Machine Kinematics
solved, then drawn
Concept

Contact order — where it appears

How many derivatives a curve and a substitute agree in, measured as the exponent by which their separation grows with the step. It is a statement about curves as sets rather than about how fast either is traversed, which is why measuring it against a constant-speed arc gives the wrong answer.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

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 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 Four positions brought together. Burmester's construction for four prescribed positions gives a curve of points that can be fixed pivots. Bring the four positions together and it has to become the cubic of stationary curvature, because a circle through four coalescing positions is a circle of four-point contact. Measured along one ray from the pole: the finite curve crosses at one radius and the cubic at another, and the gap between them falls as the square of the spread — fitted exponent 2.024, and 7.9e-5 coupler lengths at a spread of 0.01 radians. The finite construction is the same concyclic test the site's four-position synthesis is built on, so this compares that machinery against the infinitesimal one rather than against a re-implementation of either.

Four positions brought together

Burmester's construction finds the points of a moving plane whose four prescribed positions lie on a circle. Bring the four positions together and the curve it draws has to become the cubic of stationary curvature — and the gap between the two closes as the square of the spread, with the exponent measured 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 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 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 Three laws in the band: a third, a sixth and a sixth. For bands from 3% to 10 ppm of the output's swing, three things measured on the symmetric six-bar itself: the longest dwell at any angle at the rocker pin, the dwell at the sixth-order angle γ = 120.8024°, and how far below γ the best angle sits. The dots are measurements and the lines are the two-term model's predictions from two coefficients read off the output's even part. Over the four tightest bands the fitted exponents are 0.173, 0.172 and 0.333, against a sixth, a sixth and a third. At 10 ppm the best angle is 0.490° below γ against a prediction of 0.488°, and the dwell at γ is 26.43° against 26.23°. The loosest bands sit above the lines, where the terms the model leaves out are no longer small.

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.

curves · Coupler

Named alongside it

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

Cubic of stationary curvatureOsculating circlePath curvatureMoving planeBall pointBurmester pointBurmester theoryCircle-point curveInflection circleInstantaneous centrePolynomial fitStructural error

All concepts