load was its design margin: the band grows with the spring's pull, about one unit of load per hundred of pull.

 Every surface tried, and the group it permits. The census the six lower pairs come out of. Each row is a surface, sampled at 240 points; the freedoms column is six minus the rank of a matrix with one row per point, saying that the velocity a twist gives that point is tangent to the surface. Nothing is fitted and no shape is recognised — the surface's own normals write the matrix down. Three different surfaces of revolution give the same group, which is the content of the classification; two surfaces give nothing, which is what almost every surface gives. Eleven surfaces, six groups. The last column is the ratio of the smallest singular value kept to the largest discarded, so a row reading 10¹⁵ is not near being reclassified by anybody's tolerance. As built

The pair a catalogue sells

A plain bearing is a cylindrical pair and a catalogue calls it a bearing. Add two thrust faces and it is a revolute pair, which is a different joint and changes every mobility count downstream. The kinematic identity of a bought part is decided by which surfaces touch, and the catalogue's word for it is not the same information.

 Where the two analyses cross. The band on the output angle at a crank angle of 57°, with every hole on every part given a position error of 0.010, against how much of that error each pair of holes shares. The flat line is what an analysis on the four lengths gives, which is the same number whatever the answer to that question. They cross at 0.53 and nowhere else: below it the lengths-only answer is optimistic, reaching 1.414× at holes located independently, and above it pessimistic, reaching zero when the error is entirely common and the distance between two holes is perfect however badly the pair is placed. The crossing is at one half because two holes contribute √2 and the surviving fraction is √(1 − shared). Numbers that were measured

Where the two analyses cross

Tolerance the lengths and you get one number whatever the shop does. Tolerance the holes and you get a curve, running from 1.414 times that number when nothing is shared to zero when everything is. They meet at a shared fraction of exactly one half, and the crossing does not depend on the machine, the tolerance or which length is being asked about.

 Shortening a centred slide's rod evens the dragged ram until the rod is about two cranks long, and then it stops helping. A centred slider-crank with crank 1 and its rod swept from 20 cranks down to 1.305, where its worst transmission angle reaches 40°, plotted against crank ÷ rod. For each rod, the ratio of the ram's fastest to slowest speed across the central 80% of its working stroke: driven directly, which worsens from 1.701 to 2.752; and with the most even of 821 drag links, at whichever phase angle between the stages suits it, that still gives a combined time ratio of 2, which improves from 1.435 to 1.246 at a rod of 2.0 (worst angle 60°) and then worsens again to 1.401 at the 40° limit. Dashed: the best design found on any offset slide, 1.181, at an offset of 0.6 and a worst angle of 65°. Linkages

A centred slide has a best rod, and an offset beats it

Behind a drag link that brings a shaper's ram to a time ratio of 2, a centred slide's rod is a knob with a best setting. Shorten it from twenty cranks and the most even cut improves from 1.435 to 1.246 at a rod of two cranks, then worsens to 1.401 before the transmission angle reaches 40°, because the rod's skew moves the ram's speed peak where a drag link can phase against it only until the peak stops moving and just grows. An offset beats it: at every worst transmission angle up to 65° some offset slide is more even than any centred one, the best at 1.181, and on the most offset slide the rod hardly matters.

 A taller, narrower field carries the same line. The field of action of two 17–51 pairs, drawn to one scale with the path of contact stretched 7 times: across, the transverse path of contact; up, the face width; the dark lines are the contact lines in it at one instant, a new one entering every base pitch. Left: the spur pair whose four knobs give the least worst-root sliding at a transverse contact ratio of 1.2, 0.834, on a face of 10 modules. Right: a 30° helical pair whose four knobs are chosen for a transverse contact ratio of 0.4, worst root 0.209, on a face of 26.5 modules, which is what carries the same mean length of line, 12.00 against 12.00 modules. Its overlap ratio is 4.21 and its total 4.61. At this instant 10.00 and 12.10 modules of line are in contact. Teeth

A helix pays for the root with face width

A spur pair's least worst-root sliding, 0.834 on a 17-tooth pinion driving a 51-tooth wheel, was bought by spending contact ratio down to the floor of 1.2 that continuity needs. A helical pair gets its continuity from the overlap along the face, so its transverse contact ratio can fall below that floor, and the root's sliding falls with it, nearly in proportion: 0.209 at a third of the contact and 0.102 at a sixth, on the pitch point's law π εα (1/z₁ + 1/z₂) / tan αw. The helix angle itself buys under a tenth. What pays is the face: the mean length of line in contact is the transverse ratio times the face, so carrying the spur pair's load on as much line at a third of the contact takes a face 2.65 times as wide.

 A tilted face and a shorter parallel bracket hold the same line. A flat face on an arm pivoted 70 from a cam's centre, at 70° of cam rotation, the symmetric cycloidal programme with a lift of 20, base circle 33.725. Dark bracket: the face held 43 off the pivot and tilted 30° to the bracket's normal. Light bracket: the face held parallel, 37.24 off the pivot, which is 43 cos 30°. Both put the face on the same line at every angle of the cam — the support from the cam's centre agrees to 2e-14 and the normal's direction to 9e-16 radians over a turn — so the cam, the contact and the force on the pivot are the same; the tilted arm only swings 30° further round. Prescribed motion

An arm that lifts against the cam can beat a slide

A flat face on a swinging arm seemed to offer three choices: how far its pivot is from the cam, how far the face stands off the pivot, and the angle the face makes with its bracket. The angle is not a choice. A line fixed to a body turning about a pivot is fixed, up to the turning, by its distance from the pivot, so a face tilted β on a bracket c long is the face on a parallel bracket c cos β long, the same line at every instant. The real third choice is which way the arm lifts. On a symmetric programme the two ways are mirror images. On a quick rise, an arm lifting against the cam's turning needs a base circle of 6.6 to 13 where a sliding face needs 32.85, and one lifting with it needs 41 to 59, or has no convex cam at all.

 The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at -40° of driver, with the driver's locking disc of radius 13.5 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, short of the 6 flat chords cut across it, one between each pair of slots, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 126.8°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot. Drawn wrongly

Six things an indexer is not

More pins make a Geneva smoother, a finer ratchet is a more precise one, a locked wheel is a still wheel, a curved slot gives a smooth start, a cam indexer is always gentler than a Geneva, and a mechanism whose states are all in one piece can go from any of them to any other. Six things said about indexing mechanisms, each the natural reading of something true, each answered by a number the timing essays measured. They fail the same way: each judges the mechanism by its motion, and an indexer is decided at its switches — the instants a pin enters, a pawl drops or a lock takes over.

 Three wheels on one road, each bridging a different share of its texture. Two metres of a class C road built from 110 sines of random phase from 1 cm to 90 m wavelength, heights in millimetres and drawn far taller than they are, and the lowest points of three wheels of 0.3, 0.6, 1.2 m radius rolled along it together. Where a wheel's line lies on the road the wheel touches it; where it lifts off, the wheel bridges a trough. On this stretch they touch 4%, 2%, 1% of the way. Each counts its own centre's path, so the three counts differ by how much texture each bridges, and the differences carry the road's spectrum. Wheels, and where they may not go

Three wheels read the road they measure

A wheel's overcount on a real road falls as a power of its radius that the road's spectrum sets, so two wheels cannot extrapolate it away without being told the power. Three can fit it. Rolled together over 150 m of a class C road textured to 1 cm, wheels of 0.3, 0.6 and 1.2 m read a power of 0.64 against the law's 2/3, read the road's class from its level, and extrapolate the distance to 2 × 10⁻⁵ where the smallest wheel alone is out by 10⁻³. The standard band's floor, which no practical wheel bridges, turns out to lie on the same law and is extrapolated with it. The method is worth five times on a smooth road and five hundred on a rough one, and near the end of the road's texture it reads the end instead.

 Two planetaries joined at two places make one four-shaft lever. The sun–carrier, carrier–ring join: planetary 1, a sun of 26 in a ring of 52 (ratio 2.00), and planetary 2, a sun of 41 in a ring of 71 (ratio 1.73), each on its own lever with sun, carrier and ring at 0, K and K + 1, laid so that the joined members coincide. The four shafts below are the two joints and the free member of each. Their cross-ratio is 4.732, enough for a compound split to beat every split on one planetary up to a variator share of a half; its best split there spans 1.544, with the engine, output and variator where marked, and no member of either planetary carries more than 0.80 of the engine's power. More than one input

The compound split has gearsets of its own

A compound power split depends on its gearset through one number, the cross-ratio of the four shafts on its lever, and a variator share of a half needs 4.67. Simpsons stop at 3.87 inside a ring of 100 and Ravigneaux reach it only with 13-tooth planets. Two simple planetaries can be joined into four shafts in eighteen ways, twelve of them distinct gearsets, and ten of the twelve reach a half with ordinary ring-to-sun ratios. The cross-ratio is then no longer the question. Three joins reach any cross-ratio by setting two shafts nearly together and pay with 2.7 to 4.2 times the engine's power inside, and one, carrier to ring both ways, never reaches a half. A sun-to-carrier, carrier-to-ring join builds the lever with planetaries of 26/52 and 41/71, and nothing inside it carries more than 0.80 of the engine's power.

 A pretension chosen for the loss, and what a loss leaves. A three-by-three grid braced with twelve cables in a Latin square (XRF / FXR / RFX), pretensioned in the self-stress that keeps the most load capacity after any one cable goes slack, the largest cable pretension at one; line widths are in proportion to force. All twelve are taut: the cables carry from 0.81 to 1.00, and the smallest load, in any direction, that slackens another is 1.098, in units of the largest pretension. Intact it is 1.098; after its worst loss 0.404. Many of one thing

A spare is worth what its pretension leaves

Thirty-six ways of placing twelve cables on a three-by-three grid survive a slack one, and the count at each node cannot tell them apart. Their loads can. Pretensioned at its best, every one of them takes a load of about 1.1 times its largest pretension before a cable goes slack. Lose a cable and the pull it carried is shared out elastically, and what survives is between a quarter and a four-hundredth of that. A pretension chosen for the loss instead of for the intact grid lifts the worst case: to 0.32–0.40 on the Latin squares, 0.22–0.27 when four cells are crossed, and never above 0.096 when six are. The family to build is the Latin square, and even among Latin squares three kinds of spare are not equal.

 Near its threshold, the disc's flank sharpens to a point under the pin. One root of a twelve-pin cycloidal disc, pins of radius 5 on a ring of 60, cut with the eccentric at 0.97 of the threshold 0.9740 R/N. The line is the disc's edge; the circle is a pin touching it at the flank's sharpest point, 31.0° of the eccentric from the root. There the edge bulges towards the pin with a radius of 2.23, the flank's bend 7.23 less the pin's 5 (small circle, centre marked), so the contact is convex against convex, and a line contact's pressure per unit load there is 1.80 times a pin on a flat. At the root itself the edge is a hollow round the pin. The shape is the unknown

A disc cut at its threshold grows a point

A cycloidal disc's roots stay untrimmed up to an eccentric where the pin path's flank bends with the pin's own radius, and it looked as though a disc cut exactly there would have its most conformal contact, the pin and the flank sharing a curvature. It has its least. The disc's edge is the path offset inwards by the pin's radius, so where the path bends round a centre on the disc's side the edge bulges towards the pin with the path's radius less the pin's: at the threshold, a point. A line contact's pressure per unit load there rises as the inverse square root of the distance to the threshold, and the pins passing it are carrying load. Each of four drives has an eccentric of least contact stress at 0.67 to 0.80 of its threshold; cut at 0.97, the peak pressure is 9% to 80% higher for a lever 21% to 46% longer.

 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. Members that pull

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.

 The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out. Machines you have met

The lamp needs three springs

A desk lamp with a tilting head, balanced exactly with springs of no free length, was built with four springs whose peak pulls add to 162 N. Searched over every attachment point inside the same outline, the least total is 63.4 N, and it is reached with three: the head spring hung at the threshold where the upper arm's spring stops being needed, 77% of the way to the head's centre of mass. Beyond the threshold a second spring is the first one split in two, and splitting a spring never makes it smaller. A design with no parallelogram and six springs is never close: 213 N at its best.

 Past the play, a caged joint shares its load and a grooved one leans on one ball. Two six-ball Rzeppa joints bent 20°, at an input angle of 15°, each twisted 0.03 of a window clearance past its free lag, with every contact a spring of the same stiffness and the cage and balls settled where the stored energy is least. Bars: each ball's share of the total contact load. Left, grooves offset 0.1, where the cage sets the play: 0.17, 0.20, 0.19, 0.16, 0.14, 0.15, 5.91 balls' worth. Right, grooves offset 0.7, where the grooves do: 0.00, 1.00, 0.00, 0.00, 0.00, 0.00, 1.00 balls' worth. Dotted: an even sixth. What a joint is

A tight groove leans on one ball

A six-ball Rzeppa joint's free play is set by the tighter of its cage and its offset grooves, and the two limits look alike from outside. They are not alike once the play is taken up. With every contact made a spring of the same stiffness and the joint twisted past its free lag, a joint whose cage binds spreads the first load across four to six balls' worth and, a third of a window clearance on, gives every ball between 0.17 and 0.19 of it. A joint whose grooves bind puts the first load on one ball alone over most of a ball pitch, and two clearances on it still spreads it over fewer than six. At the same torque its hardest-pressed ball carries 1.5 to 3.0 times the caged joint's. Tightening the tracks to cut the play buys the play with uneven loading.

 A straight move ends where its posture does. The six-joint arm with its stops on, moving its tool in a straight line at a fixed orientation in the posture it starts in (back · elbow up · wrist flipped). Faint: the arm at the start. Solid: where the move ends, 66.0 cm along, stopped by the base joint at its stop. Dashed: the same line continued, which this posture cannot follow. Of the arm's usable postures at the start, the best would have gone 125.5 cm. One path to the tool

A straight move ends at a stop

A six-joint arm asked to move its tool in a straight line at a fixed orientation keeps the posture it starts in, because changing posture means passing a singularity. Followed from three hundred random poses inside the arm's joint stops, in random directions, the starting posture carries the tool a median of 19.5 cm. Half the moves end at a joint stop, two in five where the arm stretches straight and its posture runs out, and three in a hundred near the wrist singularity, the one the literature on straight-line moves is about. The best of the arm's other usable postures would have gone a median of 34.5 cm, and in a third of the moves it would have gone further. On the way, an earlier count of how many postures an arm can hold was found to be reading its base angle a turn out.

 A spring that pulls the same at every pose holds the band for less. The five prescribed poses of the rocking slide's body, with the arm point the earlier essays found lightest, (-0.60, -1.20) in the body's frame, and the spring that keeps the block on + − + − + for every working load from nought to the design load. The shaded sliver is every attachment point from which any spring that pulls can choose those walls at all. A constant pull of 44.6 from (3.10, 2.05), no stiffness, pulls at most 44.6 times the unit load. The problem backwards

A constant pull holds the band for less

A spring with free length was the obvious way to make the rocking slide's spring cheaper, and it does nothing: offered one, the lightest design takes none, at every arm point tried. The spring's law never mattered to where it may hang, because the walls depend only on which side of each crank pin the spring's line passes, and that cuts the attachment points to a polygon whatever the spring is. What the law decides is the price. Three poses bind at the lightest attachment point, and a spring whose tension follows its length must pay for the longest of them. A pull that is the same at every pose pays for none of them, and holds the band from nought to the design load for 44.6 units against 52.0. Designing for the band directly halves the hundred units the earlier essay read off by scaling.

 A cam cleat whose pin swings on a sprung link. A cam cleat at a lean of 2°, below its pin's lock lean of 2.56°, its pin carried on a link that swings about a frictionless pivot 2.2 pin heights above the rope, held by a spring. The link's lever is chosen so that, with the spring a quarter as stiff as the rope at that lever, cam and link turning together squeeze the rope 12.53 times the pull, the pin's own bottom edge. Along a path the pull rises to one, falls to nought and rises again. At this step the pull is 0: a fixed pin squeezes 12.53, this link 0.00, and a link at half that ratio 6.27. Contacts that only push

A cleat on a swinging pin has two leans

Carry a cam cleat's pin on a sprung link and, while the pin holds, cam and link turn together about the link's pivot: a second cam, frictionless, whose squeeze per unit pull is the cot of its own lean softened by the spring against the rope. A rising load is met at the pin's edge and a falling one at the link's, so a slack cleat keeps only what the link falls short of the edge — 12.5 times the pull on a fixed pin at 2°, 6.3 on a link at half the edge, nothing on a link at the edge itself. Nothing in it depends on the load's size: every force doubles with the pull.

 The same part, dimensioned two ways. Three holes at 0.00, 1.40, 3.50, each located to ±0.010 by the process. Dimensioned as a chain, each hole is placed from the last, so the two gaps are ±0.010 each and the errors accumulate over the whole span to ±0.0141. Dimensioned from a baseline, every hole is placed from H0, so the span is ±0.010 and the gap between H1 and H2 — which touches the datum at neither end — is ±0.0141. The two schemes are exactly √2 apart in opposite places. Nothing about the part decides this and nothing about the process does; the drawing does, and a tolerance analysis that starts from the lengths has already thrown the information away. Numbers that were measured

The same part, dimensioned twice

Three holes, one process, two drawings. Dimensioned as a chain the errors accumulate and the span is ±0.0141; dimensioned from a baseline the span is ±0.0100 and the gap between the second and third holes is ±0.0141. Exactly √2 apart, in opposite places, and nothing about the part or the process decides which — the drawing does.

 A centred slide's time ratio is its drag link's split of half a turn. The output angle of a drag link against its input angle over one turn: ground 1, input crank 4.019, coupler 3.920, output crank 1.339, the double crank that splits half a turn of its output most unevenly while keeping a worst transmission angle of 40.0°. A centred slider-crank driven by its output reaches its two dead centres at output angles exactly half a turn apart, whatever its rod, so its forward and return strokes take the input angles between the two dotted levels and outside them. At a phase of 66.3° between the stages they are 104.5° and 255.5° of input, a time ratio of 2.443. The largest over every phase angle is 2.443, against 2.304 for the best link of the quarter-step grid: no rod can raise it. Linkages

A centred ram cannot be timed past its drag link

A shaper's drag link and slider-crank were held to a combined time ratio of 2, and the centred slide's rod turned out to have a best length. Asking for a quicker return was supposed to shorten that rod and narrow the offset slide's lead. The first happens; the second runs the other way. A centred slide's dead centres are half a turn of its crank apart whatever its rod, so its ratio is the drag link's split of half a turn and nothing else: 2.304 on the grid, 2.443 off it, and no rod moves it. Below that ceiling the centred slide's best cut worsens from 1.118 to 1.466 and the offset slide's lead grows from 1% to 28%. Above it only an offset slide with a short rod reaches, and no slide keeping 45° reaches 3.

 The line in contact swings, and the least of it is what a tooth carries. A helical pair at a transverse contact ratio of 0.4 and an overlap ratio of 3.5. Left: the field of action, 0.4 of a base pitch across and the face up, with the contact lines at the moment the total length in contact is least; each line crosses the face while advancing 3.5 base pitches, so only a short piece of each lies in the field. Right: the total length in contact over one base pitch, as a share of its mean, 0.857 at its least and 1.143 at its most; the closed form b/(εβ cos βb)·[εα εβ − min(nα nβ, (1 − nα)(1 − nβ))] gives the least exactly. Dashed: the spur optimum at εα 1.2, which carries its whole face for four fifths of a pitch and twice it for one fifth, 0.833 and 1.667 of its mean. Teeth

The face carries the least line

A helical pair that gives up transverse contact to cut its root's sliding was priced by the face it needs to carry the spur optimum's mean length of contact line: 2.65 times as wide, at a third of the contact on a 30° helix. But the length in contact swings through every base pitch and the peak load stands on its least value. That least has a closed form in the fractional parts of the two contact ratios, exact against the swept field to 10⁻¹³. The spur optimum's own least is its face, a sixth below its mean, so least against least the factor is 2.21 when the overlap ratio is a whole number. Real faces land between whole numbers and need 2.21 to 2.87 times the spur's, 2.39 on a spur face of ten modules, and above 2.65 only on faces under ten. And a steeper helix is not always more line: at a fixed face, raising the overlap ratio from 1 to 1.5 takes a third of the least line away.

 Re-chosen attachment points buy back a third of the tilt at four times the springs. The tilting-head lamp with one parallelogram and the smallest springs of the earlier essay. Every spring 2 cm free and every attachment point re-chosen for the least worst moment, all springs still pulling: 0.0611 N·m, a third less, with peak pulls 29.6, 0.0, 41.6, 195.1 N (lower arm, upper arm, elbow bracket to head, wrist), 266.3 N in all. The upper arm's spring is dotted where it has no stiffness. Machines you have met

A free length is absorbed where the springs attach

Give every spring of the smallest balanced desk lamp 2 cm of free length and refit the stiffnesses, and the lamp is left with 0.093 N·m of unbalance, growing in exact proportion to the free length. The spare fourth spring that ideal springs had no use for recovers under three per cent of it, and moving the head spring beyond the threshold only makes it worse. What absorbs a free length is a spring whose length barely changes. Re-chosen attachment points buy back a third of the unbalance that way, and pay for it with four times the pull.

 One tyre pressed harder: its patch grows and averages the road it used to ride. 0.6 m of a class C road built from 110 sines from 1 cm to 90 m, heights in millimetres and drawn far taller than they are, under a 0.3 m tyre. The tyre is a circular tread on a bed of vertical springs; at each position its centre height is the one at which the springs carry the load. The grey line is a rigid wheel's lowest point, riding the peaks. The coloured line is the level the loaded tread's patch rests at, 9 mm deflection, a patch 146 mm long on a flat floor: it passes over the short texture as an average of it instead of following its tops. Wheels, and where they may not go

A load reads the road a radius cannot

Press a tyre harder and its contact patch grows, and a patch averages the road where a rigid wheel rides its peaks. Averaging is linear, so the overcount it leaves falls as the patch's first power and scales with the road's roughness in exact proportion, not as the two-thirds power three wheel sizes read. One 0.3 m tyre at deflections of 3, 9 and 27 mm fits that law and extrapolates the distance to 3 × 10⁻⁷ on a smooth road and 6 × 10⁻⁶ on an average one, against 2 × 10⁻⁵ for three rigid wheels. On the roughest roads its lightest patch lifts off in places, and it does worse than they do.

 Two cables slack, and what the other sixteen keep. A three-by-three grid with both diagonals of every cell cabled, the only placement of eighteen that stays rigid when any two cables go slack, pretensioned in the self-stress that keeps the most load capacity after the worst pair, the largest cable pretension at one; line widths are in proportion to force. Here the worst pair has gone slack together and the rest have taken up their pull elastically: the cables carry from 0.18 to 1.38, and the smallest load, in any direction, that slackens another is 0.242, in units of the largest pretension. Intact it is 1.695; after its worst pair 0.242. Many of one thing

A grid that loses two crosses every cell

To stay rigid when any two of its cables go slack, a cable-braced grid needs three arrows into and out of every row and column of cells. On three by three that is eighteen cables, both diagonals of every cell, and no placement of seventeen survives. On four by four it leaves 576 placements of 24, and for once the count is the whole rule. Pretensioned for the intact grid, 570 of them lose a cable outright to some pair of losses. Pretensioned for the pairs, all of them keep something, and the bracings with no empty cell keep the most, up to 0.198 of the largest pretension.

 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. Members that pull

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.

 A V carried on a slide and pressed onto the nose. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at 70° of crank, its coupler's round nose of radius 0.3 held in a V of two flat faces. The V rides on a slide along its own axis and a spring of force F presses it onto the nose, so it can be pushed back as the crank turns. Each face's normal is turned 45° off the V's axis, and misses the line to the coupler's pole at its contact by 47.6°. The shaded wedges are the friction cones for a coefficient of 0.2. Turning the crank past the V takes a moment of 1.39 F; released, the V pushes the crank back with 0.64 F. The chain before the lengths

A V pressed onto its nose has a window

A V of two faces holds a four-bar's coupler both ways and returns it, as long as each face is aimed off the coupler's pole by more than the friction angle. Squeeze the nose between both faces and two different things can happen. A rigid V cut a hundred-thousandth tight cannot lock, because the crank moves the nose across the V and never into it; it only stops short of home, inside a dead band that is μ/tan ψ of the play an equally loose fit would leave. A V pressed on by a spring, as a detent is, rides back along its own axis, and that is the direction a wedge locks in. It returns from an aim of arctan μ at the contact and locks at 90° less arctan μ about its axis, so its aim has a window, and the V that is most nearly reversible inside it is a right angle.

 Each fixed gear is a straight speed line through a braked shaft. The sun–carrier, carrier–ring join of a 26-tooth sun in a 52-tooth ring and a 41-tooth sun in a 71-tooth ring, its four shafts at their lever positions: sun₂ at 0.000, sun₁ + carrier₂ at 0.536, carrier₁ + ring₂ at 0.845, ring₁ at 1.000. The output stays where the compound split has it, on sun₁ + carrier₂. In the gear shown the engine drives carrier₁ + ring₂ and sun₂ is braked, so the shafts' speeds lie on a straight line through nought at the brake and one at the engine; the output turns at 0.634 of the engine's speed, a ratio of 1.58. More than one input

The winning join is a gearbox too

The sun–carrier, carrier–ring join of two simple planetaries builds a compound power split's half-share lever with ordinary teeth. Locked out of the split and given clutches and brakes, the same four shafts give five forward ratios below the split's range, from 2.73 to direct drive, and an overdrive of 3 above it. The two gears that brake a variator shaft sit 1.76 and 1.74 outside the split's ends, and three gearsets with the same cross-ratio put them at the same steps. Its planets space evenly, three and four to a set, and its meshes lose 0.41% of the engine's power along the split where the Ravigneaux that builds the same lever loses 1.01%. What none of these gearsets gives, with the output where the split needs it, is a usable reverse.

 Two stations, each shared by its own tools, and an arm that visits both. A cell with two work stations 3 apart. Station A is fed from the left and pressed from above; station B is pressed from above and emptied to the right; an arm pivoted below and between them picks a part up at A, swings it to B and parks midway. Every tool is 0.3 wide and reaches 0.15 past its station's centre, so the shaded squares are the only ground any two tools share, each square shared by its own station's tools and the arm. At 230° station A holds nothing and station B nothing. Links with a width

Two stations share a cut when they hand over together

A cam machine's region test needs one cut of its drive for every time its shared ground changes hands. A cell with two work stations has two pieces of shared ground, each with its own handovers, and a single cut can serve both if it falls when both stations stand empty. So the cell needs somewhere between its busier station's count and the two stations' sum, and its cams decide where. Out of step, a press-and-transfer cell needs four cuts; with the same tools and strokes put in step, two. The transfer arm that carries the part between stations makes one shared cut for free, because while it swings it is at neither.

 A relief as deep as the contact's own approach, spread across the flank. One root of the twelve-pin disc cut at 0.97 of its threshold, pins of radius 5 on a ring of 60, with the edge relieved inwards by a cosine bump of depth 0.01 and half-width 1 rad of eccentric centred on each flank. The relief is drawn 200 times deeper than it is: at true scale it is 0.2% of the pin's radius, the size of the contact's own approach at the design torque, and invisible. The thin line is the unrelieved edge; the pin touches it at the flank's sharpest point, where the unrelieved edge's radius is 2.23. The shape is the unknown

A relief is sized for one torque

A cycloidal disc cut near its trimming threshold has long levers and a point on each flank, where the pins carrying most of the load press hardest. Relieving the edge a little near the flanks can take most of that point away. On a twelve-pin drive at 0.97 of its threshold, a relief 0.01 deep spread across a radian either side of each flank lowers the peak contact pressure by a quarter and keeps every lever, to within 7% of the disc cut at its least-stress eccentric, whose levers are a fifth shorter. The relief does it by unloading the flank, not by reshaping it, so its depth must be the contact's own approach at the design torque. At a quarter of that torque it is worse than no relief at all, and a relief of hundredths of the pin's radius, which is what was expected to work, takes pins out of the load.

 Tighten the windows and the cage's limit drops below the grooves'. The six-ball Rzeppa joint bent 20° with its grooves offset 0.7 and a track play of one: across a ball pitch, the free lag the grooves alone allow (solid) and the cage alone allows with windows of clearance 0.5 (dashed). The joint's play is the smaller of the two. The cage is the limit wherever its curve is the lower, here at every input angle; it becomes the limit everywhere once the windows' clearance is below 0.559 of the track play. The largest play over the pitch is 0.163. What a joint is

A tight cage spreads the load until the tracks join in

A Rzeppa joint whose offset grooves set its play loads one ball first, and at equal torque its hardest-pressed ball carries up to three times a joint whose cage sets the play. Tighten the cage's windows below 0.559 of the track play and the cage becomes the limit at every input angle: the first load spreads over 3.6 balls' worth or more everywhere, the play falls, and at low torque the worst ball carries no more than a caged joint's. The cage does not stay in charge. Once the twist reaches the gap between the two limits the tracks engage, unevenly, and the worst ball climbs back towards the untightened joint's. A tighter cage widens the gap and pushes that torque up, so the windows' clearance is a choice about the torque at which even loading should end.

 A seventh joint steers the base roll off its stop. The S-R-S arm with stops on every joint, moving its tool in a straight line at fixed orientation from the same start (faint). With its upper-arm roll locked, a six-joint arm, the move ends 27.0 cm along at the base roll at its stop (dashed arm). With all seven joints and the self-motion steered at every step to move the joint nearest its stop away from it, it ends 61.5 cm along at the base roll at its stop (solid arm). Shown 60.0 cm along, with the six-joint arm stopped where its move ended. One path to the tool

A seventh joint cannot move the elbow

A seven-joint arm can change its shape while its tool stands still, and on a straight move that freedom can be spent keeping its joints off their stops. Spent that way, it lengthens the median move from 23.3 cm to 33.3 cm, against the same arm with its extra joint locked. But the gain is all or nothing. Half the moves a stop used to end gain under a tenth, and a quarter gain more than two and a half times. One stop it never gets round: the elbow's, whose angle is set by the distance from shoulder to wrist and is the same, to 10⁻¹⁵ rad, whatever the other joints do. Nor does it ever reach past the edge of the arm's reach.

 The smallest springs that balance the lamp, and the one that is not needed. The tilting-head desk lamp with one parallelogram, its springs of no free length placed to make the total of their peak pulls over the working range least, inside an outline of brackets 0.15 m high, attachment points 0.08 m past the wrist and along the head's 0.15 m. Peak pulls 22.7, 0.0, 23.2, 17.5 N (lower arm, upper arm, elbow bracket to head, wrist), 63.4 N in all and 23.2 at most. The head spring meets the head at 0.0615 m, the threshold, and the upper arm's spring has no stiffness at all: it is left out. Drawn wrongly

Six things a balancing spring is not

More springs share the load, a spring that must push wants a compression spring, a stiffer spring balances better, a refitted stiffness repairs a free length, least squares finds the balance, and the best spring is one of no free length. Six things said about springs that balance a mechanism, each the natural reading of something true, each answered by a number the essays on the desk lamp and the rocking slide measured. They fail the same way. Each treats the spring as the thing to be sized, and a balance is decided first by where the spring is hung and how its error is judged.

 Three steered pins hand a six-slot wheel from slot to slot. A 6-slot Geneva wheel with no locking disc and a driver carrying 3 pins, each in a radial guide whose radius a stationary cam sets. Each pin is in some slot for 120° of every driver turn: it runs in along a resting slot, indexes the wheel on a modified sine law over 40°, and runs out along the slot it moved. The solid curve is the path every pin follows while it is in a slot; the dashed circle is the crank's. At the moment shown pin 1 is running in, pin 2 is clear of the wheel, pin 3 is clear of the wheel; the wheel is held by one slot. Motion that stops

A wheel held by its slots needs three pins

A steered pin can hold a Geneva's wheel by running along a resting slot, but every pin is in some slot for exactly the crank pin's window, 180° less 360°/n of every turn, because it can leave a slot only by the mouth it will reach. Two windows of less than half a turn can never cover a turn, so two pins cannot hold a wheel without its disc at any slot count. Three can, on six slots and more, and four on four. At the Geneva's own share of index to dwell the index is no harder than the crank pin's, and the slots hold the wheel tighter than any lock in the field. The disc's work moves to the steering cam: on a run the cam's pressure angle is the run's own angle, and the six-slot drive's runs are 40°.

 a crank rocker with a post: the closest pair at one position. The same four-bar with a post bolted to the frame, just clear of the coupler's path. Every joint is where the solver put it, exactly as in the linkage field; the material is the only thing added. The heavy segment joins the two closest points over every pair of parts that is tested — which excludes pairs sharing a pin, since their material surrounds that pin by construction — and its length is the gap: 0.1809 here, between coupler · post. A negative value is a penetration depth, the distance the pair would have to be moved apart, and it is drawn in the warning colour. As built

Where the boundary moved

An earlier essay drew a line around what the essays here compute and listed one thing on the far side as a gap rather than a boundary: interference between links, which needed no new physics, only a body and a test. Here is what that turned out to cost and what it turned out to open.

 Which feature to hold tight. A budget of 0.020 divided between the four lengths in inverse proportion to their sensitivities, and the same budget divided between the features those lengths are derived from — a frame jig-bored in one setup at 0.90 shared, a crank drilled twice at 0.10, a coupler at 0.60 and a rocker at 0.30. The tightest tolerance moves from b to a. And the null result matters as much: with one process for the whole machine the two allocations are identical to the last digit, because the transmission factor is then common and divides out. Feature-based tolerancing changes the answer when the parts are made differently and not merely when they are made. Numbers that were measured

Which feature to hold tight

A budget divided between four lengths in inverse proportion to their sensitivities gives one answer. The same budget divided between the features those lengths are derived from gives the same answer exactly — unless the parts are made differently, in which case the tightest tolerance moves from the rocker to the crank and the frame's loosens by a factor of two.

 Either side of the crossing, only W₂ is somewhere else. Dixon's framework, pinned at B₁ with the bar B₁W₁ held in direction, a distance s = 0.3 along its motion before its first crossing (solid) and the same distance beyond it (dashed, W₂'s three bars only). The motion is Dixon's closed form with t = 1 − s²: every joint but W₂ is a function of s², so it is in the same place on both sides — here to the last digit — and W₂ is at s and −s, on opposite sides of the line through the black joints. The push on B₂ is the same at s and −s and is largest at the crossing, a fold; W₂'s side of the black line passes through nought. What can move

A push turns back at the crossing

Dixon's framework as built has play enough to go on past its first crossing onto either family, and the question left open was which one a push on B₂ chooses: the family whose self-stress is cheaper, or something else. Neither. Dixon's own formula says that near the crossing every joint but W₂ moves as the square of the distance along the motion, so the push is at a fold there on both sides, and no push can carry the framework through. Ten frameworks as built, pushed to their turning point and pulled back, all retrace, including the two that are cheaper beyond. Driven instead by W₂'s side of the black line, all ten go through, needing exactly the worst price on their path, and the price is continuous through the crossing.

 A ratio of 3 costs both stages forty degrees. The largest quick-return ratio of a shaper's drive against the worst transmission angle it must keep, from 30° to 60°: an offset slider-crank alone, 1.448, 1.360, 1.286, 1.224, 1.172, 1.128, 1.092; a drag link alone ahead of a centred slide, 2.867, 2.632, 2.443, 2.284, 2.130, 1.968, 1.809; and the two in series, both stages keeping the angle, 3.975, 3.427, 3.006, 2.696, 2.432, 2.184, 1.957, against the product of the separate ceilings, 4.153, 3.580, 3.143, 2.796, 2.496, 2.221, 1.976. The train reaches 3 at 40.0°. Marked: 40°, where the train reaches 3.006. Linkages

A ratio of three costs forty degrees

A shaper's two-stage drive, a drag link ahead of a slider-crank, could not be given a return three times as quick as its cut while its drag link kept 40° and its slide 45°. Drawn as a frontier, the largest ratio against the worst transmission angle both stages keep, a ratio of 3 costs 40.0° in both. The two stages in series reach 95.7% to 99.0% of the product of their separate ceilings at every angle from 30° to 60°, so two stages relax one stage's bound very nearly by a product. Shared unequally, every degree the drag link gives up buys the slide between one and a half and three: the drag link is the stage to let lean. And the trains on the frontier cut badly, the ram's fastest speed in the middle of its cut 3.2 times its slowest at a ratio of 3. On the way, the drag link's own ceiling at 40° turns out to be 2.443, not the 2.365 a single search had found.

 Where on the field the root band sits, and when it is loaded most. A helical pair at a transverse contact ratio of 0.4 and an overlap ratio of 1.5. Left: the field of action, the path across and the face up, with the bands a tenth of the path wide at each end shaded, where the sliding is worst, and the contact lines at the moment the share of the load in those bands is largest, 1.36 times its even value. Right: over one base pitch, the share of the load in the root bands over its even value (solid), and the total length in contact over its greatest (dashed), least 0.667 of its mean. Teeth

The least line carries the root

A helical pair's length of contact line swings through every base pitch, and the peak load stands on its least value. The roots, where the sliding is worst, sit at the two ends of the path of contact. The share of the load carried there swings too, and in the limit of a thin root band it reaches its peak while the line in contact is least: half as much again as an even share at overlap ratios just above one, a quarter above two, a sixth above three. A whole-number overlap ratio removes both swings at once. So the overlap that makes the least line largest is also the one that spreads the root's load evenly, and a face sized on the least line alone understates the root's worst moment by the root's own swing.

 Allowed 100 newtons, the lamp spends them on its attachment points. The tilting-head lamp with one parallelogram, every spring 2 cm free, its attachment points chosen inside the earlier outline for the least worst moment on any joint with the springs' peak pulls capped at 100 N in total. Worst moment 0.0620 N·m; peak pulls 22.5, 28.7, 13.9, 34.9 N (lower arm, upper arm, elbow bracket to head, wrist), 100.0 N in all. Each spring's zigzag is drawn heavier the harder it pulls, and dotted where it pulls less than half a newton. Machines you have met

The first pull goes to the spring the lamp did without

Give every spring of the balanced desk lamp 2 cm of free length and let it have more pull, and the worst moment it is left with falls from 0.086 N·m at 62 N to 0.052 at 227 N, the most any attachment points inside the outline can do. Half of that improvement costs the first 18 newtons; the second half costs the next 147. Spent on stiffness alone, with the attachment points held, extra pull buys nothing at all. And the first place it goes is the upper arm's spring, the spare that ideal springs had no use for, hung lower on its bracket while the head spring moves out past the threshold. The wrist spring, which carried almost all of the earlier re-attached design's cost, takes the largest share only after 140 N.

 B₂ and W₂ go round like the ends of a trammel bar. Dixon's framework in its own frame: the black joints on the horizontal line, the white on the vertical, nine bars joining every black to every white. Round the whole motion, all four families joined at their crossings, B₂ is at √2 cos ψ on the black line and W₂ at √2 sin ψ on the white, so the bar between them (heavy) slides with its ends on two perpendicular lines and turns once as the framework goes once round. The corner of the rectangle on B₂ and W₂ (small dot) is at angle ψ on the dotted circle of radius √2, and goes round it once. Here ψ = 60°: B₂ at 0.707, W₂ at 1.225. The arc at W₂ is the motor's angle, from the bar W₂B₂ to the bar W₂B₃, 37.2°. What can move

A drive that goes round is an angle

Dixon's framework moves round a loop through four crossings, and a push on B₂ folds at two of them while a drive on W₂'s side folds at the other two. A lever moving both was proposed as the drive that would go all the way round. It cannot, and no drive that is one number read off the joints can: round the whole loop B₂ and W₂ sit at √2 cos ψ and √2 sin ψ on their two lines, the ends of a trammel bar, so the loop is a circle and every such number has a top and a bottom on it. A lever only moves its two folds off the crossings. What goes round is an angle. Four of the framework's eighteen joint angles turn once as it goes once round, never stopping, and a motor at W₂ takes ten frameworks as built all the way round at the worst price on the loop — 1.00 to 1.46 times their worst crossing — after which every turn repeats the first.

 A slide turned about a point is free in a band round that point. The double slider of slides at 70° and a rod of 2, with a third block and a fourth at the opposite point of the rolling circle (radius 1.064), each on the line its pin draws through the crossing. Each extra slide is set to pass through its pin at one common pose of the rod, 45° from either pin's crossing, so a turn error of 0.01 rad pivots it about that pin's place then: -1.505 along the third's line and -1.505 along the fourth's. With a clearance of 0.01 the block is free while its pin is within c/sin ε = 1.000 of the pivot (the shaded bands). At the angle shown the third pin is at -1.505 and the fourth at -1.505: the rod is free. The paths points trace

Fit every extra slide at one pose

A double slider's extra slide, turned slightly wrong, leaves its block free in a band of the slide centred on the point it was turned about, and the rod free in windows where the pin passes through that band. Turned about the crossing, a third and an opposite fourth block have their windows a quarter of a turn apart and lock the rod from 0.0066 rad. Set each slide through its own pin at one common pose of the rod and every error becomes a turn about the place its pin has then, so every window contains that pose and the rod can never lock, with two blocks or six. Each slide pivoted where it alone is freest does better while the errors are small and locks from between 0.02 and 0.05 rad, where the common pose 45° from either crossing comes within a tenth of the best pivots there are.

 Five degrees below forty buy back an even cut at a ratio of three. At each worst transmission angle kept by both stages of a drag link ahead of an offset slide: the evenness of the cut, the ram's fastest speed over its slowest across the middle four fifths of the working stroke, for the train with the largest time ratio at that angle (dashed), which reaches 3.98, 3.75, 3.52, 3.43, 3.33, 3.24, 3.16, 3.08, 3.01 and cuts at 6.11, 5.21, 4.70, 4.35, 4.14, 3.90, 3.65, 3.46, 3.25; and for the most even train that still reaches a ratio of 3 (solid): 1.22, 1.26, 1.32, 1.42, 1.64, 1.90, 2.53, 2.73, 2.97, at 30, 32, 34, 35, 36, 37, 38, 39, 40°. The shaded band is 1.12 to 1.47, the best ratio-2 trains of the earlier essays. Linkages

Five degrees buy an even cut at a ratio of three

A shaper's two-stage drive reaches a quick return of 3 only by keeping no more than 40° of transmission angle in both stages, and the one train that does cuts unevenly, its ram 3.25 times faster at one point of the cut than at another. Searched for the most even cut that still returns three times as fast, the trains at 40° do little better, 2.97. But the price falls steeply below it: 1.90 at 37°, 1.42 at 35°, 1.22 at 30°. By 35° a ratio-3 shaper cuts as evenly as the best ratio-2 trains did. The even trains leave the long, nearly folded drag links that buy the most ratio for compact ones, and not one of them takes a longer rod than its angle requires.

 The bores, and the one line that has to pass through all of them. A hinge of 6 knuckles, its bores drawn at the distance each was made from the nominal axis in units of the bore tolerance. The leaf is a rigid body, so its pins are on one straight line — two parameters of position and two of direction — and it assembles when some line passes within the clearance of every bore. The line drawn is the one whose largest miss is smallest, and that miss is 0.875 of the tolerance. 2 of the 6 bores are at that distance and hold the fit; the rest are slack and could have been bored anywhere inside it without changing the answer. As built

A piano hinge is not forty door hinges

A three-knuckle hinge works because the misfit its bore errors create is smaller than the play already in its pins. A piano hinge has forty knuckles and thirty-nine of them are redundant, so the obvious reading is that it needs thirteen times the play. It needs two and a half times, and it can never need more than the bore tolerance itself — because a rigid leaf has one axis and a line through the middle of the errors misses every bore by at most the largest of them.

 The identification Jacobian of a four-bar, read by protractor. One row for every number the instrument reads and one column for every parameter that might be wrong. Each cell is the derivative of that reading with respect to that parameter, drawn to the right of its centre line when positive and to the left when negative, with the largest entry in the whole matrix at 4.09e-1. 14 rows against 4 columns: far more equations than unknowns, which is what makes an identification a least-squares problem rather than a solve, and what makes the question of which combinations of columns cancel a real one. These are the same derivatives the tolerance field computes one at a time — the same matrix read down instead of across. Numbers that were measured

A calibration is a synthesis with more equations

The site's second field prescribes three input–output pairs and solves a 3 × 3 linear system for a linkage. This one measures thirty pairs and solves the same system in the least-squares sense. Same matrix, same coefficients, same closed form — and the only structural difference produces every question this field is about.

 What each instrument recovers. The same twenty poses of the same four-bar, read three ways. A protractor on the output link recovers 3 of the four lengths and leaves the fourth exactly invisible, because its readings are dimensionless in the lengths and scaling the machine does not move them. A coordinate machine on the tracing point recovers all six parameters — the four lengths and the two that say where the tracer sits — at a condition number of 162.3. Using both recovers the same six at 26.1, 6.2 times better, which is the case for putting two instruments on one machine: not more parameters, better-conditioned ones. Numbers that were measured

Two instruments disagree about the worst

A protractor recovers three of a four-bar's parameters at a condition number of 5.2. A coordinate machine recovers six at 162. Neither number says which parameter is worst recovered, and when both are asked, they name different ones — because a condition number is a summary of a list and the list is what a report needs.

 The identification Jacobian of a four-bar, read by coordinate machine. One row for every number the instrument reads and one column for every parameter that might be wrong. Each cell is the derivative of that reading with respect to that parameter, drawn to the right of its centre line when positive and to the left when negative, with the largest entry in the whole matrix at 3.14e+0. 24 rows against 6 columns: far more equations than unknowns, which is what makes an identification a least-squares problem rather than a solve, and what makes the question of which combinations of columns cancel a real one. These are the same derivatives the tolerance field computes one at a time — the same matrix read down instead of across. Numbers that were measured

What a model is allowed to change

Before a calibration runs, somebody decides which numbers it may move. Leave one out and the fit absorbs it into the others; put one in that the instrument cannot see and the fit returns whatever the damping preferred. Both decisions are made before any measurement, both are checkable in advance, and neither is usually checked.

194 essays in 2 parts: part 1 · part 2

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