Machine Kinematics
solved, then drawn
Concept

Acceleration pole — where it appears

The point of a moving plane whose acceleration momentarily vanishes, which is a different point from the one whose velocity does. It is the one quantity in the field that depends on how the input is being driven, and on an ordinary four-bar it is never closer than a coupler length to the velocity pole.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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

Named alongside it

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

Instantaneous centreMoving planeAccelerationInflection circlePath curvatureBall pointCentrodeSecond orderVelocity ratioContact orderPole tangentVirtual work

All concepts