Machine Kinematics
solved, then drawn
Concept

Hypocycloid — where it appears

The curve a point on a circle traces as that circle rolls inside another, degenerating to a straight diameter when the radii are in the ratio two to one. It is one half of the classical cycloidal tooth pair, and its degenerate straight-line case is the geometry behind more than one exact straight-line mechanism.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

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 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 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.

CentrodeConjugate-actionCuspDescribing circleEpicycloidInstantaneous centreRouletteBacklashCentre distanceContact normalCoupler curvecycloidal motion

All concepts