Machine Kinematics
solved, then drawn
Concept

Rotor — where it appears

A body turning inside a housing it stays in contact with, whose shape is the envelope of that housing rather than a curve anybody drew. Its corners are where two branches of that envelope meet, and they are the only parts that touch the wall.

Named by 4 essays across one field — 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 Three chambers, 15,257 mm² between them. Each chamber is bounded by one flank of the rotor and the arc of housing between the two apexes that seal it, so its area is a polygon integral over two curves that are both known exactly. At this shaft angle they are 6309, 1147, 7801 mm². They add to 15257, and the housing's area less the rotor's is 15258 — a number that cannot depend on where the rotor is, and does not, to 9.2e-6 of itself. The engine's displacement is that constant shared out differently, and the three chambers are one chamber counted at three phases of the same cycle. positioned by solving, not by drawing.

Three chambers and a constant

A rotor with three corners divides its housing into three chambers, and as the shaft turns they trade area: one grows exactly as fast as the other two shrink. The total does not move — and the fact that it does not is a measurement of whether the corners are actually touching the wall.

meshing · Meshing One shaft angle, five places along a twisted rotor. A mismatched pair — a cycloidal rotor against a circular-tipped one of the same height, which are not each other's conjugates — with the rotors twisted by 1 lobe pitch along their length. The section at a given place along the shaft is the flat pair at an input angle shifted by the twist so far, so at one instant the five sections shown are at five different phases of the same mesh: the two bodies are into each other by 2.42 at the tightest section and 2.34 apart at the widest, with a mean of -0.408 across the whole seal. All five are drawn at one scale, and nothing here is meshed twice: one planar profile is read along a window.

A twist steadies what it cannot tighten

A helical rotor's sections are at different phases of the same mesh, so the clearance a machine has at one instant is a window along its own profile rather than a point on it. A wrap of exactly one lobe pitch holds the seal's open area constant through the turn — every harmonic at once, whatever the profile — and leaves its average, its tightest place and its widest place exactly where they were.

meshing · Meshing Nine contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 9 pins are in contact with the disc at this instant, and each one's line is the common normal, which passes through the pitch point on the pin circle. A pin can only push, so the sense in which it turns the disc is decided by which side of the disc's own centre its normal passes: 5 of the contacts turn it one way and 4 the other, with the largest arm in each sense 49.9 and 45.0 on a pitch offset of 55.0. A pair with one contact has no such choice, which is the whole of why two identical rotors cannot drive each other.

What a second contact is for

Two identical rotors that are exactly each other's conjugates cannot drive each other, and the reason has nothing to do with conjugacy. A ring of pins and the disc they generate is just as exactly conjugate, has nine or ten contacts at once instead of one, and never loses more than a fifth of the arm its geometry allows.

meshing · Meshing

Named alongside it

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

Conjugate-actionApexContact ratioEccentricityEnvelopeEpitrochoidLobeMeshing equationPitch pointClearanceCompression ratioContact point

All concepts