Concept

Cognate linkage — where it appears

One of the two other four-bars that trace the same coupler curve as a given one. The three are found by a construction on the original's lengths, and fitting each trace separately for the polynomial that vanishes on it gives one equation for all three.

Named by 9 essays across 4 fields — each of them below, with the objects they name alongside it.

3 four-bars, one coupler curve. Three different four-bars, with different ground pivots, different link lengths and different proportions — 0.692 and 0.799 times the size of the first. Every one of them draws this same curve. Roberts's theorem says there are always exactly three, and the construction is one complex multiplication: write the coupler point as λ = (P − A)/(B − A), put the third fixed pivot at O₂ + λ(O₄ − O₂), and the other two linkages fall out with their bars' roles permuted — what is a coupler in one is a crank in another. The curves here were traced separately, each from its own solver runs, and agree to 1.3e-5 against a sampling resolution of 1.4e-5 — which is to say, as closely as the comparison can tell.

Three linkages, one curve

Every coupler curve is drawn by three different four-bars, not one. The other two can be constructed from the first with a single complex multiplication, they have different proportions and different ground pivots, and the roles of their bars are permuted — what is a coupler in one is a crank in another.

synthesis · Synthesis

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

Roberts's theorem says three different four-bars draw the same coupler curve. Fitted separately for the sextic that vanishes on each trace, on a common normalisation, the twenty-eight coefficients agree across all three to 5 × 10⁻⁷ — a test of the theorem that shares nothing with the construction the cognates came from.

curves · Coupler

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

Every coupler curve has three finite double points, and an odd number of them are real, so no coupler curve has none. On the standard crank-rocker the only real one is a point of the curve that neither assembly ever visits, that no contour plot can find, and that sits on the circle through the three pivots of Roberts's cognates.

curves · Coupler

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 theorem gives every four-bar two others that draw the same coupler curve, and so enclose the same areas. Measured, they do, to 10⁻¹² — but each keeps the area in a different place: the crank-rocker in its crank pin's circle, the double rocker in its coupler's turn, the third machine in its output pin's circle. What is left over has no closed form, and it is one number in all three.

curves · Coupler

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

Roberts's construction hands a four-bar two others that draw its curve, and which of the eight kinds those two are is settled by the kind of the first — not by its lengths within that kind, and not by where the tracing point sits. Twenty-four thousand chains at five tracing points produce no exception, and the reason is one line: the tracing point enters the construction only as a scale, and a region is scale-blind.

curves · Coupler

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.

metrology · Parameter

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.

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.

algebra · Algebra

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 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.

curves · Coupler

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

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.

curves · Coupler

Named alongside it

The objects these essays reach for when they reach for this one.

Coupler curve the Roberts–Chebyshev theorem Signed area Grashof's condition Slider-crank Crank-rocker Double point Double rocker Genus Implicit equation Least-squares Sextic

All concepts