Field

The shape is the unknown

Everywhere else here a body has a shape and the question is where it goes. Put two bodies on fixed centres, require them to stay in contact, and the second shape stops being a choice: it is the envelope of the first one's positions, and there is exactly one of it. One routine cuts a gear tooth out of a straight edge, a cam out of a roller and a rotary engine's rotor out of its housing.

One contact, and the point the normal has to pass through. Two wheels on fixed centres, turning in the ratio 24 : 36, with one flank of each drawn. The contact is found by solving n·(v₁ − v₂) = 0 along the first flank — the two velocities are formed from the two rotations and subtracted, and nothing in that calculation knows where the pitch point is. The pitch point, marked with a cross, is computed separately as the one place where the two bodies' material points have the same velocity. The common normal misses it by 8.44e-15 of a millimetre, which is the law of gearing arriving as a measurement rather than as an assumption. positioned by solving, not by drawing.

The second shape is not a choice

The meshing equation, solved from two velocities rather than from the pitch point, and the law of gearing arriving as a measurement — with the second profile of a pair computed as an envelope rather than drawn.

Any shape has a partner

An arbitrary profile generated against its conjugate, the round trip back, and the measurement that a conjugate belongs to a pair rather than to a shape.

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

The sliding velocity at a gear contact measured by subtracting two material velocities, its single zero at the pitch point, and the specific sliding that is worse on the smaller wheel.

A straight edge cutting a 24-tooth wheel. The rack's flanks are straight lines and its pitch line rolls on the wheel's pitch circle without slipping — 48 mm of travel per radian, which is the only number in the whole process. The dots are the contacts the meshing equation has returned so far, and they are the flank being cut: an involute, produced by a tool that has no curve anywhere on it. The corner of the rack traces the fillet below, which is a different curve for a different reason — a corner is a point, and a point has no envelope but its own path. positioned by solving, not by drawing.

The tool is the definition

A straight-sided rack generating a gear tooth as an envelope, with the flank measured against the involute, the line of action fitted to the contacts, and the base circle recovered from them.

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

Involute and cycloidal tooth pairs generated at a series of centre distances, with the shape each would then require measured against the shape it has.

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

Pitch curves computed from a demanded output rate, the elliptical pair that rolls exactly on its own foci, and the ratio range that follows from the eccentricity alone.

The demand that cannot be met

The closure condition on a demanded velocity ratio, measured across a family of demands, with the positivity and convexity limits that rule out the rest.

A rotor that was not drawn, at 52° of shaft. The housing is an epitrochoid — the only shape here that was written down — and the rotor is the envelope of it, seen from a body that turns at a third of the shaft's rate about a centre orbiting at the eccentricity. Nothing about the rotor was chosen. Its three apexes come out at radius 100.000000, which is the generating radius R exactly, at 0° and ±120°; the middle of each flank comes closest to the centre at 72.0000, which is R − 2e. The apexes are the only part of the rotor that touches the housing, which is why a rotary engine's sealing problem is three lines rather than a ring. positioned by solving, not by drawing.

A rotor nobody drew

A Wankel rotor generated as the envelope of its housing, with the apex radius and the flank's closest approach measured against the generating radius and eccentricity.

Three chambers, 15,257 mm² between them. Each chamber is bounded by one flank of the rotor and the arc of housing between the two apexes that seal it, so its area is a polygon integral over two curves that are both known exactly. At this shaft angle they are 6309, 1147, 7801 mm². They add to 15257, and the housing's area less the rotor's is 15258 — a number that cannot depend on where the rotor is, and does not, to 9.2e-6 of itself. The engine's displacement is that constant shared out differently, and the three chambers are one chamber counted at three phases of the same cycle. positioned by solving, not by drawing.

Three chambers and a constant

The three chamber areas of a rotary engine computed as polygon integrals over the generated rotor and its housing, with the invariant total and the compression ratio the pure geometry gives.

Eleven lobes from twelve pins

A cycloidal reducer's disc generated as the envelope of a single pin, with the lobe count, the root and tip radii and the reduction measured against the pin count.

What is left of a 12-tooth flank. The flank the rack generated on a 12-tooth wheel, with every point tested against the cutter at every other position of the cut. The pale dots survive; the dark ones are inside the cutter at some later instant and are not on the finished tooth — 39 of 203 of them, the deepest by 5.19e-2 mm. Undercutting is not a shape, it is a removal: the same corner that leaves the fillet comes back for flank the straight edge had already generated, and what is lost is the part of the involute nearest the base circle, which is exactly the part the mating wheel's tip needs. Drag to change the tooth count. positioned by solving, not by drawing.

The cutter takes back the tooth

Undercutting measured as material removal by sweeping the cutter over the generated flank, with the tooth-count threshold bisected from the geometry and matched against the classical formula to ten places.

The ratio survives; the continuity does not. An involute pair holds its ratio at any centre distance, and that is not the same as working at any centre distance. The contact ratio — the length of the contact path divided by the base pitch, which counts how many pairs of teeth are engaged at once — starts at 1.647 for this 24 : 36 pair and falls as the shafts move apart, because the useful part of the line of action is bounded by the two tip circles. It reaches one at 2.82 mm, and below one a pair of teeth lets go before the next has picked up: the drive stops being continuous and becomes a series of arrivals. That is the real limit on the involute's indifference, and it is a limit on the teeth rather than on the tooth form.

Where two shapes stop touching

The limits of a conjugate pair — the arc of contact, the contact ratio, the interference that ends it and the centre distance at which continuity fails, each measured on generated shapes.

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

The rotors of a Roots blower computed rather than drawn: the root a lobe's tip demands from an identical mate, the height at which a circular tip's root folds, the moment arm of the single contact about the driven shaft measured through a lobe pitch and predicted by the describing circle, and why no choice of tip lets the pair drive itself.

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

The gap between two Roots rotors measured over a lobe pitch three ways: the conjugate pair eroded by a common offset, giving a flat profile at twice it to a part in a thousand; the same clearance attempted by opening the centre distance, giving a ratio of twenty-four to one at its worst; and a mismatched pair that interferes and gaps in the same turn.

A twist steadies what it cannot tighten

The seal of a twisted Roots pair computed from one planar profile read along a window: the open area as a moving average whose swing falls from 4.757 to 2.4 × 10⁻⁴ at a wrap of one lobe pitch, the window gain sin(πmτ)/(πmτ) that predicts every fractional wrap including the hump between zeros, and the extremes that do not move at all.

Nine contacts at once, and the sense each turns the disc. The 12-pin drive at 40° of its eccentric. 9 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: 5 of the contacts turn it one way and 4 the other, with the largest arm in each sense 49.9 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

Every contact in a cycloidal drive measured as an arm about the disc's own centre, computed at the contact and again at the pitch point where no contact appears; the largest arm available in each sense through a turn, against the Roots pair's single arm falling to nought twice a lobe pitch; and six pin counts that all keep one.

A disc cut undersize, turned until a pin touches. A twelve-pin cycloidal drive whose disc was cut as if its pins were 1 larger than they are, drawn at 10° of the eccentric with the disc turned a further 1.082° about its own centre — the turn at which the first pin touches. One pin touches, pin 0, marked in its own colour. The pins within half the clearance of touching are drawn in a second colour and the rest plainly: 5 of the twelve are within half the clearance and 7 further off. The clearance is exaggerated so that it can be seen; at every clearance measured, one pin touches first.

A clearance leaves one contact

A twelve-pin cycloidal drive with its disc cut undersize: the disc as the material its pins leave, measured to 10⁻⁶ against the definition; the one pin that touches first when the disc is turned, each way, at every angle; the arm of that pin against the exact disc's best, and the root sockets that make it better; the free turn per unit of clearance through a pin pitch; and the pins within reach of carrying as a function of how far the contacts may give, for two clearances. Includes the correction of the exact drive's contact count and arm floor.

The same drive with a smaller eccentric: every pin touches. Twelve pins on a ring of 60, pin radius 5, and the disc exactly as those pins generate it, at 10° of the eccentric. Left, the eccentric at exactly R/N = 5, whose roots are round sockets: 9 pins touch (solid) and 3 stand clear (dashed), the pins near the roots. Right, the eccentric at 0.8 of that, 4.00, whose roots are smooth: 12 of twelve touch. Over a pin pitch the socketed disc touches 9 or 10, and every smaller eccentric measured touches all twelve.

A disc whose roots are not sockets

A twelve-pin cycloidal drive with its eccentric at k times the pin circle's radius over the pin count, for k from a half to one: the exact disc at k = 1 against a smaller eccentric, with the pins that touch; one root close up at four eccentrics, where the socket becomes a trimmed point and then a smooth offset below k = 0.974; the table of pitch offset, contacts, exact arm floor, first contact under a clearance, free turn and pins within reach; the first contact's share of the pitch offset against k; and the free turn per unit of clearance against 0.0365/k.

The flank bends towards the disc, and the pin's radius is measured against that bend. One root of a twelve-pin cycloidal drive, pin radius 5 on a ring of 60, with the eccentric at 0.97 of R/N. The thin line is the curve the pin's centre traces in the disc's frame, its tip pointing at the disc's centre; the dashed line is that curve offset towards the disc by the pin's radius, which is the disc's edge wherever nothing trims it. Either side of the tip the flank bends the other way, towards the disc, most tightly 22.6° of the eccentric from the root, with a radius of 5.36 (the circle, centre marked). The pin's radius, 5, is measured along the same normal to the offset point. The offset point stops short of the centre of curvature, so the offset does not fold and the root is untrimmed. The two meet at 0.9740 of R/N.

The flank that trims the root

The eccentric at which a cycloidal disc's roots begin to be trimmed, for any pin count and pin radius: one root with the flank's tightest bend and the pin's radius measured along the same normal; the flank's least radius against the eccentric, closed form and second differences; the design curve of the largest untrimmed eccentric against pin radius for five pin counts, against the offset test; twenty fat pins at the classical eccentric and just below their threshold; and four drives' contacts and levers at both.

Near its threshold, the disc's flank sharpens to a point under the pin. One root of a twelve-pin cycloidal disc, pins of radius 5 on a ring of 60, cut with the eccentric at 0.97 of the threshold 0.9740 R/N. The line is the disc's edge; the circle is a pin touching it at the flank's sharpest point, 31.0° of the eccentric from the root. There the edge bulges towards the pin with a radius of 2.23, the flank's bend 7.23 less the pin's 5 (small circle, centre marked), so the contact is convex against convex, and a line contact's pressure per unit load there is 1.80 times a pin on a flat. At the root itself the edge is a hollow round the pin.

A disc cut at its threshold grows a point

The contact between a cycloidal drive's pins and its disc as the eccentric approaches the trimming threshold: one root with a pin at the flank's sharpest point and the disc's own radius there; the pressure factor along a lobe at three eccentrics; the flank's pressure against the distance from the threshold for four drives; the drive's peak contact pressure per unit torque against the eccentric, with each drive's least; and a ledger of thresholds, least-stress eccentrics and what the last few per cent of eccentric cost.

A relief as deep as the contact's own approach, spread across the flank. One root of the twelve-pin disc cut at 0.97 of its threshold, pins of radius 5 on a ring of 60, with the edge relieved inwards by a cosine bump of depth 0.01 and half-width 1 rad of eccentric centred on each flank. The relief is drawn 200 times deeper than it is: at true scale it is 0.2% of the pin's radius, the size of the contact's own approach at the design torque, and invisible. The thin line is the unrelieved edge; the pin touches it at the flank's sharpest point, where the unrelieved edge's radius is 2.23.

A relief is sized for one torque

A twelve-pin cycloidal disc cut at 0.97 of its trimming threshold with its edge relieved near the flanks: one root with the relieved edge, depth magnified; the pressure factor along a lobe with no relief, a narrow one and the best; the drive's peak contact pressure per unit torque against the relief's depth for five widths; the peak per root torque against the torque, beside the unrelieved and least-stress discs; and a ledger of peaks, levers, loaded pins and lost motion.

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