Machine Kinematics
solved, then drawn
Concept

Specific sliding — where it appears

The sliding speed at a contact divided by the speed of the surface being asked about. It is different for the two flanks of one contact, and the smaller wheel of a pair always gets the worse of it.

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

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 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 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

Named alongside it

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

CentrodeConjugate-actionInvoluteMeshing equationVelocity ratioContact normalDescribing circleEnvelopeEpitrochoidInstant centreInstantaneous centrethe law of gearing

All concepts