Machine Kinematics
solved, then drawn
Concept

Ball point — where it appears

The one point of a moving plane whose path has four-point contact with its own tangent line, lying on both the inflection circle and the cubic. It is where a linkage draws its straightest line, and it is generally not where a designer put the tracing point — the site measures how much is left on the table.

Named by 5 essays across one field — each of them below, with the objects they name alongside it.

Ball's point at 66°. Two curves and one point. The circle is the locus of points whose path is momentarily straight; the cubic is the locus of points whose path curvature is momentarily not changing. A point on both has a path that is straight and staying straight — four-point contact with its own tangent line, one order better than anything else in the plane. That is Ball's point. Walking the circle and looking for a zero of κ′ turned up 1 sign changes here and 1 of them survived being checked for continuity — the others are the asymptote the pole itself produces. Its measured order of contact with the line is 3.95. positioned by solving, not by drawing.

The straightest point there is

Where the circle of points going straight meets the cubic of points whose curvature is standing still, there is one point whose path is straight and staying straight. Neither Watt nor Chebyshev put their tracing point there, and moving Watt's to it more than halves the error over the whole stroke.

curvature · Curvature Three points, three orders of contact. Replace the coupler point by a crank pivoting at the centre of its path's osculating circle, drive the linkage away from the instant, and measure how far the point gets from that circle. The distance grows like a power of the step, and the power is the number of derivatives that agreed. An ordinary point gives 3.00; a point of the cubic of stationary curvature gives 3.94; a Burmester point, where the curvature is stationary and its rate of change is too, gives 4.97. Nothing in the measurement knows which kind of point it was handed. This is what the two special curves are for: they are where a single pivot can replace a whole linkage for longest. positioned by solving, not by drawing.

How long a pivot stands in for a linkage

Throw the four-bar away and replace a coupler point by a crank pivoting at the centre of its path's curvature. The two separate like the cube of the crank step for an ordinary point, the fourth power on the cubic, and the fifth at a Burmester point — three exponents that are the whole field, measured with no derivative anywhere in the measurement.

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.

Inflection circleAccelerationAcceleration poleInstantaneous centreMoving planePath curvatureContact orderCubic of stationary curvatureStructural errorBurmester pointOsculating circleParameterisation

All concepts