Coupler — the series
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What a coupler point draws
A point rigidly attached to the coupler of a four-bar traces a curve of degree six. Move the attachment a little and the curve changes a great deal. For most of the twentieth century the practical way to find the linkage that draws a wanted curve was to look it up in a book of printed atlases.
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The straight-line problem
Before 1800 a long true flat surface was harder to make than almost anything else, so guiding a piston straight without a slide was worth solving. Watt's answer was an approximation. Measuring how good an approximation, over how much of the stroke, turns out to be a more interesting question than whether it is exact.
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Where the coupler is turning
At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.
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Peaucellier and the exact answer
Eighty years after Watt settled for an approximation, a French army officer found a linkage that draws an exactly straight line from pin joints alone. It works by inversion in a circle, the product it holds constant is measurable, and solved here it comes out straight to 10⁻¹⁶ of its span.
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Where a hinge pin can go
A cabinet door's front corner moves sideways as it opens at a rate equal to how far the pin sits behind it, so with the doors touching, no pin behind the door's face can open one without going through the next. Three millimetres of gap buys nine and a half millimetres of depth, and that is the whole reason a concealed hinge has four bars instead of a pin.
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Holding an axle still
A Panhard rod moves the axle 3.56 mm sideways over 80 mm of travel and a Watt's linkage moves it 34 microns — a hundred times better, and by a higher power. The Panhard's error is quadratic in the travel and the Watt's is fifth order, which is a much stronger statement than "the Watt is better" because it says how the comparison changes with the suspension.
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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.
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The curve the other assembly draws
A crank-rocker's two assemblies do not share a coupler curve. Each draws a whole closed oval of its own through a full turn of the crank, the two ovals never meet, and both are the zero set of one sextic, so the equation a machine's own motion determines also describes a second machine it can never become.
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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.
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A sextic that comes apart
A parallelogram chain's coupler curve is not one curve. Its sextic divides exactly by a circle, leaving a quartic, and each factor is one of the two things the machine can do. The division leaves rounding and nothing else, a coupler one millionth too long leaves a remainder a million times larger than rounding, and the two factors meet at the two places where the machine has to choose.
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A symmetric curve from a lopsided machine
A four-bar with a crank of 1, a ground of 3 and a rocker of 2.5 has no symmetry anywhere in it. Make the rocker, the coupler and the arm from the rocker pin to the tracing point one length, and the curve it draws is its own mirror image to 4 × 10⁻¹⁵, about a line through the rocker pivot turned from the ground line by exactly half the coupler's angle at that pin.
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The area a coupler point encloses
Trace a point on the line through a crank-rocker's two moving pins and the region its curve encloses has area (1 − u)πa²: the crank pin's own circle, scaled by how far along the line the point sits. No ground, coupler or rocker length appears in it. Four machines that share only a crank enclose 2.199115 each, to 3 × 10⁻¹⁴.
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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.
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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°.
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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.
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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.
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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.
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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.
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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.
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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.
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A third slide moves only through the crossing
Every point of a double slider's rolling circle runs on a line through the slides' crossing, so a rod can ride a third block on a third slide there, and a fourth and a fifth. The count gives nought, then minus one, then minus two; the rod turns freely every time, because each slide past the second is implied by the first two. Build the third slide slightly wrong and the count wins, in a way that depends on the error's kind: moved parallel, the rod turns freely inside the block's clearance and does not assemble outside it; turned by ε about the crossing, it moves only in two windows a turn, of 2·arcsin(c/(2r sin ε)) each, where the third pin passes through the crossing.
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A fourth block can only take freedom away
A rod on a double slider can carry a third block on a third slide through the crossing and still turn, and a slide turned slightly wrong leaves it free only in two windows a turn. The hope was that a fourth block, at the point of the rolling circle opposite the third, would travel in opposition and cancel the error. It cannot, for two reasons. The opposite pin is the other end of a diameter: it runs on the perpendicular slide, a quarter of a turn out of phase, so its windows fall exactly between the third's. And whatever its place, a block only adds a condition the rod must also meet, so the rod's freedom is an intersection and can only shrink. With the double slider's own blocks exact, the third and the opposite fourth turned 0.0066 rad each lock the rod, where the third alone still moves at any error.
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Fit every extra slide at one pose
A double slider's extra slide, turned slightly wrong, leaves its block free in a band of the slide centred on the point it was turned about, and the rod free in windows where the pin passes through that band. Turned about the crossing, a third and an opposite fourth block have their windows a quarter of a turn apart and lock the rod from 0.0066 rad. Set each slide through its own pin at one common pose of the rod and every error becomes a turn about the place its pin has then, so every window contains that pose and the rod can never lock, with two blocks or six. Each slide pivoted where it alone is freest does better while the errors are small and locks from between 0.02 and 0.05 rad, where the common pose 45° from either crossing comes within a tenth of the best pivots there are.