Machine Kinematics
solved, then drawn
Concept

Cusp — where it appears

A corner of a traced curve, occurring where the tracing point is momentarily still and therefore where it crosses the moving centrode. It is a statement inside the moving plane alone: the frame, the ground pivots and the fixed centrode are not involved, and the site measures the tracing point's speed there against its speed a fifth of a radian later.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

A rotor that was not drawn, at 52° of shaft. The housing is an epitrochoid — the only shape here that was written down — and the rotor is the envelope of it, seen from a body that turns at a third of the shaft's rate about a centre orbiting at the eccentricity. Nothing about the rotor was chosen. Its three apexes come out at radius 100.000000, which is the generating radius R exactly, at 0° and ±120°; the middle of each flank comes closest to the centre at 72.0000, which is R − 2e. The apexes are the only part of the rotor that touches the housing, which is why a rotary engine's sealing problem is three lines rather than a ring. positioned by solving, not by drawing.

A rotor nobody drew

The rotor of a rotary engine has three corners, three flanks and one job: to stay in contact with a housing while turning at a third of the shaft's speed about a centre that orbits. Given the housing and that motion, the rotor is not designed. It is computed, corners and all, by the routine that cuts a gear tooth.

meshing · Meshing 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 Four points of one connecting rod: two crossings, none, two isolated points, two crossings again. The closed curves drawn over both assemblies by four points of the connecting rod of a slider-crank with a crank of 1 and a rod of 3 on a slide through the crank's pivot, which is marked in each panel. Each curve is a quartic and owes two double points. (0.12, 0.15), 0.19 rods from the crank pin: two crossings, at (±0.65, ±0.36), ringed; (0.4, 0.5), 0.64 rods from the crank pin: a complex pair, so the curve crosses itself nowhere; (0.75, 0.94), 1.20 rods from the crank pin: two isolated points at (±4.06, ±3.00), dotted, off the drawn curve; (1.1, 1.0), 1.49 rods from the crank pin: two crossings, at (±2.66, ±2.95), ringed. Each panel has its own scale. The second curve has no visible double point at all, and the third has two that no configuration of the machine ever reaches.

The double points that leave for infinity

A slider-crank's connecting rod draws a quartic that owes two double points. Close to the crank pin they are two real crossings. Between a crank's length and a rod's length from it they are a complex pair and the curve crosses itself nowhere. At exactly one rod both run off to the same real point at infinity and meet there as a tacnode whose branches are complex. Past that they come back real. Offsetting the slide changes the two circles by (R² − a²)(R² − b²) + e²R² = 0 and sends only one of the pair to infinity.

algebra · Algebra A growing four-bar walks away from its own double points. A four-bar with a rod of 3, its crank and rocker both of length t, pivoted so that both pivot circles pass through the slides' crossing, with the coupler point at (0.3, 0.2) in the rod's units. Drawn in units of t: the two pivot circles (dotted), their second meeting (ringed), the real coupler curve the machine draws (solid), and its real double points, filled where two real configurations meet and hollow where none reaches. At size t = 7 the curve's real double points are isolated (2.02), isolated (3.32), crossing (0.93); the number in brackets is how far the line of rod orientations through each passes from the unit circle's centre, a crossing below one and isolated above. As t grows the curve shrinks onto the crossing and the second meeting and the double points stay where they are.

A growing four-bar reaches none of its departing double points

A four-bar growing into the elliptic trammel loses three double points, and wherever they are real the question is whether the machine passes through them. In the limit it passes through none: a configuration that reaches a point a crank's length away needs its rod along an isotropic direction, which no real rod has. How fast each point leaves the machine's reach is a ratio of four distances, √(1 + λ²)·|Z|·|Z − Zₛ| / (2b·|Z + 1|·|Z + iλ|), and it vanishes only at the pivot circles' second meeting. Every crossing is lost at a finite size: through a cusp, when the law predicts it, or by meeting another crossing and leaving as a complex pair, when it does not.

algebra · Algebra

Named alongside it

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

CentrodeCoupler curveDouble pointHypocycloidInstantaneous centreRouletteApexConfiguration spaceConjugate-actionEccentricityElliptic trammelEnvelope

All concepts