Field

The paths points trace

A point on a coupler draws a curve of degree six. Choosing the linkage that draws the curve you want is the oldest hard problem in the subject, and straight lines were the hardest of all.

Five points on one coupler. The same four-bar, with a tracing point rigidly attached to the coupler at five different places. Each curve is a sextic — degree six — and moving the attachment point a little changes it a great deal. That sensitivity is the reason coupler-curve synthesis was done with atlases of printed curves for most of the twentieth century: there is no simple inverse, so the practical method was to look one up.

What a coupler point draws

Coupler curves as sextics, why they are so sensitive to the attachment point, how tracing points are attached rigidly rather than as free joints, and why synthesis is hard.

$How straight, over how much of the stroke. The deviation from a straight line, as a fraction of the traced span, against how much of each mechanism's stroke is used. The vertical axis covers fifteen decades. Watt's and Chebyshev's linkages are excellent over a short stroke and degrade as more is used; Peaucellier's sits at the bottom of the plot at every fraction, because it is not an approximation. The gap at full stroke is about fourteen orders of magnitude, and it is the difference between a mechanism that is nearly right and one that is right.$

The straight-line problem

Watt's and Chebyshev's straight-line linkages measured against the chord, why the error depends on how much of the stroke is used, and what an approximation is worth.

Peaucellier's cell: exact straight-line motion from pin joints. The rhombus and the two long arms hold |OP| · |OQ| constant at 16 = 5² − 3², which is inversion in a circle about O. Inversion carries circles through the centre to straight lines, and the link CQ makes Q run on exactly such a circle — so P travels on a line, with no approximation anywhere. Measured over 160 solved positions the deviation is 6.5e-16 of the span, which is arithmetic noise rather than a small error.

Peaucellier and the exact answer

The Peaucellier–Lipkin cell, the inversion that makes it exact, the constant product it maintains, and why the working range is set by the rhombus closing.

Where the coupler is turning

The instantaneous centre by Kennedy's construction and independently from the solved velocity field, the fixed and moving centrodes, and the rolling checked by measuring arc length along both.

Three linkages, one polynomial. Roberts's theorem says three different four-bars draw the same coupler curve. Here each one is traced, and each trace is fitted separately for the sextic that vanishes on it — on a common normalisation, or the comparison would be between three polynomials in three coordinate systems. The twenty-eight coefficients agree across all three to 5.0e-7. The three traces are drawn on top of one another and the curve is the same object each time; the test shares nothing with the construction that produced the cognates, which is why it is a test.

Three linkages, one equation

The cognate theorem checked by implicit equation rather than by overlaying traces: three linkages, three independent fits, one polynomial.

Two ovals of one sextic. A four-bar with ground 4, crank 1, coupler 3.5, rocker 3, its coupler point solved at 720 crank angles on each assembly. Each assembly closes on its own oval through a full turn of the crank, and the two ovals never meet: they come no closer than 1.712. Every solved point satisfies the one eliminated sextic to 6.6 × 10⁻¹⁶ of its largest term. The machine drawn solid and the one drawn faint are the same four bars at the same crank angle of 60°, and taking a pin out is the only way from one oval to the other.

The curve the other assembly draws

The two ovals of a Grashof four-bar's coupler sextic, one per circuit, measured on solved positions; the triple rocker whose two assemblies share one curve; and a census that sorts the two cases by Grashof's condition.

Three double points, and the one that is real is never visited. The coupler curve of a four-bar with ground 4, crank 1, coupler 3.5, rocker 3, coupler point at u = 0.45, v = 0.50, both ovals solved. The dashed circle is where the coupler's orientation can fail to be fixed by the point it carries; it passes through both fixed pivots and through the third pivot of the cognate construction, centre (2.000, 0.010), radius 2.0000. The curve's three finite double points are on it. The one that is real is isolated — a point of the curve no oval passes through, at (3.964, −0.368). The other two are a complex-conjugate pair and have no place in the plane.

A point the machine never reaches

The double points of a four-bar's coupler sextic found by construction on the circle where the coupler's orientation is undecided, classified as crossings and isolated points, counted over a census, and set against what a sign scan and a tolerance search report.

A parallelogram's sextic is a circle and a quartic. The parallelogram with ground 4, crank 1.5, coupler 4, rocker 1.5, coupler point at u = 0.45, v = 0.50. Its eliminated sextic divided by the circle of radius 1.5000 about (1.800, 2.000) leaves a remainder of 6.3 × 10⁻¹⁶ of its largest coefficient, and the quotient is a quartic whose leading form is exactly a multiple of (x² + y²)². Of the configurations solved across a full turn on both assemblies, 720 are on the circle — the machine as a parallelogram, coupler parallel to the ground — and 720 on the quartic, the machine crossed. The two factors meet at four finite points: 2 are change points, marked solid, where one configuration belongs to both; the other 2 are places the two drawings merely cross.

A sextic that comes apart

The reducible coupler sextics of the parallelogram and the kite, divided exactly by their circles; which motion each factor is; where the factors meet and which meetings are change points; and why a solver's branch label is not a factor.

Crank angle θ and crank angle −θ, joined. The coupler curve alone, with 12 chords, each joining the point the tracing point reaches at a crank angle θ between 0° and 180° to the point it reaches at −θ on the same assembly. Every chord is square to the dashed axis, to 7.5 × 10⁻¹⁶ in the cosine, and every midpoint lies on it. Measured at 48 pairs, the reflection of one point misses the other by at most 4.2 × 10⁻¹⁵. The crank's own angle is the pairing: the two places the curve crosses its axis are crank angles 0° and 180°, the only angles equal to their own negatives.

A symmetric curve from a lopsided machine

The symmetrical coupler curve derived from an inscribed angle, confirmed by pairing crank angle θ with −θ, refused off the condition by a test that is told no axis, shown not to be necessary by a Roberts cognate, and read for what it gives a dwell.

One crank, four machines, one area. Four crank-rockers sharing only a crank of length 1, with grounds, couplers and rockers all different, each tracing the point 30% of the way from the crank pin to the rocker pin. The curves have different shapes and sizes and different places in the plane, and the area each one encloses is 2.199115 = (1 − 0.3)·π·1² — on both assemblies of every one, with a worst difference of 3.2 × 10⁻¹⁴.

The area a coupler point encloses

The enclosed area of a coupler curve as a Holditch-type formula in the areas the pins enclose and the turns the coupler makes, checked against the traced polygon across all four Grashof classes, with the one term off the coupler line that has no closed form here.

Three machines drawing one oval, and where each keeps its area. The crank-rocker with ground 4, crank 1, coupler 3.5 and rocker 3, its tracing point at (0.45, 0.5) of the coupler, and the two other four-bars Roberts's construction gives for the same curve, each drawn holding the same point of one oval and each with its input pin's path dashed in the input colour and its output pin's in the output colour. The shaded oval encloses 2.116354 for all three. In the crank-rocker the crank pin goes round and carries 1.727876 of it; in the double rocker neither pin goes round and the coupler's turn carries 1.727876; in the rocker-crank the output pin goes round and carries 1.727876. The remaining 0.388478 is the same in all three.

Where three machines keep one area

Roberts's three four-bars compared by the area each encloses: every term of the area formula for all three machines and both ovals, the one line of algebra that makes three closed forms one number, the machine a classification misnames, three integrals with no closed form in fixed ratios, and how finely an area can detect a cognate built wrong.

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 flattest dwell is not the longest

The dwell six-bar built on the vertex of a symmetric coupler curve: the angle at which the fourth-order term vanishes and the contact becomes sixth order, the output that keeps the curve's symmetry to high order, and the longest dwell at each tolerance found below that angle rather than at it, against the searched dwell six-bar.

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.

A null space of fifteen is not noise

Implicit fits to a parallelogram four-bar's two motions, whose null spaces of fifteen, six and one are the multiples of each factor; the same count of multiples above degree six on a crank-rocker's whole curve; and fits to shrinking arcs of one oval, whose counts change with the arc and whose singular values slope rather than drop.

What region each four-bar's two cognates land in. One row per region of length space. Against each, the regions of the two four-bars Roberts's construction gives for the same coupler curve, and how many of the 24000 chains in the census landed in that region. Every row has one entry: across the whole census, and at each of 5 tracing points, the original's region decides its cognates' regions with nothing left over. The four Grashof regions are above the rule and the four triple rockers below it, and no row crosses it — a crank-rocker has a double rocker and a rocker-crank, a double crank has a double crank and a double crank, a rocker-crank has a rocker-crank and a double rocker, a double rocker has a crank-rocker and a crank-rocker, a 0–π rocker has a 0–π rocker and a 0–π rocker, a π–π rocker has a π–π rocker and a π–0 rocker, a π–0 rocker has a 0–0 rocker and a 0–0 rocker, a 0–0 rocker has a π–0 rocker and a π–π rocker.

The kind is decided before the lengths are

The eight regions of four-bar length space under Roberts's construction: a census of 24,000 chains finding the cognates' regions fixed by the original's, the three signed permutations that explain it, why the map does not close until the driven pivot is quotiented out, the four sets of three that result, and the region in which three machines carry the same three areas permuted.

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 dip that buys the dwell

The two-term model of a sixth-order dwell six-bar's output near its vertex, solved in closed form for the best fourth-order term at a given band; its three predictions — the best angle's offset as the cube root of the band, the dwell as the sixth root, and a fixed ratio of (27/4)^(1/6) between the detuned and the sixth-order dwell — measured on the machine from three per cent of the swing to ten parts per million; the output's shape at the optimum; and the band at which an ordinary fourth-order vertex stops competing.

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.

A closed form belongs to a rotation

The closed-form area terms of Roberts's three cognate four-bars: the substitution that maps each cognate's crank, coupler and rocker terms onto the original's, a census over every Grashof region agreeing as sorted multisets and by direction to 10⁻¹⁵, the three directions of rotation in Roberts's figure turning together to 0.07°, and why the double crank's permutation was the general case.

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.

A slide exchanges the crank and the rod

Roberts's cognates in the slider-crank limit: the second cognate receding with the rocker, the first becoming the exchanged machine A′ = λ(B − A), B′ = λB, a census of 240 slider-cranks in every kind agreeing configuration by configuration and closed form by rotation, the frozen third rotation, and the map of kinds as a reflection.

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.

A second slide keeps only the rod

The double slider's cognates: the exchange failing at the second limit, the Cardan rolling circle whose every pair of points is a double slider on the same ellipse, the two circles of radius (a + b)/2 and (a − b)/2, a census of 200 machines and 1,600 cognates traced on their own slides, the rod's rotation carried in both senses, and the area as the rod's term and the mixed integral, πL²(|λ|² − u − v cot α).

A rod on three blocks, on three slides through one point, and it moves. The double slider of slides at 70° and a rod of 2, with a third block on the rod at a point of its rolling circle (radius 1.064, drawn at this instant) and a third slide along the line that point draws, through the same crossing. Five bodies and six joints of one freedom each: the count gives 3·4 − 2·6 = 0. The rod turns anyway, all the way round, every pin on its slide, because each point of the rolling circle runs on a line through the crossing and the third slide is that line.

A third slide moves only through the crossing

A rod on three, four and five blocks on concurrent slides through a double slider's crossing: the count against the rank of the slide conditions, the third pin's distance from a misplaced slide through a turn, the free share of a turn against a turn error for six pin places with the closed form 4·arcsin(c/(2r sin ε))/2π, the parallel offset that is all or nothing, when in the turn each place's windows fall, and the share freed when the error is pivoted away from the crossing.

The pin opposite the third runs on the perpendicular slide, a quarter turn behind. The double slider of slides at 70° and a rod of 2, with a third block at a point of its rolling circle (radius 1.064) and a fourth at the opposite point, each on the slide its pin draws through the crossing. The two extra pins are the ends of a diameter of the rolling circle, so their slides are at right angles and their positions along them are 2r cos(φ − φ₃) and −2r sin(φ − φ₃): here 1.864 and 1.027. Four blocks and the count gives minus one; the rod turns, and the fourth pin passes the crossing a quarter turn after the third.

A fourth block can only take freedom away

A rod on a double slider with a third and a fourth block on slides through the crossing: the third and the opposite fourth pins on perpendicular slides; the two pins' places through a turn and each block's windows; the free share against the extra slides' turn error, alone and in quadrature, as closed forms and swept; the free share against the fourth pin's place round the rolling circle; and a ledger with clearance in every block, solved exactly at each rod angle.

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

Fit every extra slide at one pose

A double slider with a third and an opposite fourth block whose slides are turned about chosen points: each slide's free band about its pivot; the two blocks' windows through a turn, turned about the crossing and about one common pose; the free share against the common pose at three errors; the free share against the error for four rules of pivoting and the best pivots searched; more blocks set at one pose; and a ledger with clearance in every block.

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