Machine Kinematics
solved, then drawn
Concept

Contact point — where it appears

The place two bodies touch at an instant, which travels along both surfaces as they move. It is not a material point of either body, which is why a contact slides even when the pitch circles roll.

Named by 6 essays across 4 fields — each of them below, with the objects they name alongside it.

A flat follower needs a face, and how wide is not a matter of taste. The cam profile the cycloidal programme produces on a base circle of 30, at 70° of rotation, with the follower's face across the top and the contact marked. The contact is not on the follower's axis: it sits at ds/dθ from it, so it wanders sideways as the cam turns and the face has to reach from the most negative value of that to the most positive. Here that is 19.099 — measured off the drawn profile as 19.099, the two routes agreeing to 1.0e-16. A face cut to the lift, or to the base circle, or to whatever looked right, is a face the cam runs off.

A follower needs a face

A flat-faced follower does not touch the cam on its own axis. The contact wanders sideways as the cam turns, by exactly ds/dθ, and a face cut to the lift or to the base circle or to whatever looked right is a face the cam runs off.

cams · Cam 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 Where the pad touches, and the rule that fixes it. The contact point's travel across the valve tip through one cam event, for a rocker squared up with the valve shut and for the same rocker squared up at mid-lift. The first wipes 1.35 mm and the second 0.34 mm — a factor of 4.0 for a shim under a stud. The rule an engine builder is taught is that the geometry is symmetric about wherever it is set square, so setting it square at rest puts the entire lift on one side of the symmetry. It is folklore that turns out to be geometry, and the same shim brings the peak lift back to 12.84 mm against 12.78.

Where the pad touches

A rocker's pad slides 1.35 mm across the valve tip through one cam event when the geometry is squared up with the valve shut, and 0.34 mm when it is squared up at mid-lift. The engine builder's rule about shimming a rocker stud is folklore that turns out to be geometry, and the same shim brings the peak lift back to what the ratio promised.

applied · Cam Two rotors cut undersize, and the gap that is their seal. The conjugate pair at three positions of its turn, each rotor cut 3 undersize — its boundary moved that far inward along its own normal, with the full-size outline dashed behind it. At full size the two are in contact at every angle, so cutting both back by 3 leaves exactly 6.0 between them wherever they were touching. Measured over a whole lobe pitch the gap runs from 5.9965 to 6.0000 against a prediction of 6, a worst departure of 3.46e-3.

The clearance that is the seal

Two rotors that are each other's conjugates touch at every angle of their turn, so cutting both back by the same amount leaves exactly twice it between them, everywhere. Open the shafts by the same amount instead and the gap runs from four per cent of it to all of it. And a pair that is not conjugate has no seal to cut: over one lobe pitch it swings from two and a half units inside itself to two and a third apart.

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 The taut inner, and the walls it is actually held by. A sheath of two 8° bends on a radius of 40, its bore drawn at a clearance of 4, with the shortest path from ferrule to ferrule inside it. Nothing about a bend goes into that path: it is the shortest route the tube allows, found by tightening a funnel against every cross-section in turn, and the places it reaches the wall are where it is held. At this clearance the two bends hold 53% and 53% of their own turn, and the inner is short of the centreline by 0.854 against the closed form 1.117 — 76% of it. Widen the bore and the path lifts off.

Which walls a strand is held by

A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.

strands · Strand

Named alongside it

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

Conjugate-actionEnvelopeBacklashClearanceCycloidal driveDerivativeDesign ruleFollowerLiftLobeToleranceCam

All concepts