Machine Kinematics
solved, then drawn
Concept

Moving plane — where it appears

The whole plane rigidly carried by a moving link, of which the link itself is one small part. Thinking of the whole plane rather than the link is what makes the pole, the centrodes and the inflection circle available, since all of them are points of it that are not points of the mechanism.

Named by 13 essays across 2 fields — each of them below, with the objects they name alongside it.

The coupler's motion at 66°. Every point drawn as a stub is a point of the coupler's own plane, and the stub is that point's velocity — solved, not sketched. They all point different ways and they are all consistent with one statement: at this instant the whole plane is turning about a single point, the pole, marked with a cross. It is off this frame at 1.5 coupler lengths from the crank pin, which happens whenever the coupler is close to translating. The arrow through it is the pole's own velocity, which is a quantity about the motion rather than about any point of it, and half of everything in this field follows from its direction and its size. positioned by solving, not by drawing.

The mechanism drops out

Every other field here is about a machine. This one is about the motion a machine makes — a plane sliding over a plane — and near any instant that motion is a handful of numbers with no linkage in them. Two mechanisms that agree on those numbers make the same motion, and one of them can always be thrown away.

curvature · Curvature The circle of points going straight, at 66°. Every point on this circle is, at this instant, travelling in a straight line: its path has zero curvature there. The circle passes through the pole — where the point is not moving at all — and its diameter is 47.64, which is the pole's own speed divided by the plane's angular rate. Nothing here was assumed to be a circle. The locus is the zero set of a quadratic whose |w|² coefficient is φ′³, a real number with no cross term and no difference between its two square terms, and a general conic fitted to the sampled locus returns those coefficients at 8.1e-15 and 6.4e-15. At this position the coupler is close to translating, the pole has run off the canvas and the circle with it — the figure is the size of the mechanism, and δ here is 13.6 coupler lengths. positioned by solving, not by drawing.

The circle of points going straight

At any instant some points of a moving plane are travelling in a straight line. They form a circle — not nearly a circle, a circle — and the reason is one real coefficient in a quadratic. A coupler curve has an inflection exactly when that circle sweeps over the tracing point, which turns out to be rare.

curvature · Curvature Two poles at 66°, and only one of them is still. The stubs here are accelerations, not velocities, and they vanish at a different point from the one where the velocities vanish. The cross is the velocity pole — the point the plane is turning about — and the square is the acceleration pole, the point that is momentarily not accelerating. Over a full turn of this crank the two are never nearer than 1.49 coupler lengths and get as far apart as 22.2. The instantaneous centre is about velocities and about nothing else: the point that is not moving is being accelerated, usually hard, and a body pivoting about it in the ordinary sense would not be. positioned by solving, not by drawing.

The other pole

The instantaneous centre is the point of a moving plane that is not moving. There is a second point that is not accelerating, it is somewhere else entirely, and over a full turn of one four-bar the two are never closer than one and a half coupler lengths and get as far apart as twenty-two.

curvature · Curvature Where the curvature is standing still, at 66°. The cubic of stationary curvature: every point on it traces a path whose curvature has stopped changing at this instant, so its osculating circle fits for one order longer than an ordinary point's. It passes through the pole — twice, with a double point there — and through both moving pins, which is the plainest case there is, since a pin traces an exact circle and a constant curvature is certainly a stationary one. The expression whose zero set this is looks like a quartic and its fourth-degree terms cancel to 1.2e-14 of the third-degree ones. positioned by solving, not by drawing.

Where the curvature stands still

One derivative further on there is a second locus: the points whose path curvature has momentarily stopped changing. The expression whose zero set it is looks like a quartic, its fourth-degree terms cancel, and the cancellation is measured at ten to the minus fourteen rather than assumed.

curvature · Curvature Two poles at 66°, and only one of them is still. The stubs here are accelerations, not velocities, and they vanish at a different point from the one where the velocities vanish. The cross is the velocity pole — the point the plane is turning about — and the square is the acceleration pole, the point that is momentarily not accelerating. Over a full turn of this crank the two are never nearer than 1.49 coupler lengths and get as far apart as 22.2. The instantaneous centre is about velocities and about nothing else: the point that is not moving is being accelerated, usually hard, and a body pivoting about it in the ordinary sense would not be. positioned by solving, not by drawing.

Six things a centre is not

The instantaneous centre is the most over-read object in this subject. Six claims about it are in circulation, three are false, two are true of something else, and one is nearly right — and each of them comes with a measurement of how badly it goes wrong.

wrong · Misconception A corner in a curve, and the reason for it. This coupler point traces a curve with a cusp — a corner, where the curve stops and reverses rather than turning. The reason is that a cusp happens where the tracing point is momentarily still, and the only point of a moving plane that is momentarily still is the pole. So the cusps of a coupler curve are the instants when the pole passes through the tracing point: they are the crossings of the tracing point by the moving centrode, which is a statement about two curves in the coupler's own plane with the fixed plane not involved at all. This point was chosen by taking the pole's body coordinate at one instant, so it is on the moving centrode by construction; at the corner its speed is 9.9e-15 against 2.87 a fifth of a radian later. positioned by solving, not by drawing.

Where a curve has a corner

Some coupler curves have corners in them — points where the curve stops, turns round and comes back. A corner happens where the tracing point is momentarily still, the only point of a moving plane that is momentarily still is the pole, and so the corners of a coupler curve are decided entirely inside the coupler's own plane.

curvature · Curvature Two Burmester points, or none, depending where the crank is. A Burmester point's path stays on one circle to fifth order. A planar motion has at most four of them; two of this mechanism's are always its own moving pins, whose paths are exact circles and satisfy every order at once. The other two are real for 67 per cent of the turn and complex for the rest, and the count changes without anything about the mechanism changing. The window matters and is stated: points beyond a hundred coupler lengths from the pole are not counted, and widening the window from six to four hundred moves the count of positions-with-two from 191 to 245 out of 360. A silent cap here would read as an absence.

The circle a point stays on longest

One condition further on are the points whose path holds a circle to fifth order. A planar motion has at most four; two of them are always the mechanism's own pins, and the other two are real for two thirds of a turn and complex for the rest — a count that changes while nothing about the mechanism does.

curvature · Curvature 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 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.

curvature · Curvature The acceleration pole walks round the inflection circle. The four-bar of this field — ground 4, crank 1, coupler 3.5, rocker 3 — at a crank angle of 240°, with its inflection circle, the locus of coupler points whose paths are momentarily straight, of diameter 9.23. The cross is the velocity pole and the ring is the far end of the diameter from it, the inflection pole. The square is the acceleration pole when the crank is driven with an angular acceleration of 0.323 times its angular speed squared. It is on the circle, at 30.0° from the diameter, which is arctan of the coupler's angular acceleration over its angular speed squared. The circle does not move when the drive changes; only the square does. Dragging changes the crank's acceleration.

The other pole is on the circle

The point of a moving plane with no acceleration depends on how fast the input is driven, which is why it was set apart from the rest of this field as the one object that is not geometry. It is not free, though. It always lies on the inflection circle — the geometric locus of points going straight — at an angle from the circle's diameter whose tangent is the plane's angular acceleration over its angular speed squared. A drive chooses one angle, and the geometry has already chosen everything else.

curvature · Curvature The third-order pole goes where the drive puts it. The four-bar of this field at a crank angle of 60°, with its inflection circle. The cross is the velocity pole and the ring Ball's point, where the circle meets the cubic of stationary curvature. The square is the third-order pole — the coupler point whose jerk is nought — when the crank is driven with an angular acceleration of 1 and a jerk of 6, each per the right power of its speed. It is at (0.492, 10.663). The faint curve is where it goes as the jerk alone is varied: a circle through the velocity pole. Every point off the inflection circle is the third-order pole of exactly one drive; on the circle only the velocity pole and Ball's point can be. Dragging changes the drive.

The third pole goes anywhere but the circle

The velocity pole is fixed by the geometry and the acceleration pole by the geometry up to one number, its place on the inflection circle. The next one, the point of a moving plane whose jerk is nought, depends on two numbers of the drive — and two numbers are enough to put it anywhere. Every point of the plane off the inflection circle is the third-order pole of exactly one drive, because a body point's jerk is affine in the crank's acceleration and jerk. On the circle only two points qualify: the velocity pole, and Ball's point, where the path is straight to third order.

curvature · Curvature The crank's acceleration fixes its jerk, so the two numbers a point needs cannot both be met. For the coupler point (0.45, 0.5) of the standard four-bar, over the crank's stroke from 30° to 150°: the second schedule number λ = θ⃛/θ̇³ that, with the first κ = θ̈/θ̇², makes the point's jerk nought at each angle; and the λ that a crank actually following that κ has, κ′ + 2κ², since both numbers come from one function, the crank's speed against its angle. Their difference, the mismatch, reaches 1.77 and is nought only at 39.8°. Wherever it is not nought no crank holds the point's jerk at nought.

No drive holds the third pole

At one crank angle a drive's acceleration and jerk can put the coupler's third-order pole on almost any point, making that point's jerk nought. Over a stretch of the stroke they cannot. A crank's schedule is one function, its speed against its angle, and its jerk number is fixed by its acceleration number: λ = κ′ + 2κ². So the two numbers a point needs at every angle must satisfy that relation, and on the standard four-bar they miss it by as much as the numbers themselves, agreeing only at isolated angles. The best schedule a crank can follow takes a fifth to a third off a coupler point's jerk, and no more.

curvature · Curvature Where on the coupler a scheduled crank is smoothest. The standard four-bar's coupler plane, A at the origin and B at (1, 0), shaded darker where a point's jerk per unit of its own travel over the stroke from 30° to 150° is smaller, with the crank's speed scheduled to make that point's jerk least: from 1.34 to 5.12, measured in time over a stroke of fixed duration. The smoothest scheduled point is (−0.02, −0.80), at 1.263 against 4.555 with a steady crank; the smoothest with a steady crank is (−1.00, 1.25), at 3.397. Ball's point at the stroke's middle lies at (1.55, 3.27), off the map, where it scores 1.702 scheduled.

The smoothest coupler point is chosen in time

A crank's speed can be scheduled to cut a coupler point's jerk, though never to nought. Which point should a designer free to choose it pick? Measured as the schedule essays measured jerk, per crank speed cubed, the answer is a point whose schedule swings the crank's speed four to one, and it is a cheat: that measure divides by the speed it changes. Measured in time over a stroke of fixed duration, per unit of the point's own travel, the smoothest point of the standard four-bar is (−0.02, −0.80) in the coupler's frame. Its schedule cuts its jerk by 72% with a speed swing under two to one, and it is 63% smoother than the best a steady crank finds anywhere. Ball's point, the instant's straightest, lies in a second, shallower valley four coupler lengths away.

curvature · Curvature A crank that must come round: slow through the stroke, quick through the return. The crank's speed over a whole turn, as a share of its mean, under periodic schedules that leave it at the speed it started with and give the working stroke, 30° to 150° (shaded), a third of the cycle: smoothest stroke point, return nearly free (−0.02, −0.80), from 0.66 to 1.87; smoothest stroke point, return weighted a thousandth (−0.02, −0.80), from 0.83 to 1.46; the turn's own point, return weighted a thousandth (1.44, 1.89), from 0.81 to 1.35. Highlighted: the turn's own point, return weighted a thousandth.

A crank that must come round picks another point

A crank scheduled over one stroke can cut the smoothest coupler point's jerk by 72%. A crank running continuously must come back to the speed it started at, and its working stroke must keep its share of the cycle. Held to both, the same point keeps two-thirds of its gain only by putting five and a half times a steady crank's jerk into the return, and no schedule beats a steady crank on both strokes. Choose the point for the whole turn instead and a gentle schedule, its speed swinging only 1.66 to one, cuts the working stroke's jerk by 74% and the return's by 18% together. That point is three coupler lengths from the first.

curvature · Curvature

Named alongside it

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

Instantaneous centrePath curvatureAcceleration poleCentrodeInflection circleAccelerationBall pointContact orderCoupler curveKinematic geometryPole tangentConic fit

All concepts