Machine Kinematics
solved, then drawn
Concept

Lobe — where it appears

One repeating unit of a rotor or disc profile: a bulge outward from its pitch curve followed by a hollow inward. The number of lobes sets a ratio or a count of chambers, and the shape of one lobe is usually computed from the part it meshes with rather than drawn.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

Eleven lobes from twelve pins, at 46° of eccentric. The pins are circles and the disc is their envelope. One turn of the eccentric moves the disc back by one lobe pitch — a reduction of 11 : 1 from a ring, a disc and an offset bearing, with no gear teeth anywhere — and the profile's 11 lobes are counted off the generated curve rather than put there. The roots sit at 50.00 mm and the tips at 60.00, which are R − r ∓ e: the eccentricity is the lobe height, twice over. positioned by solving, not by drawing.

Eleven lobes from twelve pins

A ring of round pins, a disc on an eccentric, and a reduction of eleven to one with no gear teeth anywhere. The disc's profile is not designed: one pin generates one lobe of it, the other ten lobes are the same curve, and the count that decides the ratio is a count of lobes on a shape nobody drew.

meshing · Meshing Two identical rotors in mesh, and the one place they touch. Two rotors, each with 2 cycloidal lobes on a pitch circle of radius 50, on centres 100 apart and turning at the same speed in opposite senses, drawn with the first turned −20°. Each rotor's roots were computed from its tips, and the mate is the same rotor turned. At this position they touch at one point, on the first rotor's tip, and the common normal there misses the pitch point by 2.2 × 10⁻¹⁰. The contact sits on the describing circle tangent to both pitch circles, within 8.5 × 10⁻¹⁰, so the normal is the chord from the pitch point to it. The normal's moment arm about the mate's shaft is 32.14, positive when the contact turns the mate forward.

Rotors that mesh and cannot drive each other

Two identical lobed rotors on shafts turning one to one are each other's conjugate: give half of a lobe and the meshing equation computes the other half so exactly that the rotor is its own mate. The pair holds its ratio at every instant and still cannot drive itself, because the one contact between them pushes the driven rotor backwards for exactly half of every turn.

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 Eight contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 8 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: 4 of the contacts turn it one way and 4 the other, with the largest arm in each sense 42.2 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 eight to eleven contacts at once instead of one, and never loses more than thirty per cent of the arm its geometry allows.

meshing · Meshing

Named alongside it

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

Conjugate-actionContact pointContact ratioCycloidal driveDead centreEnvelopeEpicycloidPitch pointRotorBacklashClearanceContact normal

All concepts