Field

The motion, not the mechanism

Everything else here is about a machine. This is about the motion a machine makes, near one instant — where the moving plane is turning, which of its points are going straight, which of them could be replaced by a single pivot and for how long. Two linkages with nothing in common that agree on those numbers are interchangeable, and the mechanism turns out to be one of infinitely many ways of producing a motion rather than the thing the subject is about.

The mechanism drops out

A planar motion described near an instant without reference to the mechanism that produced it — the pole, the pole tangent and the derivatives of a loop equation, obtained exactly rather than by differencing a solve.

A point, its pole, and the centre it is turning about. The tracing point is on the coupler at (0.45, 0.5) of its length. The cross is the pole, the faint curve is the path the point traces over a whole turn, and the circle is the one that path is momentarily on — centre marked, radius 0.267. The point, the pole and the centre are collinear, which is not an accident of this position: a point's centre of curvature always lies on its own ray from the pole, and Euler and Savary's relation says where on it. Here that relation puts the centre 5.6e-16 of a unit from where differentiating the loop equation three times puts it. positioned by solving, not by drawing.

Every point has a centre

Euler and Savary's relation, checked against the centre of curvature obtained by differentiating a loop equation — two routes that share no arithmetic, agreeing to fifteen digits over twenty-eight comparisons.

The circle of points going straight

The inflection circle obtained as the zero set of path curvature and then measured to be a circle, with a general conic fitted to it and its cross term reported, and the inflections of a coupler curve counted two ways.

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 velocity pole and the acceleration pole of a planar motion, measured against each other over a full turn, and six things an instantaneous centre is commonly taken to be and is not.

The collineation axis at 66°. The coupler line extended and the frame line extended meet at Q, and the line from the pole through Q is the collineation axis. Bobillier's theorem is that the axis and the pole tangent make equal angles with the two rays PA and PB, in opposite senses — so having the axis gives the pole tangent, which is otherwise the one quantity here that needs the motion differentiated. Measured over 50 pairs of conjugate points the relation holds to 2.5e-14 radians. positioned by solving, not by drawing.

A construction with no arithmetic in it

Bobillier's theorem measured on fifty pairs of conjugate points, the pole tangent recovered from the coupler and frame lines alone, and the inflection circle constructed from three points and compared with the computed one.

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

The cubic of stationary curvature derived as the zero set of the rate of change of path curvature, with the quartic terms measured to cancel and the double point at the pole confirmed along every ray.

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

Ball's point found as the single non-pole intersection of the inflection circle and the cubic of stationary curvature, its order of line contact measured at four, and Watt's and Chebyshev's classical tracing points measured against it.

Four positions brought together

The finite four-position circle-point curve compared against the infinitesimal cubic of stationary curvature as the positions coalesce, with the convergence exponent fitted at 2.02 and two wrong ways of spacing the positions recorded.

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

Cusps of a coupler curve located as the crossings of the tracing point by the moving centrode, with the point's speed at the corner measured at ten to the minus fourteen against 2.87 a fifth of a radian later.

Two circles, and an exact straight line. The moving centrode of this motion is a circle of radius 1.5 and the fixed one is a circle of radius 3.0, measured to 8.9e-16. The small circle rolls inside the large one, and a point on its rim traces a diameter of the large one — exactly, with no error term. Watt's linkage is straight to nine per cent of its span and Chebyshev's to twelve; this is straight to 0.0e+0, and the difference is not one of degree. It is the difference between a curve that approximates a line and two centrodes whose rolling produces one. positioned by solving, not by drawing.

Exact because two circles roll

The elliptic trammel's motion described by its centrodes, with the rod ends measured to stay on the axes exactly, the two circles measured to their nominal radii, and the whole thing set against the approximate straight-line linkages.

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

The Burmester points of five infinitesimally separated positions, found on the cubic of stationary curvature, counted over a full turn, with the contact order measured at five and the window the count depends on stated.

A curvature is a size with a minus sign

What a scaling does to the curvature field's objects, why the constructions are similarities and the values are not, and what an angle-only measurement therefore recovers.

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

Contact order measured directly by replacing a coupler point with a crank at its osculating centre and fitting the separation exponent, at 3.00, 3.94 and 4.97, with a first version that returned 2 for every point and why.

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

The moving plane's rotation and translation recovered from the two centrodes alone, checked against the linkage's, with the arc-length identity and the unbounded branches of a four-bar's fixed centrode.

Three circles at one point, at 50°. The osculating circle of the fixed centrode at the pole, the osculating circle of the moving centrode there, and the inflection circle. All three pass through the pole and all three have their centres on the common normal — which is what makes the relation between them a relation between three numbers on a line. The two centrode radii are 16.180 and 9.730, signed along that normal, and the inflection circle's diameter is 24.398 against the 24.409 the other two give. Nothing in that arithmetic knows there is a linkage. The window is 30.25 wide and the larger osculating circle is 32.36 across, so it is cut at the edge: what is drawn is the neighbourhood of the pole, where the three curvatures are the same three numbers however far the circles carrying them reach.

The two numbers are the curves' own

The inflection circle's diameter computed from circles fitted to the two centrodes, checked against the derivative route across a whole turn, with the sign convention derived rather than chosen.

Holding the coupler still: the frame's inflection circle is the mirror of the coupler's. The four-bar with ground 4, crank 1, coupler 3.5 and rocker 3 at a crank angle of 225°, redrawn in the coupler's own frame, so the coupler is the horizontal bar and the ground link is what moves. The circle on one side of the pole is the ordinary inflection circle — coupler points travelling straight — carried into this frame. The circle on the other side is the inverse motion's: frame points travelling straight when the coupler is held. It is computed by differentiating the swap and, separately, by solving the coupler-held four-bar, and both routes put it at the first circle reflected through the pole, to 7e-15 on a radius of 4.45. The line through the pole is the common tangent. Dragging the crank angle moves both circles and they stay mirror images.

The frame seen from the coupler

The inverse motion of a four-bar's coupler, obtained by differentiating the exchange of the two planes and by solving the chain with its coupler held as a four-bar in its own right; its inflection circle shown at seven crank angles to be the ordinary circle reflected through the pole; frame points on that circle measured to have no path curvature; conjugate points whose roles swap to 10⁻¹⁶; and the Grashof kinds of the machines on either side of the swap.

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 acceleration pole of a four-bar's coupler placed on its inflection circle by a crank acceleration; the measured angle at the velocity pole against arctan of the angular acceleration over the angular speed squared, over 1,275 instants on five motions and five drives; that angle round a constant-speed turn, with the two crank angles where the acceleration pole is the inflection pole; the gap between the two poles as the diameter times the cosine of the angle; Bresse's two circles meeting at the acceleration pole; and the ledger for every motion in the field.

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 third-order pole of a four-bar's coupler: the pole under several drives, with the circle it runs round as the jerk alone varies; the poles of 625 drives, spread over the plane and absent from the inflection circle; the two pencils of circles, through the velocity pole and through the acceleration pole; the size of drive a point needs against its distance from the circle; and the only two points on the circle a drive can reach.

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

Whether a crank schedule can keep a chosen coupler point's jerk at nought over a stroke of the standard four-bar: the jerk number each point needs at each angle against the one its acceleration number forces, for four points; the point's jerk over the stroke with a steady crank and with the best schedule; the crank speed each best schedule asks for; and a table of the mismatch, the best schedule's share of the steady jerk and a time-domain check of the jerk formula.

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

Which coupler point of the standard four-bar a scheduled crank keeps smoothest over the stroke from 30° to 150°: the coupler plane shaded by each point's jerk per unit travel in time, steady and scheduled; the score along the line from Ball's point to the smoothest point; the crank speed each schedule asks for; the smoothest point's jerk through the stroke in time against the other measure's choice; and a ledger of five points.

A crank that must come round picks another point

Periodic crank schedules on the standard four-bar, the working stroke from 30° to 150° held to a third of the cycle: the crank's speed over a turn under three schedules; the working stroke's jerk against the return's as the return's weight grows, with the turn's own point beside them; the open stroke's point's jerk through a turn in time; where the two points sit in the coupler's plane; the neighbours' and the turn's point's jerk under one schedule; and a ledger of two points and four schedules.

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