Machine Kinematics
solved, then drawn
Concept

Describing circle — where it appears

The circle whose rolling traces the two halves of a cycloidal tooth pair, and the curve the contact path of such a pair turns out to be. It exists only while the two pitch circles touch, which is why the pair is so sensitive to the centre distance.

Named by 6 essays across 3 fields — each of them below, with the objects they name alongside it.

A flank nobody would draw, and the partner it forces. On the left, a driving flank made up on purpose: a rising curve with a nine-cycle wobble on it, chosen to be nothing in particular. On the right, in the driven wheel's frame, the shape that has to mate with it. There was a contact at all 121 sampled positions and the worst residual of the meshing equation was 3.13e-10. Conjugate action does not single out the involute — every profile has a partner that holds the ratio exactly, and the reasons for preferring one tooth form over another are all somewhere else. Generating back the other way returns the original flank to 2.33e-5 mm over 61 points. positioned by solving, not by drawing.

Any shape has a partner

Conjugate action does not pick out the involute. Hand the construction a flank invented on purpose to be nothing in particular and it returns a mate that holds the ratio exactly — so the question a tooth form answers is not whether it can transmit motion, and every real reason for choosing one is somewhere else.

meshing · Meshing The measurement every gear cut since 1900 is a consequence of. Take a pair that is conjugate at its design centre distance, move the shafts apart, and ask what shape the driven wheel would have to be for the ratio to hold. The involute wants the same shape at every distance — 2.91e-8 mm at two millimetres out, which is the comparison's own noise floor — because its shape is fixed by its base circle and the centre distance is not one of that circle's arguments. The cycloidal pair is conjugate at nought and wants a shape 6.71e-3 mm different at five hundredths of a millimetre out, because its describing circle has to roll between two pitch circles that are no longer touching. A bearing that wears, a housing bored a little wide, a case that warms up: all of them are this axis.

The shape that does not mind where the shafts are

Two tooth forms, both exactly conjugate, both in use for centuries. Move the shafts five hundredths of a millimetre apart and one of them wants a different shape and the other does not — and that single measurement is close to the whole reason every gear cut since about 1900 is an involute.

meshing · Meshing A cam is the envelope of a roller. The follower's roller drawn in the cam's frame at 15 angles of the cam: the roller slides in a straight guide and the cam turns under it, so from the cam's point of view the roller travels round it on a path set by the lift law. The cam's surface is the envelope of those circles — the same computation as a rack generating a tooth, with a circle instead of a straight edge and a lift law instead of a rolling condition. The cams field builds the same surface by offsetting the roller-centre path inward by the roller radius, and the two agree to 4.1e-4 mm. positioned by solving, not by drawing.

A cam is a conjugate pair

A cam is built by offsetting the path of the follower's centre inward by the roller radius. It can also be built by asking what shape stays in contact with a circle that slides in a stated way — the same computation that cuts a gear tooth — and the two surfaces agree to sixteen millionths of a millimetre.

cams · Cam 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 The rotor is not a Reuleaux triangle. The generated rotor, with the Reuleaux triangle through the same three apexes drawn over it — three circular arcs, each centred on the opposite corner. They part company by 1.21 mm on a rotor of generating radius 100, and the flank is the one that is right: it is the envelope of the housing, and a Reuleaux flank would foul the wall. The measurement that cannot be argued with is the width. A Reuleaux triangle has the same width in every direction — that is what it is for — and this rotor's runs from 150.0 to 174.1, a spread of 16 per cent. positioned by solving, not by drawing.

Six things a shape is not

A rotor that is not a Reuleaux triangle, teeth that are not the same shape as each other, a fillet that is not an arc, a mesh that does not roll, a conjugate pair that cannot be built and a tooth form that was not deduced. Six claims in circulation, each with the number that kills it.

wrong · Misconception 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

Named alongside it

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

Conjugate-actionEnvelopeMeshing equationContact normalInvolutethe law of gearingPitch pointEpicycloidHypocycloidPitch curveTransmission errorUndercutting

All concepts