Machine Kinematics
solved, then drawn
Concept

Circuit — where it appears

A connected piece of a mechanism's configuration space: the positions it can move between without being taken apart. A four-bar has one or two, decided by Grashof's condition, and each draws its own closed coupler curve.

Named by 7 essays across 4 fields — each of them below, with the objects they name alongside it.

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

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.

curves · Coupler 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

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.

curves · Coupler 20,000 four-bars, and not one with three circuits. 20,000 four-bars with the ground at one and the other three lengths drawn from 0.05 to 3, each counted exactly. crank-rocker: 3,031, of which 3,031 have two circuits, 0 have one and 0 cannot be assembled; double crank: 3,086, of which 3,086 have two circuits, 0 have one and 0 cannot be assembled; Grashof double rocker: 1,497, of which 1,497 have two circuits, 0 have one and 0 cannot be assembled; triple rocker (non-Grashof): 12,386, of which 0 have two circuits, 9,528 have one and 2,858 cannot be assembled. The most circuits any linkage has is 2.

Never three circuits

A four-bar has one circuit or two, and twenty thousand random four-bars counted exactly contain no exception. The reason is a count of four points where the two assemblies merge, which makes the configuration curve a curve of genus one, and Harnack's theorem allows a real curve of genus one two pieces and no more. A six-bar's curve has genus five or seven, and its circuits go to four and six.

algebra · Algebra Eight four-bars, one from each region the three signed sums cut. One linkage from each of the eight sign patterns of T₁ = g + c − a − b, T₂ = g + b − a − c and T₃ = b + c − a − g, each solved at a crank angle in the middle of its range. The thick arc round the left pivot is where the input's pin can go and the arc round the right pivot is where the output's can: crank-rocker + + +, input 100% of a turn and output 22%; double crank − − +, input 100% of a turn and output 100%; rocker-crank − + −, input 22% of a turn and output 100%; double rocker + − −, input 23% of a turn and output 21%; 0–π rocker + + −, input 76% of a turn and output 72%; π–π rocker + − +, input 84% of a turn and output 51%; π–0 rocker − + +, input 84% of a turn and output 85%; 0–0 rocker − − −, input 67% of a turn and output 91%. Grashof's condition names four of these and calls the other four one thing; the arcs show that the four triple rockers differ in which way along the ground line each rocker swings through.

Eight kinds of four-bar

Grashof's condition gives a four-bar one of four names and calls every linkage that fails it a triple rocker. Three signed sums of the lengths give eight, and a census of four thousand random linkages finds every one moving exactly as its signs say — because the planes where those sums vanish are the only places a four-bar's motion can change its kind.

linkages · Fourbar Four offset slider-cranks, one from each region the limit leaves. One slider-crank from each region of T₂ = b − a + e and T₃ = b − a − e, with the slide the dashed vertical line a distance e from the crank's pivot and the ground line dashed across. The thick arc round the pivot is where the crank's pin can go and the thick stretch of the slide is where the slider can: crank-rocker + +, crank 1, rod 3.5, offset 0.8, the crank reaching 100% of a turn and the slider between 2.37 and 4.43 on each side of it; double rocker − −, crank 3.5, rod 1.2, offset 0.6, the crank reaching 22% of a turn and the slider between 2.22 and 4.66 on each side of it; 0–π rocker + −, crank 2, rod 2.5, offset 1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line; π–π rocker − +, crank 2, rod 2.5, offset −1.6, the crank reaching 65% of a turn and the slider running from −4.21 to 4.21 through the ground line. These are the four of the eight four-bar kinds in which T₁ is positive.

Four kinds of slider-crank

An offset slider-crank is a four-bar whose output bar and ground have grown without bound, and in that limit the three signed sums that sort four-bars into eight kinds lose one of their signs. Four kinds survive. A census of four thousand finds every slider-crank moving as its region predicts, the textbook condition for a full crank turn turns out to be one region exactly, and each of the four kinds that vanish is carried, at a length that can be written down, into the survivor that shares its other two signs.

linkages · Fourbar The chain's whole configuration space, drawn on its two angles. A slider-crank asks one thing of its two angles: a cos θ + b cos φ = e, with θ the crank's angle and φ the rod's. Each panel is the square of those two angles from −π to π, with the curve that equation cuts out, followed by arclength on the equation alone. The number under each panel is how many whole turns θ, φ and φ − θ make around a circuit. A curve that crosses the square from side to side carries a turn in that angle; a closed loop inside the square carries none. The two regions where a member turns have two circuits each and the two where none does have one, which is what the four-bar regions these came from predict.

Three rotations, and four benches

The slider-crank chain's four inversions are four famous machines, and they are not four classifications. Four links make six pairs, one of those pairs cannot rotate at all because a slide is a rotation of nought, and the five that are left are three quantities between them. So an inversion chooses which two of the chain's three rotations sit at its bench, and the four regions of length space already say what all three do.

linkages · Fourbar The configuration space of a crank rocker. Every point of the square is a pair of angles — the crank's and the rocker's — and the curve is where the coupler is exactly the right length to join them. That curve is the mechanism: one equation in two angles leaves one freedom, which is the mobility. It has 2 components, and that is the two assembly branches. A built linkage cannot cross between them, because there is no path in the set to cross by — and each goes all the way round, which is what makes the crank a crank. The square's left and right edges are the same line, and so are its top and bottom.

Branches were components all along

Seven words have been used for one thing. An assembly branch, a circuit, an assembly mode, a working mode, a posture and a branch defect are all statements about the connected components of a mechanism's configuration space — and once that is said, a four-bar's two circles, a platform's six modes and a synthesis defect stop being three subjects.

serial · Algebra

Named alongside it

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

Grashof's conditionAssembly branchConfiguration spaceChange pointClassificationConnected componentCoupler curveCrank-rockerDouble pointDouble rockerLimit positionSextic

All concepts