Machine Kinematics
solved, then drawn
Concept

Centrode — where it appears

The locus of a moving body's instantaneous centre, traced in the fixed plane or in the moving one. The two roll on each other without slipping, so every planar motion is one curve rolling on another whatever mechanism produced it.

Named by 13 essays across 5 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 Two surfaces in contact, sliding everywhere but one place. The sliding speed at the contact of an involute pair, formed by taking the velocity of each body's material point at the contact and subtracting. It is zero at exactly one position — the instant the contact is at the pitch point, measured here at 7.30e-15 mm per radian — and grows linearly on both sides of it, at the rate the relative angular velocity says. Gears roll at one point of the tooth and slide everywhere else, which is why a tooth wears into a shape with a band of polish across it rather than uniformly. What is not claimed is any consequence of the sliding: friction, wear and heat need forces, and there are none here.

Rolling at one point only

Two wheels in mesh are usually described as rolling. Their pitch circles are — those are centrodes, and centrodes roll — but the surfaces that are actually touching slide against each other everywhere except at one instant, and the sliding is the largest velocity in the mechanism.

meshing · Meshing 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 coupler is pivoting, at 70°. At any instant the coupler is turning about one point — not a pin, and usually not on the mechanism at all. Kennedy's theorem finds it: the crank and coupler share the pin at A, the coupler and rocker share B, so the coupler's centre relative to the frame must lie on both O₂A extended and O₄B extended, and it is where they cross. The dashed lines are that construction. The cross is a completely different route to the same point — the place where the coupler's solved velocity field is zero, computed from the Jacobian and knowing nothing about Kennedy. Across 119 positions the two agree to 2.7e-15. At this instant the centre lies outside the frame — the two construction lines are nearly parallel, the coupler is close to translating, and the pivot has run off rather than gone missing.

Where the coupler is turning

At every instant the coupler of a four-bar is rotating about a single point — not a pin, and usually not on the mechanism at all. Track that point in two different frames and you get two curves which, rolled on each other without slipping, reproduce the coupler's motion exactly. The bars are one way of producing it and not the motion itself.

curves · Coupler Two ellipses on their foci, at a ratio of 0.603. Each wheel is an ellipse turning about one of its own foci, with the centres a major axis apart. The focal property does the work: the two radii from the two foci add to the major axis, so the contact stays on the line of centres by construction, and the rolled arc lengths agree to 2.5e-8 of their length. The ratio at this instant is 0.6033; over a turn it runs from 0.538 to 1.857, a range of 3.4490 against the ((1+e)/(1−e))² = 3.4490 the eccentricity predicts. And one turn of one wheel is exactly one turn of the other, which is the condition that makes it a pair of wheels rather than a pair of curves. positioned by solving, not by drawing.

A ratio that is a function of the angle

A gear pair is usually two circles rolling. Ask instead for an output that runs forty per cent fast for half a turn and forty per cent slow for the other half, and the two shapes that deliver it are not a design decision — the demand fixes both pitch curves completely, and the only question left is whether they close.

meshing · Meshing 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 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

Watt's straight line is straight to nine parts in a hundred and Chebyshev's to twelve. Here is one that is straight to nothing at all — no error term, no working range, no approximation anywhere — and the reason is that its two centrodes are circles, one rolling inside the other at exactly half its radius.

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.

A curvature is a size with a minus sign

Everything in the curvature field is a similarity invariant in shape and a reciprocal length in value. Scale a moving plane and its inflection circle scales, its cubic of stationary curvature scales, and every curvature it computes is divided by the factor — so a bigger machine traces gentler paths than its drawing suggests.

curvature · Curvature An open belt, whose wraps make one turn. Two pulleys of radius 40 and 24 mm with their centres 160 mm apart, with the strand running the same way round both. The two tangency points on each pulley are marked; the run between them is computed from the signed radius difference and nothing else. The wrap angles are 191.48° and 168.52°, and they add to exactly one turn — the turning number of a simple loop. Length 522.6633 mm, against the textbook formula's 522.6633. positioned by solving, not by drawing.

The road a wheel carries with it

A taut strand on a pulley is a rolling contact: the material at the tangency is at rest against the surface, and the ratio between two bodies on one span is the ratio of their arms. But this rolling constraint integrates, where a wheel's does not — and the difference is that a strand rolls along a line and a wheel rolls across a plane.

rolling · Rolling 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 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

Every essay in this field runs on the inflection circle and the pole tangent, and every one has taken them from the motion's derivatives. They are the two centrodes' radii of curvature at the pole, combined harmonically — measurable off the curves with no mechanism in the arithmetic, and carrying a sign that changes eight times in a turn.

curvature · Curvature 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

Hold a four-bar's coupler still and let its frame move, and every construction of the curvature field has a counterpart. The frame's points that travel straight lie on the ordinary inflection circle reflected through the pole — found by differentiating the swap and, independently, by solving the coupler-held four-bar, both to 10⁻¹⁴ — and a point and the centre of curvature of its path trade places exactly. The inverse motion of one four-bar is the ordinary motion of another.

curvature · Curvature

Named alongside it

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

Path curvatureInstant centreInstantaneous centreVelocity ratioMoving planeInflection circlePole tangentCoupler curveEuler savaryRolling without slippingRouletteAcceleration pole

All concepts