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The thread: Exact and approximate, part 2 of 3

Watt's straight line is not straight and Peaucellier's is. The difference is measurable, it took ninety years to close, and the error curve of an approximation is a more interesting object than the approximation.

The pose set is finite exactly when the part is held. Two arrangements, both with 0.06 of clearance on every contact, with the set of positions the part's centre may occupy drawn to scale. The one on the left holds: its pose set is a small bounded polyhedron, and every dimension of it is proportional to the clearance. The one on the right does not: its pose set runs off the page in the direction the part slides out of the vee, and giving the contacts a tighter tolerance narrows the box without ever closing that direction. A tolerance cannot buy a hold. The clearance decides how big a finite pose set is and the arrangement decides whether it is finite, and the second question has to be settled first because no amount of the first will settle it.

Contacts that only push

Held is not located

Back every obstacle off by a clearance and the permitted poses become a polyhedron — bounded exactly when the arrangement is a hold, since an unbounded direction of it would be a ray of the escape cone. So whether a part is held is whether its pose set is finite, the clearance is what gives that set a size, and the two questions have to be settled in that order because no tolerance settles the first.

Three points, three orders of contact. Replace the coupler point by a crank pivoting at the centre of its path's osculating circle, drive the linkage away from the instant, and measure how far the point gets from that circle. The distance grows like a power of the step, and the power is the number of derivatives that agreed. An ordinary point gives 3.00; a point of the cubic of stationary curvature gives 3.94; a Burmester point, where the curvature is stationary and its rate of change is too, gives 4.97. Nothing in the measurement knows which kind of point it was handed. This is what the two special curves are for: they are where a single pivot can replace a whole linkage for longest. positioned by solving, not by drawing.

The motion, not the mechanism

How long a pivot stands in for a linkage

Throw the four-bar away and replace a coupler point by a crank pivoting at the centre of its path's curvature. The two separate like the cube of the crank step for an ordinary point, the fourth power on the cubic, and the fifth at a Burmester point — three exponents that are the whole field, measured with no derivative anywhere in the measurement.

What each curve costs, in cosines. Every polynomial in the two arm angles is a constant plus a sum of terms A cos(mα + nβ + φ) with whole-number m and n, and the number of those terms is what a machine has to build. The count is not the degree and not the monomial count: a circle costs one term, a general line two, and a lemniscate — degree four — costs five, fewer than the cubic above it, because its symmetry cancels frequency pairs the cubic keeps. Each row's expansion was checked against a direct evaluation of its own polynomial at random angles, worst disagreement 1.8e-14.

The problem backwards

Prescribing a curve rather than points

A four-bar can be made to pass through nine prescribed points and no more; past nine the problem is over-determined and the answer is an optimiser's. Prescribe the whole curve as an equation instead and there is no counting to do — but the mechanism that comes back has four hundred bars where the four-bar had four.

Seven arrangements, one routine, and the two that hold. Every row is the same three steps: write down one row per contact — the moment of its normal about the origin, then the normal itself — take the convex hull of those rows, and ask whether the origin is inside it. The parts differ, the numbers of contacts differ, and the routine does not. Two of the seven hold. The other five leave the part something, and the interesting column is what: four rays of rotation for the pinwheel, a translation straight out of the vee, and for the last two a whole line rather than any number of rays, which is what a rank below three means and is the case a reader has to be warned about. Note that the four contacts of the second row are the four of the first row, on the same four edges of the same square, at the same distance along each. positioned by solving, not by drawing.

Drawn wrongly

Six things a hold is not

A rank read as a restraint, a count read as an answer, four contacts placed the wrong way round, a nullity taken for a spin, a part free in every direction and unable to leave, and a tolerance offered as a cure for an arrangement that was never a hold. Six claims, each with the number that kills it.

Why this measurement is not a derivative. The honest failure mode of the field's instrument, drawn rather than hidden. Every set looks like its own tangent space near the identity — that is what a tangent space is — so a chain sampled over a thousandth of a radian reports the dimension of its velocities, which is the number the screw system already gives. Four pins at random has four joints, and sampled over 10⁻⁹ radians its displacements occupy four dimensions; sampled over two radians they occupy six. The step is at 10⁻⁶, which is where the departure from the tangent space falls below the rank tolerance — so the position of the step is a fact about arithmetic and the two plateaux are facts about the mechanism. A group is a statement about displacements you could compose, and no derivative can make it.

What a joint is

The instrument that is not a derivative

Sample a chain of four random pins over a millionth of a radian and its displacements occupy four dimensions, exactly as a SCARA arm's do. Sample the same chain over two radians and they occupy six. The step is at 10⁻⁶, and where it sits is a fact about arithmetic while the two plateaux are facts about the mechanism.

More than one input

Two shafts that must be in line

Asking a two-stage gear train for a ratio is easy. Asking it for a ratio and for its input and output shafts to be coaxial is asking for a solution of two equations in four integers, and there is no reason for one to exist. A twelve-to-one reverted train needs a sixty-three-tooth wheel before it has any solution at all — while sixteen to one, a larger ratio, manages with fifty-six.

Half a tooth, spent three ways. Each bar is one beat of the escape wheel: exactly half a tooth pitch, 6.0° on 30 teeth, whatever the faces are cut like. The dark part is the impulse, which is the only part that does anything to the pendulum; the pale part is the drop, in which nothing is touching anything; the short tail is the lock-in run, in which the arriving tooth drags the wheel backwards as it settles. On the arc with no draw that tail is exactly zero and the budget has two terms. On every other face it is not, and the three still sum to the half pitch to twelve figures — which is the check, since the three are computed from three different contacts.

As built

What a drop cannot be smaller than

Two thirds of an escape wheel's travel is drop, and drop does nothing. The obvious economy is to cut it down, and it cannot be cut down, because every dimension it is made of has a tolerance and a drop smaller than the accumulated error is a tooth that does not clear the pallet it is leaving. The stack is 0.39°, and it barely moves when the tooth count triples.

One path to the tool

Where an error at the shoulder ends up

The same angular error at every joint of an arm, and the tool is out by 0.156 mm because of the shoulder, 0.017 mm because of the wrist roll and exactly nothing because of the last joint. The numbers are not properties of the joints. Each one is the distance from the tool to that joint's axis, measurable off the drawing with a ruler.

What a tensioner can take up, and where it stops being one. The run's length as the arm swings, over the whole interval in which the idler is actually touching the strand: 4.123 to 4.774 radians, and 1013.37 to 1030.99 mm — a range of 17.62 mm on a strand of a metre. Outside that interval the geometry still returns tangent lines and a length; what it returns is not a strand, because the path it describes cuts through the pulleys. The curve is flat at the left-hand end and steep at the right, which is the whole of a tensioner's design problem: its authority is proportional to the sine of half its own wrap, so an idler set near the edge of contact travels a long way and takes up nothing.

Members that pull

Where a strand stops touching

A body joining a run costs length as the square of how far it intrudes — exponent 2.0000, measured over four decades — so at the moment contact begins the strand's length is stationary. That is why a tensioner set at the edge of its own contact takes up 0.000245 mm of belt per millimetre it travels.

The ratio survives; the continuity does not. An involute pair holds its ratio at any centre distance, and that is not the same as working at any centre distance. The contact ratio — the length of the contact path divided by the base pitch, which counts how many pairs of teeth are engaged at once — starts at 1.647 for this 24 : 36 pair and falls as the shafts move apart, because the useful part of the line of action is bounded by the two tip circles. It reaches one at 2.82 mm, and below one a pair of teeth lets go before the next has picked up: the drive stops being continuous and becomes a series of arrivals. That is the real limit on the involute's indifference, and it is a limit on the teeth rather than on the tooth form.

The shape is the unknown

Where two shapes stop touching

A conjugate pair is exact at every instant it has a contact. It does not have one for ever: a profile is an arc rather than a curve, and both ends of that arc are somebody's decision — which is why the useful question about a pair of shapes is not whether they mesh but for how long.

Numbers that were measured

The instrument's error, multiplied

Repeat a whole calibration on independently noised readings at four levels three decades apart and the error in the recovered shape is linear in the noise, with a fitted slope of 0.9994 and a constant of 1.90. That constant belongs to the mechanism and the poses, not to the instrument — and it is bounded by one over the smallest singular value.

Three prescribed positions of a rigid body. The whole of the design problem, before any mechanism exists. A body has to occupy these three positions — each one a place and an angle, three numbers — and what carries it between them is not yet decided. A forward analysis starts from link lengths and finds the motion. This starts from the motion, and the lengths are what has to be found. The marked points are the poles: any planar displacement is a rotation about one point, so each pair of poses has one, and the arcs show the turn each represents through the body's own origin. A pole is a property of the displacement and not a mechanism — nothing has been chosen yet. 1 of the 3 poles lies outside this frame and is not drawn; near-parallel displacements push their pole a long way off.

The problem backwards

What a synthesis assumes it knows

Every construction in this field is handed a demand in absolute coordinates — three positions of a coupler plane, at stated places — and returns a linkage in the same coordinates. Scale the demand and the answer scales, which means the construction's whole content is about shape and its answer carries a size it was given.

A geared five-bar at 3 : 2, its coupler pin traced over 2 input turns. Two cranks of length 1 on pivots 3 apart, meshed through gears whose pitch circles, dashed, have radii 1.800 and 1.200, so the second crank turns 1.5000 times for every turn of the first and the other way. Coupler bars of 3 and 2.5 join them at the pin C, and the thin curve is where C goes over 2 turns of the input. The two crank pins are always between 1 and 5 apart and the bars can span 0.5 to 5.5, so the chain never meets a dead position. After 2 turns the machine is exactly where it began and the curve is closed.

What can move

One freedom, and a motion that never repeats

Mesh a gear on each crank of a five-bar and the count and the rank agree that one freedom is left, at every gear ratio. Whether the machine ever comes back to where it started is a different question, and neither instrument can see it: at 3 to 2 it is home after two turns, at 37 to 23 after twenty-three, and at the golden ratio never, with its nearest returns at the Fibonacci numbers.

A dwell six-bar built on the vertex of a symmetric coupler curve. The four-bar with ground 3, crank 1, and coupler, rocker and arm all 2.5, with the angle at the rocker pin set to 120.8024°, so that its coupler curve is its own mirror image about the dashed line. The curve crosses that line at crank angle 0°, and a link of the osculating radius there, 5.7691, runs from the tracing point to a pin at the centre of curvature; an output link of 3 from a third ground pivot holds that pin, square to the dwell link at the vertex. Faintly, the dwell link and output 50° of crank either side. The output swings 7.43° over a whole turn, and near the vertex its angle changes only at sixth order in the crank's.

The paths points trace

The flattest dwell is not the longest

A coupler curve that is its own mirror image has no odd terms in its distance from a circle centred on the mirror line, so one angle of the coupler can remove the fourth-order term and leave a dwell of sixth order, with no search of the curve. A six-bar built there dwells for 58.7° of crank inside 0.1% of its swing. Turned two degrees away from that angle it dwells for 80.1°, and the searched six-bar for 24.4°.

$What a pin's size costs a synthesis. Every one of the 1,176 exactly correct three-position syntheses the synthesis field's survey produces, asked a question its three verdicts cannot ask: are any two of its pins closer together than the material round them? The poses span about two units, so the horizontal axis is a pin as a fraction of the mechanism. At a boss radius of 0.05 nothing is lost; at 0.4 — a pin nearly a fifth of the pose span — 27% of the exact solutions cannot be built, and 15% of the ones that had already passed the branch, circuit and order tests go with them. The defect is real, it is rarer than the kinematic ones, and it is not correlated with them.$ The problem backwards

A defect that is not kinematic

The synthesis field's survey ends in three verdicts and all three are about which solutions a sweep visits. Here is a fourth, found by asking how much room two pins need — and it is the first defect here that a simulation cannot find, because in the equations a pin is a point.

Two permitted motions, and a composite that lifts off. A disc resting on a straight edge — a roller follower on a flat-faced cam, and the simplest higher pair there is. Two bodies touching at a point rather than over a surface, two freedoms: slide along the edge, and turn, because a disc is its own symmetry group about its centre. Both are permitted and both keep the contact exactly. Their composite does not. The faint discs are the two permitted displacements taken separately; the solid one is one followed by the other, and its centre sits 1.049 radii off the dashed line where a tangent disc's centre has to be. The excursion is exactly |t₂ sin φ₁| — the second displacement's slide times the sine of the first one's turn — derived from the two displacements rather than from the composition and agreeing to 10⁻¹⁶. A lower pair's freedoms compose and a higher pair's do not, which is why a joint's freedom count is the dimension of a group in one case and the dimension of nothing in the other.

What a joint is

A higher pair has no group

A disc resting on a straight edge may slide along it and may turn about its own centre. Both keep the contact exactly. Do one and then the other and the contact lifts off the edge by |t sin φ| — up to 1.2 radii over an ordinary range — so the two freedoms are real and the pair of them is not closed. The count is still right and there is nothing for it to be the dimension of.

A tooth flank is the end of an unwound strand. A strand wrapped on a circle of radius 45.105 mm, unwound while kept taut. Its free end traces the involute — the same curve the gears field builds from its own parameterisation, agreeing to 1.5e-14 mm over the whole flank. The strand is the important part of that sentence and not the curve: the taut portion is a tangent to the base circle, its length is the arc it has left, and both of those are statements about a strand rather than about a tooth. At this position the free length is 27.965 mm. positioned by solving, not by drawing.

Teeth

A tooth flank is an unwound strand

Unwind a taut string from a circle and its free end traces the involute, to 1.5·10⁻¹⁴ mm of the curve the gears field draws. That is not a curiosity: the line of action of an involute pair is a crossed strand on the two base circles, so the property the involute is chosen for — a ratio that does not care where the shafts are — is a belt's property rather than a curve's.

One, two, sixteen, two hundred and thirty. Every planar chain of mobility one, up to ten links, counted by enumeration rather than quoted. The pins column is forced: a chain of 10 links has one degree of freedom only if it has exactly (3n − 4)/2 pins, which is why no odd link count appears. Pass the count is how many graphs satisfy Grübler's rule, are connected, are simple and give every link at least two pins. Are chains is how many of those survive the fourth condition, that no proper subchain is already a structure — and the gap between the two columns is the whole of this field's first argument: at ten links 1,878 graphs pass a rule that 230 of them deserve. Mechanisms is larger again, because a chain is not a mechanism until a link is held still, and how many different mechanisms that gives is a question about the chain's own symmetry.

Drawn wrongly

Six things a chain is not

A count read as a verdict, a rank trusted where it is blind, a fingerprint used as a proof, a list of five taken for a complete one, a solver treated as a convenience, and a census read as a catalogue of machines. Six claims, each with the number that kills it.

How far the driven link turns is a fact about the lengths, not the chain. Each of the sixteen eight-link chains, given the arbitrary placement its own layout produces, driven from its first available choice of frame and input, and swept until a frame stops closing. 10 of the sixteen reach every angle and the rest rock through between 107° and 244°. Nothing in this chart is a property of the chains. Change the placement and the bars change; the census above them does not. It is here because it is the sharpest way to say what this field does and does not decide, and because the temptation to read a topology census as a catalogue of machines is exactly the mistake it prevents.

The chain before the lengths

The chain has no lengths

Every number in this field survives multiplying every link by a different scale factor, because there are no lengths to scale. Which is also the statement of what a census cannot decide — and the sixteen eight-link chains, each given one arbitrary set of dimensions and driven, produce a chart in which nothing belongs to the chains.

Bars against terms, over the whole catalogue. One mark per compiled machine. The bar count rises much faster than the term count, and the reason is the summing chain: term k has to have its direction carried to the k−1th vertex of the chain, one parallelogram per hop, so the carrying costs a translator for every pair of terms. Nine curves, from five bars to four hundred and thirteen, on a term count that goes from one to eighteen.

The curve as an equation

What universality is worth

The linkage exists, it is four hundred and thirteen bars, and it draws ten degrees of its curve. All three are true and only the first is in the theorem — which is the ordinary shape of a result about what exists, and the reason it was worth building one to find out.

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

Members that pull

A drum is a size, a wrap is a shape

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

A parallelogram at its flat position, exact and built slightly wrong. Left, a parallelogram with ground 3, cranks 2 and coupler 3 lying flat, where its two assemblies meet; faintly, the two ways it can go on, drawn at 30°. Middle, the same linkage with its coupler 0.05 too long, drawn at the closest it can come to the flat position: 7.49° away on either side, so the input's circle is thick where the input can go and red across the 15.0° it can never enter. Right, the input crank 0.05 too short, at the flat position: its two assemblies put the output crank 43.76° apart, and they do not meet at any input angle.

What can move

A parallelogram a micron wrong

A parallelogram linkage sits exactly where two kinds of four-bar meet, so a parallelogram that has actually been made is always one of four other machines. Make one bar a micron wrong on a 300 mm frame and the input stops a tenth of a degree short of lying flat, or the output turns round there with an acceleration that grows as one over the square root of the error. A third bar turns the square root back into a misfit of one micron.

Where a fit's singular values fall away, on each motion of a parallelogram. The 28 singular values of the degree-six fit, largest first and relative to the largest, for points traced on the parallelogram's circle motion, on its quartic motion, and on both. On the circle motion the last 15 lie below a drop of 3.7 × 10¹³, from 0.232 to 6.2 × 10⁻¹⁵; on the quartic motion the last 6 lie below a drop of 3.0 × 10¹⁰, from 6.3 × 10⁻⁵ to 2.1 × 10⁻¹⁵; on both motions the last one lies below a drop of 2.5 × 10¹², from 3.1 × 10⁻⁴ to 1.2 × 10⁻¹⁶. A drop of ten decades or more is a null space that is exactly there: fifteen sextics vanish on a circle, six on a quartic, and one on both.

The paths points trace

A null space of fifteen is not noise

Points traced on one motion of a parallelogram four-bar leave a degree-six fit with fifteen polynomials that vanish on them, behind a drop of thirteen decades: the circle times every quartic. Half an oval of an ordinary coupler curve leaves two to five, behind drops of two. The count is the same kind of number in both cases, and only the drop beside it says which one is algebra.

The closed path of a geared five-bar's pin at four gear ratios. A five-bar with two cranks of length 1 on pivots 3 apart, couplers 3 and 2.5, and a gear on each crank, drawn over one whole cycle at ratios 1 : 1, which closes after 1 input turn; 2 : 1, which closes after 1 input turn; 3 : 2, which closes after 2 input turns; 5 : 3, which closes after 3 input turns. Both assembly branches of the coupler pin are drawn, one in each colour. The curves they make together have degree 6, 10, 16, 26.

How many answers

Every rational gear ratio has a degree

Mesh a gear on each crank of a five-bar and the pin where its couplers meet draws a closed curve at every rational ratio. Its degree is 6 at 1 to 1, 16 at 3 to 2, 68 at 13 to 8 and 178 at 34 to 21: four times the larger term of the ratio plus twice the smaller, read off the span of one polynomial in one variable. Along the golden ratio's convergents the degree grows by the golden ratio at each step, and at the golden ratio itself no fit finds any.

One pin and two, on the same wheel. A 6-slot Geneva wheel at one instant of its index, driven by a crank carrying one pin and by the same crank carrying two. The wheel, the slots and the centre distance are identical; only the pin count differs. Each pin drives the wheel through one slot pitch while the driver sweeps 120°, so 1 pin gives 1 index a driver turn and leaves 67% of it at rest, 2 pins give 2 indexes a driver turn and leave 33% of it at rest. Nothing else about the mechanism changes, which is why the whole question is arithmetic.

Motion that stops

Two pins and no dwell at all

Put a second pin on a Geneva's crank and the wheel indexes twice a turn instead of once, at a quarter of the acceleration for the same output rate. Put a third on a six-slot wheel and it never rests; put a third on an eight-slot wheel and two pins meet in two slots and it jams. Both of those look like separate conditions and are one, and the boundary between them is an equation in integers with exactly three solutions.

What one point can and cannot see of a group. For each of the twelve: the group's dimension, the dimension of one point's orbit under it, and the difference — the stabiliser, the motions that leave that particular point exactly where it is. The orbit is the only picture a group has, and this table is the honest caption on it. Planar motion and spherical motion are both three-dimensional and both sweep a point over a two-dimensional surface, so each leaves one motion doing nothing at all: a turn about the plane's normal in one case, a turn about the radius in the other. A point does not see the whole group, and no drawing of one trajectory can be a complete picture of what a joint permits.

Drawn wrongly

Six things a joint is not

A freedom count read as a description, a screw system read as a group, a pair list read as a convention, a trajectory read as a determination, a nominal alignment read as a delivered one, and a higher pair read as a larger joint. Six claims, each of them what a careful person would say, each answered with a number.

Whether the part goes in is one inequality. Two of the four contacts are moved and the other two left where the drawing says; the horizontal and vertical axes are those two errors, inward positive. The shaded region is where the part still goes in and the unshaded region is where it does not fit at all — not fits badly, not is located wrongly: there is no position and no orientation the part can take. The boundary is the straight line 0.250·e₁ + 0.250·e₂ = 0, whose coefficients are the shares from the previous figure. Four probe points are marked, each checked twice — once by the inequality and once by a linear program that looks for a pose and reports the program infeasible when there is none — and the two agree at every one. A hold turns a set of tolerances into a single condition, and the weights in it are what say which contact is worth making accurately.

Contacts that only push

Which contact to make accurately

A hold turns a set of contact tolerances into one linear inequality, and the weights in it are the coefficients of the combination that cancels — a quarter each on a square held by four, and 0.144 to 0.424 on a hexagon held by five. Above that line the part goes in and below it there is no pose it can take at all: not badly located, not out of position, no fit.

The machine compiled from a rectangular hyperbola. xy − 0.5, compiled: 20 bars and 20 joints, painted by what each part is for. The two-link arm at the pivot carries the tracing point; the reflectors and means build each term's angle; the rigid offsets fix the constant φₖ and the amplitude; the translators carry those directions out along the summing chain, whose last vertex is held on a line. That last constraint is the equation. Every joint drawn is the output of a Newton–Raphson solve on 36 equations, converged to 4.2e-16, and the polynomial at the tracing point is 2.2e-16.

The curve as an equation

A compiled machine and its own scale

A linkage compiled from a polynomial has bars whose lengths are the polynomial's coefficients and joints whose angles are its phases. Scale it and every coefficient scales — so the machine computes the same polynomial multiplied by a constant, which is a different polynomial with the same roots.

Two rotors cut undersize, and the gap that is their seal. The conjugate pair at three positions of its turn, each rotor cut 3 undersize — its boundary moved that far inward along its own normal, with the full-size outline dashed behind it. At full size the two are in contact at every angle, so cutting both back by 3 leaves exactly 6.0 between them wherever they were touching. Measured over a whole lobe pitch the gap runs from 5.9965 to 6.0000 against a prediction of 6, a worst departure of 3.46e-3.

The shape is the unknown

The clearance that is the seal

Two rotors that are each other's conjugates touch at every angle of their turn, so cutting both back by the same amount leaves exactly twice it between them, everywhere. Open the shafts by the same amount instead and the gap runs from four per cent of it to all of it. And a pair that is not conjugate has no seal to cut: over one lobe pitch it swings from two and a half units inside itself to two and a third apart.

A flat face on a swinging arm, and the cam it asks for. The cam a flat-faced follower needs when its face is carried on an arm pivoted 300 from the cam's centre, with a base circle of 20 and a lift of 20, drawn in the cam's own frame at three cam angles. The heavy line is the face, the dashed line is the perpendicular from the cam's centre to it, and the dot is the contact. Its distance from the foot of that perpendicular is the offset, which for a sliding face would be the programme's own velocity and here is a quantity with the arm in it. Over the turn it runs from -20.41 to 17.95, so the face must be 38.4 wide.

Prescribed motion

A flat face on an arm is worse

Carry a cam's roller on a swinging arm instead of a slide and the pressure angle improves — a long arm beats an offset. Carry a flat face on one and the opposite happens. The cam sees the sine of the follower's rotation rather than the rotation, and the distortion costs convexity: the smallest workable base circle rises from 10.66 on a slide to 28.68 at a pivot three base circles out, and below a certain arm length no base circle works at all.

The same five-bar with its gears meshed outside and inside. A geared five-bar drawn over a whole cycle at four ratios, with its two cranks turning against each other in the top row and together in the bottom one — an external mesh and an internal one, which is the same machine with the ratio's sign changed. Both assembly branches are drawn. The curves have the same degree in both rows — 1 : 1 at 6, 2 : 1 at 10, 3 : 2 at 16, 5 : 3 at 26 — and they are not the same curves: the lower ones are the maximally circular members of their degree and the upper ones are not, which is a difference nothing about the drawing shows.

How many answers

The mesh inside keeps the half

Mesh a geared five-bar's two gears inside each other instead of side by side and the cranks turn together rather than against each other. The curve its pin draws has exactly the same degree at every rational ratio — and it passes through each circular point half that degree, which is as often as any curve can, where the counter-rotating machine manages between a third and two fifths. The sign of the ratio is the whole difference.

Motion that stops

A ratchet with no teeth

A roller clutch holds one way and runs free the other with nothing on it a tooth could be called. Whether it grips is a question about friction, which no drawing settles. How far it moves before it does is geometry: a clearance divided by the gap's own slope, so its lost motion is a length rather than a fraction of a pitch — 1.4° against a 24-tooth ratchet's 15° — and unlike a ratchet's it is not divided by adding more holding elements.

One tooth count, three pressure angles. A 20-tooth gear cut at 14.5°, 20°, 25°, drawn at a common pitch radius and overlaid on its own pitch and base circles. The three teeth have the same thickness at the pitch circle — that is what the standard fixes — and differ everywhere else: a coarser pressure angle puts the base circle lower and leans the flank over, so the tooth gains metal where it is held and loses it where it reaches: thicknesses at the base circle run 1.628, 1.756, 1.967 modules and at the tip 0.866, 0.695, 0.510. Those two run opposite ways, and the second of them ends at a hard stop — a tooth whose tip thickness reaches zero has come to a point and cannot be cut.

Teeth

The angle the standard left free

Involute geometry fixes the tooth curve and leaves one number open. Raising it buys smaller pinions and spends contact ratio, monotonically and in opposite directions, so there is no angle that is best at both — and the familiar twenty degrees is a choice with a date on it rather than an optimum.

$The gap is a straight line in the width. Three machines, four widths each, every width a fraction of that machine's own limit. Each set of points is collinear to the last bit of a double — the slopes wander by less than 10⁻¹³ across the range — because a feature-to-feature distance is linear in the corners of the two bodies and a bar's corners are linear in its width. The slope reads the contact: Chebyshev's linkage at -2.692, a crank rocker with a post at -1.350, a crank passing a stud at -1.350. A bar's boss grows 1.35 times as fast as its side, so −1.35 is a boss against something that is not growing and −2.70 is boss against boss. Extending each line to zero gives the widest link the machine will take, and the bisection that finds it the hard way agrees — 0.30113 against 0.30112, inside the bisection's own residual.$ Links with a width

The gap is a straight line in the metal

Thickening every link by the same amount subtracts the same amount from every clearance, exactly, and moves the angle at which the worst one occurs by nothing at all. So a whole swept check can be done once on bars of any width and every other width read off by subtraction — until the closest pair changes hands, and never past zero.

The two curves the pole rolls along, at 60°. The pole is a different point at every instant, and it traces one curve in the fixed plane and another in the moving one. Those are the centrodes, and the whole motion is the second rolling without slipping on the first — a statement with no mechanism in it, which is why two completely different linkages with the same centrodes produce the same motion. The moving centrode is drawn here in the position it occupies at this instant, and it touches the fixed one at the pole to 0.0e+0 of a unit. positioned by solving, not by drawing.

The motion, not the mechanism

The linkage, put back from two curves

This field opened by saying the mechanism drops out, and that every planar motion is one curve rolling on another. Both are true and neither had been measured. The rolling reproduces the four-bar's own placement to a residual that quarters when the sampling halves, and the two curves lay equal arc to a part in a billion.

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

Members that pull

The wrap that walks along the axis

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

Three joints, and the three screws that describe them. A leg of three revolute joints, drawn as its three axes, and the principal screws of the three-system they span, drawn through the system's own centre. The three principal axes are mutually perpendicular — worst cosine 1.1e-16 — and they meet at one point, missing it by 2.6e-16. Their pitches are -0.3766, -0.0338, 0.8075, and every screw the leg leaves free has a pitch the three of them give by h₁l² + h₂m² + h₃n². Nothing in the three joint axes looks like a right angle and the system's own frame is one.

Out of the plane

What a leg of three joints leaves free

Five essays of this field have computed the order of a screw system and drawn none of them. A three-joint leg spans a three-system; its three principal axes are mutually perpendicular and meet at a point, six numbers price every screw in the family, and the directions of the lines it contains form a cone.

Drawn wrongly

Six things a compiled linkage is not

A closure residual read as a verdict, a theorem read as a design, an exact answer read as an accurate one, a degree read as a cost, a construction read as a search, and a neighbourhood read as a turn. Six claims, each of them what a careful person would say, each answered with a number.

What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on.

As built

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

What each machine is sold with. Every mechanism in this field arrives with a number. The middle column is that number; the right-hand column is what the mechanism was measured to have, by the library named under the row. The coloured bar and the word under each machine are the verdict: exact means the mechanism has the number at every position, mean that it is the average of something that varies within every cycle, bounded that it is false by a stated and negligible amount, point that it is the value at one position, and quoted that the mechanism has no such quantity at all. The lit row is the one this essay is about. Of the 14 rows, 6 are quoted and 3 are exact.

Machines you have met

Which numbers survived

Fourteen machines were measured against the numbers they are sold with. Three survived exactly, and all three are counts. One is false by a stated bound, two are exact means of things that vary, three are honest values at a stated position, and six name quantities their mechanisms do not have. The pattern is not about honesty; it is about what kind of thing a number is.

$The two circuits at a flat position, and the play that joins them. The input and output angles of a parallelogram whose input is short by 1e-3, near the flat position where the exact parallelogram's two motions would cross. With no play the linkage's configurations are two curves, an upper and a lower, 0.1095 radians apart in output angle at the flat input angle — the square root of the error, not the error. Each shaded band is what a radial play of a stated fraction of the error makes reachable, and the innermost boundary is the play-free pair. At a play equal to the error the bands meet and the linkage can pass from one circuit to the other.$ What can move

A length error is undone by its own size

A parallelogram built a thousandth wrong loses its change point, and the two motions it could have chosen between end up a tenth of a radian apart — the square root of the error rather than the error. The radial play that joins them again is a thousandth exactly: not of that order, that number. It is the same number a third crank charges the same linkage in misfit, and no pin is worth more of it than any other.

Teeth

A tooth that lives on a sphere

Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.

Three circles at one point, at 50°. The osculating circle of the fixed centrode at the pole, the osculating circle of the moving centrode there, and the inflection circle. All three pass through the pole and all three have their centres on the common normal — which is what makes the relation between them a relation between three numbers on a line. The two centrode radii are 16.180 and 9.730, signed along that normal, and the inflection circle's diameter is 24.398 against the 24.409 the other two give. Nothing in that arithmetic knows there is a linkage. The window is 30.25 wide and the larger osculating circle is 32.36 across, so it is cut at the edge: what is drawn is the neighbourhood of the pole, where the three curvatures are the same three numbers however far the circles carrying them reach.

The motion, not the mechanism

The two numbers are the curves' own

Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.

What a leg of three joints permits and what it resists, on one frame. The leg of three revolute joints in grey, and the principal screws of two three-systems drawn through one centre: the twists the leg lets its platform make, and the wrenches it holds without any joint turning. They are computed separately — the second is the reciprocal complement of the first, found by its own eigenproblem — and they come out with the same centre, 5e-16 apart, and the same three axes, parallel to 0e+0. Only the pitches differ, and each wrench pitch is its twist's with the sign reversed, to 1e-15: -0.3766 against 0.3766, -0.0338 against 0.0338, 0.8075 against -0.8075.

Out of the plane

The lines a leg turns about and the lines it is pushed along

A leg of three revolute joints permits a three-system of motions and resists a three-system of loads. The two share a centre and three axes and differ only in the sign of every pitch — and the revolute axes of the first and the lines of force of the second are the two rulings of one hyperboloid, every line of one meeting every line of the other.

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

Members that pull

A strand in a tube

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

Many of one thing

The count says how many and not where

A kagome lattice has three joints and six bars in every cell and counts to exactly nothing, so a patch cut from it has as many mechanisms as its edge has lost bars: 5L − 5 for a rhombus of L cells a side, which the rank confirms at every size with no bar redundant. Straight or twisted, the number is the same. Where the mechanisms are is not: a straight patch keeps nearly half its edge weight in the middle, and a patch whose triangles are turned by 17° keeps a twentieth.

Six coupler points of one slider-crank: three at other distances from the crank pin, three exactly a rod away. A slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, drawn at one position with the closed curves six points of its rod trace over both assemblies. A point is given as (u, v) in the rod's own units — u along the rod from the crank pin, v across it. On the left (0.4, 0.5), (−0.3, 0.2), (1.2, −0.6): their distances from the crank pin are 0.64, 0.36, 1.34 rods. On the right (0.6, 0.8), (0, 1), (−0.8, 0.6), each exactly one rod from the crank pin, as far as the slider pin is. All six are quartics, all six are bounded — no real branch runs off the page — and nothing in the drawing tells the two panels apart. The difference is at infinity.

How many answers

Where a slide puts the rest of the degree

A slider-crank's connecting rod draws a quartic that passes once through each circular point, which leaves two of its four meetings with the line at infinity unaccounted for. They are not along the slide. A point u along the rod and v across it sends them to the complex slopes [2v ± i(1 − u² − v²)] / [(1 + u)² + v²], whatever the crank, rod or offset — confirmed by slicing and by the fitted equation — and they are real only for points exactly a rod's length from the crank pin, where they merge into one direction at half the point's angle.

Contacts that only push

The hold is in the corners

A disc cannot be held by frictionless contacts and a regular polygon can, so a polygon with more and more sides has to lose its hold somewhere. Searched exhaustively, the best four contacts sit at alternate ends of four edges a quarter-turn apart, and their margin is the half-edge sin(π/n) less a correction that falls as 1/n² — 74% of it at eight sides, 99.4% at sixty-four. The hold is lost as the side shrinks, not as its square, and it is carried entirely by how far a contact sits from its edge's middle.

The shape is the unknown

A twist steadies what it cannot tighten

A helical rotor's sections are at different phases of the same mesh, so the clearance a machine has at one instant is a window along its own profile rather than a point on it. A wrap of exactly one lobe pitch holds the seal's open area constant through the turn — every harmonic at once, whatever the profile — and leaves its average, its tightest place and its widest place exactly where they were.

A rectangular hyperbola, compiled from a multiple of its equation. The machine compiled from p · (1 + x² + y²) for a rectangular hyperbola, with the translators — the parallelograms that carry a direction from where it is produced to where it is needed — in their own colour. The factor 1 + x² + y² is at least one at every real point, so every equation here vanishes on exactly the same curve. The machines do not agree: 20 bars at p, 75 bars at p · (1 + x² + y²), 144 bars at p · (1 + x² + y²)². This one solves 29 positions over an arc of 0.508 radians, and at every one of them the original polynomial reads 4.93e-14.

The curve as an equation

The price is on the equation

A line costs five bars. The same line, written as its own equation multiplied by a factor that is never zero, costs fifty — and the machine compiled from the longer equation draws the same line just as exactly. Every cost this field quotes belongs to a polynomial and not to a curve, and the cheapest equation of a given curve is a quantity nobody here has.

The problem backwards

The slide a fourth pose leaves

Three prescribed poses leave a whole circle of body points that want a slide. A fourth leaves exactly one, found where two of those circles meet a second time. It had been predicted as one or three, because a real cubic meets the line at infinity in one real point or three. But both of Burmester's cubics pass through the two circular points, so only one real point is left. Four hundred random four-pose problems each have one slide and one slot pin, both found with compasses.

What a sweep at the wrong resolution reports. One machine — a crank passing a stud a tenth of a unit across — swept at six sample counts. At eight samples the smallest gap found is 0.0809 and the machine reads as clear; at twelve it is 0.0067 and still clear; from sixteen on it is negative and the crank is inside the stud by 0.0099. Nothing about the twelve-sample answer looks wrong: the curve it draws is smooth, its minimum is interior, its margin is small and positive. A sweep cannot report what it did not look at, and the repair is not more samples but a bound on what happens between them.

Drawn wrongly

Six things a body is not

A verdict read as a measurement, a hull read as a part, a sweep read as a proof, a drawing read as a configuration, a geometry read as a force, and a plane read as a place. Six claims, each of them what a careful person would say, each answered with a number.

Numbers that were measured

Three machines, one curve

Roberts's theorem says every four-bar coupler curve is drawn by exactly three different four-bars. Read as an identification problem that is a least-squares objective with three separate exact minima, whose cranks differ by sixty per cent — so an instrument that records only where the tracing point went has three answers, and no amount of data chooses between them.

A framework built off square, and the play it asks for to move at all. The nine bars with their two lines 4° from perpendicular, as built and after the first joint has been pushed 0.2 along its line. The framework has a freedom by rank and no motion, so the push cannot be taken with the bars at their lengths. With every bar allowed to be wrong by 1.509e-4 — which is a radial clearance of 7.547e-5 at each end — a placement exists, and the worst bar in it is out by 1.509e-4. The same push on the perpendicular framework needs no allowance at all, because there it is a motion.

What can move

The right angle as a tolerance

Dixon's nine bars move only when their two lines are exactly perpendicular, and a framework built a degree off square has a freedom by rank and no motion at all. Give its joints clearance and it moves a bounded distance: the play each bar needs is proportional to the tilt and to the square of the travel, one constant serves every tilt, and all nine bars end up at that play exactly. Then it reaches its first crossing, where the push is as large as the motion ever makes it, and a hundredth of a push further costs fifty-eight times the law — square or not.

Teeth

The mesh with one curvature reversed

Turn an annulus's teeth inward and the same involute law produces a different machine: a centre distance that is a difference, two base tangencies on one side of the line of action, more contact than an external pair carries, and three separate floors on the tooth counts, all of them the same statement about where an involute stops existing.

Holding the coupler still: the frame's inflection circle is the mirror of the coupler's. The four-bar with ground 4, crank 1, coupler 3.5 and rocker 3 at a crank angle of 225°, redrawn in the coupler's own frame, so the coupler is the horizontal bar and the ground link is what moves. The circle on one side of the pole is the ordinary inflection circle — coupler points travelling straight — carried into this frame. The circle on the other side is the inverse motion's: frame points travelling straight when the coupler is held. It is computed by differentiating the swap and, separately, by solving the coupler-held four-bar, and both routes put it at the first circle reflected through the pole, to 7e-15 on a radius of 4.45. The line through the pole is the common tangent. Dragging the crank angle moves both circles and they stay mirror images.

The motion, not the mechanism

The frame seen from the coupler

Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.

Two circles, one term. The curve r² = 1.44 together with r² = 2.56, whose squared radii sum to 4.00 — four times the square of the arm's link length. Their product equation expands to 1 term, which is what either circle costs on its own, so the second circle is free. The machine compiled from it has 17 bars against 11 for the single circle, runs over 5.200 radians against 1.560, and stays on the outer component throughout: its radius varies by 4.15e-13 over 240 solved positions. A mechanism moves continuously and the two circles are disjoint, so no assembly of it reaches both.

The curve as an equation

Two circles for the price of one

Search every multiple of a curve's equation by a polynomial of degree two and the cheapest is the curve's own equation, on four curves and by exhaustion. On the circle a second multiplier ties — and what it describes is two concentric circles, whose squared radii sum to four times the arm's link length squared, at exactly the cost of one.

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

Members that pull

Which walls a strand is held by

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

What each branch carries, against what the engine delivers. The power in the variator's branch and in the straight path, as multiples of the engine's own, for a planetary of K = 1.40. Both are ratios of powers, so the load cancels and nothing here is a force: the split comes from requiring the gearset to be lossless at every admissible set of speeds, which fixes the torques at 1 : K : −(1+K). The variator carries v/(K − v) and the straight path K/(K − v), and both run away at the pole. The variator first carries the engine's whole power at v = 0.700, which is exactly half the way to the pole — and the straight path is already carrying more than the engine everywhere past nought, flowing the other way through the planetary. That excess is the circulation.

More than one input

The power that goes round twice

Sliding a power split's travel towards its pole buys ratio span for nothing, on the kinematics. It is not for nothing. The variator's own branch carries v/(K−v) of the engine's power and overtakes it at exactly half the way to the pole, and the tolerance trim the span was computed from is not reached until the variator is rated for six times the engine — which no machine is.

Several legs, one platform

Tilted, near the dead yaw

A paired Gough platform held level is singular at one yaw wherever it stands. Tilt it and that stops being true — the six moments are no longer equal and the home position is held. It is a poor rescue: the rise is quadratic in the tilt, so a degree buys a sixty-fourth of what eight degrees buys, and what the tilt actually does is not remove the singularity but turn it into a surface through the workspace.

Four points of one connecting rod: two crossings, none, two isolated points, two crossings again. The closed curves drawn over both assemblies by four points of the connecting rod of a slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, which is marked in each panel. Each curve is a quartic and owes two double points. (0.12, 0.15), 0.19 rods from the crank pin: two crossings, at (±0.65, ±0.36), ringed; (0.4, 0.5), 0.64 rods from the crank pin: a complex pair, so the curve crosses itself nowhere; (0.75, 0.94), 1.20 rods from the crank pin: two isolated points at (±4.06, ±3.00), dotted, off the drawn curve; (1.1, 1.0), 1.49 rods from the crank pin: two crossings, at (±2.66, ±2.95), ringed. Each panel has its own scale. The second curve has no visible double point at all, and the third has two that no configuration of the machine ever reaches.

How many answers

The double points that leave for infinity

A slider-crank's connecting rod draws a quartic that owes two double points. Close to the crank pin they are two real crossings. Between a crank's length and a rod's length from it they are a complex pair and the curve crosses itself nowhere. At exactly one rod both run off to the same real point at infinity and meet there as a tacnode whose branches are complex. Past that they come back real. Offsetting the slide changes the two circles by (R² − a²)(R² − b²) + e²R² = 0 and sends only one of the pair to infinity.

Two jaws on a polygon, and the cones they can push within. A regular polygon of 5 sides and circumradius 1, gripped by two contacts with a friction coefficient of 0.3. Each wedge is the cone of directions its contact can push in, half-angle atan 0.3 = 16.7° about its normal. Two contacts hold the part — resist every load, given a squeeze — exactly when the line between them lies inside both cones. An odd polygon has no parallel faces, and its most nearly opposed faces are 36.0° short of parallel, so the line can be inside both cones only if their half-angle is at least half that: a friction of tan(π/10) = 0.3249. Here the part is not held. Dragging changes the number of sides.

Contacts that only push

Two jaws and a coefficient

Every hold in this field has been frictionless, and two of its essays say what that leaves out. Put friction back and a part is held by two contacts in the plane instead of four and by three in space instead of seven; a disc, which no frictionless fixture can hold at all, is held by two jaws at any friction above nought; and an odd polygon needs a friction of exactly tan(π/2n). What friction takes away is the thing that made a frictionless hold a fact about geometry: a frictional hold is a hold per unit of squeeze.

Five poses, five four-pose slides, and the compromise between them. Left, the moving body's frame with the slider point of each four-pose subproblem — the point whose four images are in a line when one pose is left out — and the compromise point whose five images all miss one guide by the same amount. Right, the fixed frame: the body in its five poses and the images of the chosen point, with its guide. Leaving out pose 1 to 5 misses that pose by 0.5304, 0.0836, 0.0383, 0.0422, 0.1288 units; the compromise misses every pose by 0.0140, on alternate sides. The body is about two units long.

The problem backwards

The slide five poses almost have

Five prescribed poses leave no slide: a body point would need five images in a line, three conditions on two coordinates. But each of the five four-pose subproblems has exactly one slider point, and each misses the pose it left out by a distance a pair of compasses can find. Those five misses price the best compromise without solving for it. The slide that misses every pose by the same amount misses by one over the sum of their reciprocals — the misses add like resistances in parallel — and giving up the best single pose instead is always between two and four times worse.

A four-bar at 1°, solved. Ground 4, crank 1, coupler 3.5, rocker 3. Every joint position here is the output of a Newton–Raphson solve on the loop-closure equations, converged to 3.3e-14 — not a placement that looked right. Grashof's condition classifies these lengths as a crank rocker, and sweeping the crank through 360° confirms it: 120 of 120 positions assemble. The transmission angle at this instant is 54.3°.

Drawn wrongly

Six things a measurement cannot tell you

A calibration with a perfect residual whose fourth number is a starting guess, a rank that says nothing about a second answer, an improvement that proves nothing about a parameter, a class with no margin, a plan scored on poses that were refused, and a model missing something no data can find. Six claims, each with the number that kills it.

Teeth

Contact that runs along the tooth

A straight tooth engages along its whole face at once, so the amount of contact at a mesh is a square wave. Slant the tooth and a second contact ratio appears that has no tooth count in it and no pressure angle — bought with face width and helix angle alone, and able to reach a value at which the total length of contact stops varying at all.

Three laws in the band: a third, a sixth and a sixth. For bands from 3% to 10 ppm of the output's swing, three things measured on the symmetric six-bar itself: the longest dwell at any angle at the rocker pin, the dwell at the sixth-order angle γ = 120.8024°, and how far below γ the best angle sits. The dots are measurements and the lines are the two-term model's predictions from two coefficients read off the output's even part. Over the four tightest bands the fitted exponents are 0.173, 0.172 and 0.333, against a sixth, a sixth and a third. At 10 ppm the best angle is 0.490° below γ against a prediction of 0.488°, and the dwell at γ is 26.43° against 26.23°. The loosest bands sit above the lines, where the terms the model leaves out are no longer small.

The paths points trace

The dip that buys the dwell

A symmetric coupler curve's six-bar dwells longest a little below the angle that makes its error sixth order, and three bands gave three numbers. Across three and a half decades of tolerance the best angle's distance from that angle grows as the cube root of the band, both dwells as its sixth root, and their ratio is (27/4)^(1/6) = 1.3747 at every band — because the best machine spends the whole band on one dip and ends its dwell where the output comes back.

The same arm with its face through the pivot and stood off it, each at its smallest convex cam. A flat-faced follower on an arm pivoted 90 from the cam's centre, delivering the symmetric cycloidal programme with a lift of 20, drawn at 60° of cam rotation. On the left the face is a line through the pivot, and the smallest base circle that keeps the cam convex is 28.679. On the right the face stands 33.56 off the pivot on a bracket, parallel to where it was, and the smallest convex base is 26.136 — 8.9% smaller. A sliding face needs 10.664, and neither arm reaches it. The dot is the contact; the small circle at the right is the pivot, and on both sides it clears the cam.

Prescribed motion

Where on the sine the arm swings

A flat face on a swinging arm delivers its programme through d sin ψ. Differentiating that twice brings in a square of the follower's speed, multiplied by sin ψ, and sin ψ has a sign. Standing the face off its pivot moves the swing down the sine until it straddles ψ = 0, where the square helps on one half of the stroke. On an arm ninety out that lowers the smallest convex base from 28.68 to 26.14, and it lets an arm of 61 have a cam where the face through the pivot needs 80. It never reaches the sliding face's 10.66.

More than one input

Every power split is on one curve

There are twelve ways to attach an engine, a variator and an output to a planetary's three shafts, and asking which is best turns out to be the wrong question. On 1,994 splits across all twelve and nine gearsets, the output's span is (1 + R·a)/(1 + a) to 5 × 10⁻¹⁵, where R is the variator's own span and a is the power the variator carries over the power the gears carry. Every split lies on one of three curves of span against the variator's largest share. The previous essay's 1.70 is 2 − 1/R on the worst of them, and a split beats its variator's own span only by making the variator carry more than the engine.

The drive in 2 arcs. Chebyshev's two arms, whose regions over the whole drive overlap though the parts never come within 0.380 of each other. The outlines are each part's region over the whole drive. The filled ground is what a test that sweeps the drive in 2 equal arcs still flags: the points some arc's sweep of one part shares with the same arc's sweep of the other. With 2 arcs that is 0.859 square units. The parts are drawn at their closest approach.

Links with a width

Four arcs and the false alarm is gone

A swept region forgets when, so two parts whose regions overlap may never meet: Chebyshev's two arms share 4.66 square units of ground and never come within 0.38. Cut the drive into arcs, sweep each part over each arc, and compare regions only arc against the same arc. The test stays sound and costs the same sweep, and it forgets less with every cut. Four arcs clear Chebyshev's arms and eight clear Peaucellier's worst pair. A certificate that needs no grid asks for three times as many, and the same arms made wide enough to touch are refused at every number of arcs.

A disc cut undersize, turned until a pin touches. A twelve-pin cycloidal drive whose disc was cut as if its pins were 1 larger than they are, drawn at 10° of the eccentric with the disc turned a further 1.082° about its own centre — the turn at which the first pin touches. One pin touches, pin 0, marked in its own colour. The pins within half the clearance of touching are drawn in a second colour and the rest plainly: 5 of the twelve are within half the clearance and 7 further off. The clearance is exaggerated so that it can be seen; at every clearance measured, one pin touches first.

The shape is the unknown

A clearance leaves one contact

A cycloidal drive's disc, exactly as its pins generate it, touches nine or ten of its twelve pins at once, and that is why it can drive where two identical rotors cannot. Cut the disc two hundredths undersize, as a real drive is cut, and a rigid disc turned until it touches meets one pin, at every angle of the eccentric and in both directions. The arm that pin gives is never worse than the exact disc's best. What a clearance takes away is load sharing, and deflection gives it back: contacts that may give by the whole clearance put six or seven pins within reach.

The one real double point a four-bar owes is the one that leaves when its rocker becomes a slide. Left: a four-bar with a crank of 1 pivoted at the origin, a coupler of 3 and a rocker of 5 pivoted at (3, −5), so that the rocker's circle touches the line y = 0 where a slider-crank's slide would run. The coupler point is (0.4, 0.5) in the coupler's own units, and the curve it draws over both assemblies is shown with the circle through the three cognate pivots, on which all three finite double points lie. Two of them are a complex pair near the crank; the third, the only real one, is an isolated point at (0.92, −5.32) that no configuration reaches. Right: the same picture divided by the rocker length for rockers of 4, 10, 20, 50, 200, 100,000, with the drawn rocker's point ringed. The machine shrinks to the origin, the pivot circle settles onto a fixed circle, and the isolated point settles onto one point of it, (−0.484, −0.580), marked with a cross: the value d(|w|² − 1)/(w̄ − 1) for w = 0.4 + 0.5i and d the direction from the slide to the rocker's pivot.

How many answers

The double point a slide cannot keep

A four-bar's coupler curve owes three finite double points and a slider-crank's owes two, so a four-bar whose rocker grows into a slide has to lose one. Parity decides which: the one that leaves is always real, and always an isolated point no configuration reaches. It runs off along d(|w|² − 1)/(w̄ − 1) times the rocker's length, a law that knows nothing of the crank, the rod, the offset or where the rocker touches the slide. On the circle one coupler length from the crank pin the law is zero, and all three double points leave together like the cube root of the rocker.

Near its mirror, a double Cardan joint's lag is an ellipse in its two errors. A W double Cardan joint whose working angles average 15°, with a phasing error δ between its middle yokes across and a difference ε between its two working angles up, both in degrees. The curves are where the predicted lag spread over a turn, tan a·√((sin a·δ)² + ε²), is 0.05°, 0.1°, 0.15°, 0.2°. The dots are 35 joints solved from their pins, labelled with their measured spread in degrees; they meet the ellipse to 0.06% at worst. The ellipse is 3.86 times as wide as it is tall: at 15° a degree of phasing error does what 0.259° of mismatch between the angles does.

What a joint is

Two errors, one ellipse

A double Cardan joint keeps step only while a mirror fixes its motion, and any real one is a little off. Split each error by what the mirror does to it and only the half it reverses costs anything: a phasing error spreads the lag at sin a · tan a per degree, a difference between the working angles at tan a, and raising both angles together costs nothing at all. The two costs leave their marks an eighth of a turn apart, so they add as the sides of a right triangle — a root sum of squares that no assumption of independence was needed for.

A two-arm lamp balanced at every pose. A two-arm desk lamp: a lower arm 0.4 m long pivoted on its base, an upper arm carrying a 0.8 kg head 0.35 m out. A parallelogram — the lower arm, a short crank at the base, a link parallel to the lower arm, and the upper arm's first few centimetres — keeps the base crank parallel to the upper arm, so the upper arm's spring can run from a post above the base pivot to that crank and see the upper arm's angle alone. The lower arm's spring runs from the same post to the lower arm. Both springs have no free length and stiffnesses matched to their arms' weights. At every pose the lamp's total potential energy is the same: here it differs from the pose with both arms upright by 3e-16 of the head's moment, and the joint torques are 0 and 0 N·m — rounding. Dragging swings the lamp.

Machines you have met

The lamp that stays where it is put

A desk lamp that stays wherever it is left is balanced, not held: its springs store exactly the energy its weight gives up, at every pose. For one arm that takes a spring with no free length, fixed above the pivot, whose energy has a cos θ term that cancels the weight's to 10⁻¹⁵. A spring with a free length L₀ leaves an unbalance of exactly L₀/s of the weight's torque, and a stiffness a fraction ε off leaves 2ε of its moment. For two arms it takes a parallelogram: without one, the upper spring sees the angle between the arms and the lamp's energy varies by twice the head's moment; with one, both springs sit at the base and the lamp is flat everywhere.

A wheel on a rough road, and the path of its centre. A road whose height varies as a sine of 10 mm amplitude and 200 mm wavelength, and a wheel of 150 mm radius rolling along it without slipping. The dashed curve is the path of the wheel's centre; the thick stretches of road are the parts the wheel touches. The road's tightest curve, at the bottom of each dip, has a radius of 101 mm; a wheel smaller than that touches all of it, and a larger one bridges the dips. Over one wavelength this wheel turns through 1.3621 radians, and that times its radius is 204.308 mm — the length of its centre's path, not the road's 204.847 mm and not the 200 mm it has travelled. Dragging changes the wheel.

Wheels, and where they may not go

What a wheel counts on a rough road

An odometer counts a wheel's turns and multiplies by its radius, and on a flat road that is the distance travelled. On a rough road it is not, and it is not the road's length either. It is the length of the path the wheel's centre follows — exactly, because the contact is the wheel's instantaneous centre. A wheel smaller than the road's tightest dip rolls over all of it and counts the road's full length; a larger one bridges the dips, pivots over the crests, and counts less, falling towards the true distance as λ³/24(R + ρ)². On a road 2.4% longer than its wavelength, a 50 mm wheel overcounts by 2.4% and a 2.4 m one by 0.03%.

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

Members that pull

One belt for every step

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

The cheapest curve of each degree. The curve of degree 6 whose equation expands into the fewest cosines of the arm's two angles: cos 3θ = 0.3: 3 concentric circles about the pivot. It costs 1 term — cos(3α − 3β + …) — and the machine compiled from it has 27 bars. The dashed circle is the arm's reach. The dots are the machine's tracing point over 140 solved positions; it draws one component and stays on it. At every even degree the cheapest curve costs one term and at every odd degree two, whatever the degree.

The curve as an equation

The cheapest curves at each degree

Search every equation of degree up to six, not the multiples of one curve, for the fewest cosines a compiled machine has to build, and the answer does not depend on the degree at all: one term at every even degree, two at every odd one, each reached by exactly one set of frequencies. The cheap curves are concentric circles placed where Chebyshev's polynomials put them — k of them for the price of one — and, at odd degree, a line through the pivot with circles that come free. One symmetry of the arm forbids anything cheaper: its two links can be swapped without moving the tip.

The compromise slide needs its block on alternate walls; a working load chooses otherwise. The five prescribed poses of a body about two units long, driven by the one crank of Burmester's four that turns through them in order — centred at (1.531, -0.730), radius 1.492, turning 149° — and guided by the compromise slide, whose block has play 0.0140 either side of the guide's centre line; the two walls are drawn ten times further apart than they are. At pose 3 the compromise needs its block on the + wall (hollow ring). A working load along the guide, opposing the block's motion, presses it onto the − wall (filled). The walls needed at the five poses are + − + − +; the walls that load presses are + − − − −, and the body's own weight presses + − − + +.

The problem backwards

The load picks the wall

The best slide for five poses misses each by 0.0140, on alternate sides, so a block with that much play either side reaches all five — if it bears on the right wall at each. Nothing in a slot says which wall. The load does: the block presses whichever wall the load's moment about the crank pin sends it to, and changes walls only where that moment changes sign. Four changes are needed. The body's own weight gives two, in the wrong places, and play then makes the slide worse; a working load gives one, and needs nineteen times the play.

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.

What can move

Every change point lies flat

A parallelogram a thousandth wrong is rejoined by a pin clearance of exactly a thousandth, at any of its four bearings. The obvious guess is that an ordinary change point — a linkage on one Grashof boundary with no equal bars — would need a clearance with a constant in front and would reveal which bearing is loose. It does neither, because every change point has its four joints on one line. What does acquire a constant is the angle: 2√(2bδ/c(g + a)) when the circuits separate, and a stall constant with no coupler or output in it at all.

Three machines drawing one curve, with every member coloured by the rotation it shares. The double crank with ground 1, crank 3.2, coupler 3.6 and rocker 3.4, tracing point (0.45, 0.5), and the two cognates Roberts's construction gives for its curve, each drawn holding the same point. Each member is coloured by which of three directions of rotation it carries: 1 is the original's crank, turning with the first cognate's coupler and the second cognate's rocker; 2 is the original's coupler, with both cognates' cranks; 3 is the original's rocker, with the first cognate's rocker and the second's coupler. Under each name are that machine's three closed-form area terms, in the colour of their direction: the same values, 17.693, 0.102, 16.343, in a different column of each machine and the same colour in all three.

The paths points trace

A closed form belongs to a rotation

The three four-bars that draw one coupler curve carry the closed-form part of its area in different terms of the formula. Substituting Roberts's construction into those terms shows why: every term of a cognate is a term of the original, moved to another column. Over 480 random chains in all eight regions the three machines' closed forms agree as multisets to 10⁻¹⁵ and disagree column by column. Each closed form belongs to one of three rotations, and each machine carries each rotation in exactly one of its members.

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.

Every shared cell asks for a cut between two visits; 2 cuts answer all of them. Chebyshev's two arms: each of the 5,510 cells of ground both parts occupy at some point of the drive, as the stretch of drive angle between one part's last visit to it and the other part's first — every such cell is visited by each part in one unbroken stretch, so each gives exactly one interval. Sorted by where they end, one in 21 drawn. A partition of the drive clears the pair exactly when every interval holds a cut. The vertical lines are the fewest cuts that do, placed greedily at the earliest unstabbed end: 1.072 and 1.428, making 3 arcs of 0.360, 0.356, 0.274 radians. The thicker intervals are the ones the rule cut at; they do not overlap, so no partition clears the pair with fewer cuts.

Links with a width

Three arcs, cut in the right places

Equal arcs clear Chebyshev's two arms at four and Peaucellier's long arm and rhombus bar at eight. Each cell of ground the two parts share is visited by each part in one unbroken stretch of the drive, so each asks for exactly one cut, somewhere between the two visits — and the fewest cuts that answer every cell are found greedily from one sweep and proved least by the intervals that forced them. Three arcs clear Chebyshev's arms and five clear Peaucellier's pair, and halving only the arcs that still flag finds four and five at three and four times the sweeping.

A slider-crank whose crank becomes a slide loses both its double points. Left: a slider-crank with a crank of 5 pivoted a crank's length before the point where its crank pin crosses the slide, a rod of 3 and the slide along y = 0, drawn with the slides' eventual crossing at the origin. The rod point (0.3, 0.2) in the rod's own units draws the curve shown, and its two double points are the dots: isolated points, at (−7.90, 2.47) and (−2.10, −2.47). Right: the same double points divided by the crank, with the crank's pivot at the origin and the crossing at (1, 0), for cranks of 5, 10, 30, 100, 300, 1000. The pair settles symmetrically onto the crosses at ±(0.595, −0.505) — the law ± i(|w| − e^(i·arg w))/√(1 − |w|²) for w = 0.3 + 0.2i — and leaves in proportion to the crank.

How many answers

The trammel takes both double points

Grow a slider-crank's crank without bound and its crank pin's circle flattens into a second slide: the machine becomes the elliptic trammel, whose curves are conics with no double points at all. Both of the slider-crank's double points have to go, and parity cannot say how. They go together, symmetric through the crank's pivot, at ±i(|w| − e^(i·arg w))/√(1 − |w|²) crank lengths — a law that knows nothing of the rod or the slide's offset. Inside the one-rod circle they are two real isolated points, outside it a complex pair. And on the circle with the rod as its diameter, where the trammel draws straight lines, one of them does not leave at all: it settles on the crossing of the two slides.

More than one input

A second planetary lets the variator carry a difference

On a single planetary every power split lies on one of three curves of span against the variator's share of the engine's power. Build the split on a four-shaft Simpson or Ravigneaux instead and nothing changes while the fourth shaft is free: 3,588 splits, all on the curves. The one new arrangement is the variator between two of the gearset's shafts. That compound split is an input-coupled split and an output-coupled split in series, and its variator carries only the difference of their shares — so at every share below one it can span more than any single planetary, the square of (√R + k)/(1 + k√R). It needs a lever with the variator's shafts exactly its share apart, which real gearsets do not have beyond a share of about a third.

A seat round the stop's nose holds the four-bar if it is tighter than the nose centre's path allows. The four-bar of frame 4, crank 1.2, coupler 3.4 and rocker 2.8 at a crank angle of 70°, its coupler carrying a round nose of radius 0.3 centred at (1.593, 2.160). The nose sits in a hollow seat on the frame of radius 0.8, touching it at (1.650, 2.454), on the line from the nose's centre to the coupler's pole at (2.505, 6.883), off the top of the drawing. The dashed circle is where the nose's centre may go without the nose entering the seat: a disc of radius 0.50. The curve through the nose's centre is the path the four-bar gives it, with a radius of curvature of 0.893. The seat's disc is tighter than the path, so the path leaves it either way and the four-bar cannot turn: the stop holds. The threshold is a seat of radius 1.193, the path's radius plus the nose's.

The chain before the lengths

A stop aimed at its pole holds by curvature

A stop whose normal passes through its pole forbids nothing at first order, and whether it holds at all is decided by curvature. For a round nose the deciding curve is not the one it touches: the seat that just holds is the path of the nose's centre moved out by the nose, 1.193 on a four-bar where the contact point's own circle says 0.776. Past it, one contact holds both ways — loosely, as the square root of its clearance.

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.

A quintic of eighteen terms, and the nearest curve that costs a few. The quintic x^5 + 0.4x^2y − y = 0, whose equation expands into 18 cosines of the arm's two angles, inside the arm's reach (dashed). Drawn over it, the curve of the same degree with at most 4 terms that comes nearest to it inside the reach, found by trying every set of terms of that size: it keeps 0,1|1,0, 1,-2|2,-1 and is at most 2.77e-2 from the quintic and the quintic at most that from it — 1.4% of the reach. The slider steps through the budgets at which the nearest curve improves.

The curve as an equation

Four terms of eighteen draw the quintic

A quintic whose equation costs a compiled machine eighteen terms is drawn to 1.4% of the arm's reach by four of them and to 0.02% by ten, found by trying every set. The four are not its four largest. They are a line through the pivot bent by the square of the radius, the family the cheapest curves come from, and a curve close to that family is nearly free. An ellipse, a parabola, a lemniscate and a folium are not close to it, and each costs its whole equation to come within four per cent.

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.

Dixon's nine bars with every joint somewhere inside its clearance. Dixon's framework — joints at -2, 1, 3 on one line and -1.5, 1, 2.5 on a line square to it, every joint of one set barred to every joint of the other — with each joint moved to a seeded point inside a disc about its nominal place. The discs are drawn with a radius of 0.12 so that they can be seen; the framework measured has joint errors about a hundred times smaller. The nominal joints lie on two lines, which is one conic, and that is what lets nine bars on six joints flex. The joints as built lie on no conic: at an error of ρ, seed 2's smallest singular value is 0.1746ρ where the nominal framework's is nought, so by rank it is a structure.

What can move

The clearance that makes the framework generic

A clearance lets Dixon's nine bars move, and it also moves their joints off the two lines that let them move at all. Six joints off a conic make the framework rigid by rank, with a smallest singular value proportional to its distance from a conic. The same clearance that did the damage pays for it fourteen times over: a framework built anywhere inside its clearance discs is pushed to its first crossing using at most 7% of the allowance those discs give. The tilt of the lines decides how far it goes; its own joint errors change that by about 1%.

A slider-crank and its one cognate draw one curve, and each carries the crank's rotation on a different member. Left: a slider-crank with a crank of 1, a rod of 3, its slide 0.5 from the crank's pivot, and the tracing point (0.4, 0.5) in the rod's own units; both of its circuits are drawn. Right: the only cognate that survives when a four-bar's rocker becomes a slide, a slider-crank with a crank of 1.921, a rod of 0.640 and an offset of 0.320, its slide turned by 51.3°. Both are drawn holding the same point. Members are coloured by rotation: 1 is the original's crank and the cognate's rod, 2 the original's rod and the cognate's crank, and 3, the slide, cannot turn in either. Under each name are that machine's closed-form area terms for the circuit through the held point, each computed from its own traced circuit read anticlockwise: the original's crank carries 1.885 and the cognate carries 1.885 on its rod; the other terms are nought, because neither machine's rotation 2 goes round.

The paths points trace

A slide exchanges the crank and the rod

Roberts's construction gives a four-bar two cognates. Grow the rocker into a slide and one of them leaves with it; the other becomes a slider-crank whose crank is the original's rod and whose rod is the original's crank, the whole machine turned and scaled by the tracing point. So a slide keeps two of the figure's three rotations and freezes the third, the closed forms of the area move between crank and rod, and every slider-crank whose crank turns has a cognate whose crank cannot.

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.

Links with a width

Four cuts serve the whole cell

A region test between moving parts needs its drive cut where each pair's two visits to shared ground separate, and one pair of Peaucellier's arms needs four cuts. The cell has six such pairs, and cut one at a time they would need sixteen. Cut together they need four, the same four moments, because every pair crosses its shared ground on either side of them. Built from a whole-drive region test the cell would take six planes; the four cuts give back the true four, where equal arcs need eight to do it.

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.

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 curved slot bends at its mouth with the crank's circle, or against it. A six-slot Geneva wheel whose slots are curved: mouth curvature 0.5 of the crank's. The driver's pin runs round the dashed circle and enters each slot along its mouth, which is radial so that the wheel starts from rest. At entry the pin's acceleration is square to the slot, and the part of it the slot's own bend takes up does not turn the wheel. Solved over the index, the wheel starts with an acceleration of 0.289 and stops with a deceleration of 0.866, in units of the crank's speed squared, against 0.577 at both ends for a straight slot; the two always add to 1.155. Its peak is 1.649 against the straight slot's 1.350.

Motion that stops

A curved slot moves the step

A Geneva's pin arrives with an acceleration of tan(π/n) of the crank's speed squared, and no choice of entry direction removes it. A choice of slot shape can: bend the slot's mouth with the crank's own circle and the slot absorbs the pin's acceleration instead of handing it to the wheel. But the pin leaves by the mouth it came in by, and there the same bend adds to the step. Solved at four slot counts and seven curvatures, the entry and exit steps are tan(π/n)(1 ∓ κ̂) and always sum to 2 tan(π/n). A curved slot moves the step from one end of the index to the other, every bend raises the peak, and no slot tried beat the straight one.

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.

Out of the plane

A shake travels as a root of the play

Two three-revolute legs that share a twist let their platform shake along it: the gap between their nearest poses grows as the square of the push, or the cube once one joint is tuned, or the fourth power once two are. With play in the joints, the pair moves until that gap uses up what the play can take, and that is a root: the square, cube or fourth root of the play. At a play of a millionth the three pairs travel 0.013, 0.066 and 0.139 along their twist. A tuned joint is worth more the tighter the joints are, five times the travel at a millionth and eleven at a hundred-millionth.

The seat at the pole, and the friction cone at its contact. 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 in a seat of radius 0.8 on the frame, aimed through the coupler's pole. At the contact the shaded wedge is the friction cone for a coefficient of 0.1: every force the seat can press with lies within 5.7° of the normal. The normal itself runs through the pole, so it has no lever about it; the cone's edges do. Under a moment of 0.001 on the crank against a contact of stiffness 10⁶ the crank turns 1.70e-4 radians, against 1.48e-3 with no friction; released, it stays at 1.70e-4.

The chain before the lengths

A seat with friction stays where it is pushed

A stop aimed through its pole holds a four-bar by curvature, because its normal has no lever about the pole. Friction has one, and the obvious guess is that it lets such a stop hold at first order. It does something narrower and stranger. It cannot hold what curvature releases, since a gap that opens carries no force for friction to be a share of. It makes a seat that holds up to twenty-six times stiffer, turning its cube-root give into a square root. And it makes any stop whose pole lies inside its friction cone stay where a load pushed it: released, the seat at the pole does not come back.

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.

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'.

How many answers

A four-bar becomes a trammel along a family of laws

A four-bar becomes the elliptic trammel when its crank and its rocker both grow without bound, their pivots receding at right angles, and its three double points must all leave. Taken one link at a time the limit sends one point off along the four-bar's law and then the other two along the trammel's. Taken in one step it does something neither order sees: all three leave together, in proportion to the machine, at positions set by the coupler point and by λ, the rocker's length over the crank's. The two one-step orders are the family's two ends, λ → ∞ and λ → 0, and at equal rates the coupler point (0.3, 0.2) has its three double points up to 0.86 crank lengths from where either order puts them.

Moving the pivot adds terms everywhere but one locus. The folium x³ + y³ − 1.5xy = 0, drawn in its own coordinates, and the square of pivot positions from −1 to 1 searched at steps of 0.05 (shaded). With the arm's pivot at the origin, where the catalogue puts it, the equation expands into 11 cosines of the arm's two angles. Moved anywhere in the square off the line x + y = 0 (heavy), it expands into 12; at (0.5, 0.3), for instance, 12. 41 of the 1681 pivots tried cost less than 12, all on that locus, and none costs less than the catalogue's 11.

The curve as an equation

Moving the pivot only adds terms

A compiled machine's cost is the number of cosines its curve's equation expands into, and moving the arm's pivot gives the same curve a different equation. So an expensive curve might only be expensive because of where the arm was put. Searched over 1,681 pivots, it never is. Moving the pivot adds the equation's own derivatives at the pivot as new terms, and they cancel only on one locus per curve — a centre, or a line — on which the catalogue's pivot already sits. Anywhere else the ellipse costs 6 instead of 4, the folium 12 instead of 11, the lemniscate 9 instead of 5, the parabola its 6 everywhere, and the quintic that four terms nearly draw costs 30 instead of 18.

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.

Past the crossing, one push names two placements, one in each family. Dixon's framework, joints at −2, 1, 3 on one line and −1.5, 1, 2.5 on a line square to it, held by B₁ and the direction of B₁W₁, with B₂ pushed 0.5 along its line. Solid: the placement on the family the framework starts in (++), reached before the crossing. Dashed: the placement at the same push on the family beyond it (+−), reached after W₂ has passed the other line at a push of 0.591 — the faint framework — and the push has begun to fall again. Every bar has its drawn length in both. A framework with no errors moves from one to the other through the crossing; a framework as built has no motion and must be carried there by its play.

What can move

The price of the other family

At its crossing, Dixon's framework passes from one family of placements to another, and past it every push names two placements, one in each. A framework built with its joints anywhere inside a clearance ρ has no motion to follow, so its play has to carry it. The price of going on is proportional to ρ, from 0.002ρ to 0.20ρ across ten frameworks, exactly as a parallelogram's change point is — and reversing every joint error leaves it unchanged. Eight of the ten find the far family dearer than their own and two find it cheaper, and all of them find it far inside the 2ρ their clearance gives.

One ellipse, drawn at once by four double sliders on four different pairs of slides. The ellipse traced by a double slider with slides at 70°, a rod of 2 and a tracing point off the rod at λ = 0.3 + 0.35i: semi-axes 1.587 and 0.542. Drawn over it at one instant, four more double sliders holding the same point, each on its own pair of slides through the ellipse's centre, with rods of 1.73, 2.08, 0.93, 1.03 and slides at 54°, 77°, 63°, 80°. The two in the first colour are from the original's own rolling circle and turn with it; the two in the second are from the other and turn against it. Any two points of either circle would do as well.

The paths points trace

A second slide keeps only the rod

A slide turned a four-bar's rocker into a line and left one cognate, the slider-crank with its crank and rod exchanged. Turn the crank into a slide as well and the exchange stops producing a machine, since λ times a sliding pin's position is a vector that shrinks to nothing twice a turn. The cognates do not vanish; they multiply. Every double slider's ellipse is drawn by any two points of either of two rolling circles — a two-parameter family on each — and the one rotation left, the rod's, is carried by every one of them, with the original on one circle and against it on the other.

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.

A pin steered off its circle carries the wheel on any law. A 4-slot Geneva wheel and its driver. The pin rides in a radial guide on the driver and a stationary cam sets its radius, so its path through the index (solid) is not the crank's circle (dashed). Here it is cut for the modified trapezoid law over the circle's own window of 90° of driver: the wheel starts with an acceleration of nought and stops with nought, against tan(π/4) = 1.000 at both ends for the crank pin, and peaks at 3.112 against the crank pin's 5.407, in units of the driver's speed squared. The pin's radius moves between 0.667 and 0.708 on a crank of 0.707.

Motion that stops

A steered pin keeps only the window

A Geneva's step at entry and exit belongs to its pin's circle, not to its slots. Carry the pin in a radial guide on the driver and set its radius with a stationary cam, and the wheel can be given any law at all — the pin is simply where the driver's ray meets the slot's line. Cut for a modified trapezoid, a six-slot wheel starts and stops with no step and peaks 14% below the crank pin, while the cam moves the pin by 1.2% of the crank. What the circle cannot hand over is its window: no ray before the crank pin's entry meets a resting slot, so the index can never be longer, and every degree spent holding the wheel in its slot comes out of it as the square.

A V of two faces, each turned off the line to the pole. 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 resting in a V of two flat faces on the frame. Each face's normal is turned off the line from the nose to the coupler's pole so that at the contact it misses the pole by 12.79°, one each way, so that each face holds the crank one way at first order. The shaded wedges are the friction cones for a coefficient of 0.2, half-angle 11.3°: the pole lies outside each, so released from either side the crank springs back. Each face's moment per radian, through a contact of stiffness 10⁶, is 72 thousand.

The chain before the lengths

Return and stiffness come from the same aim

A stop with friction looked like a trade: aim it near its pole and friction multiplies its stiffness seventeen times, aim it outside the friction cone and it springs back when released. The trade is an artefact of how the gain was counted. Friction's factor multiplies a stiffness that vanishes at the pole, and measured outright a stop aimed further away is stiffer at every friction — 271 per radian at 1.1° and 104,000 at 21.3° with none, 4,631 and 184,000 at a coefficient of 0.3. Return and stiffness are bought by the same aim. What the aim costs is the other side: a face aimed off its pole holds nothing the other way, so a stop that must return both ways is a V of two faces, and across a friction band from 0.1 to 0.2 a V aimed 12.3° each way returns from both and is 8,800 times stiffer than the seat at the pole.

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.

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.

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.

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