Meshing — the series · Machine Kinematics

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

Two bodies on fixed centres, told to stay in contact. Give one of them a shape and the other one's shape is no longer available to be designed — it is the envelope of the first one's positions, there is exactly one of it, and one routine computes it for a gear, a cam and a rotary engine alike.

part 1 · meshing

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.

part 2 · meshing

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.

part 2 · meshing

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

There is no curve anywhere on the cutter that makes an involute gear. It is a straight edge, dragged past a turning blank, and the involute is what the motion leaves behind — along with a fillet that is a corner's path, a contact locus that comes out a straight line, and a base circle that is measured rather than drawn.

part 3 · 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.

part 3 · meshing

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.

part 4 · meshing

The demand that cannot be met

Ask for an output rate and the two pitch curves follow with no design step in between — so the interesting question is not how to draw them but which demands admit any pair of wheels at all. The answer is one equation on a whole function, it is about the demand's mean and nothing else, and five per cent of error leaves the output eighteen degrees out after a turn.

part 4 · meshing

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

The rotor of a rotary engine has three corners, three flanks and one job: to stay in contact with a housing while turning at a third of the shaft's speed about a centre that orbits. Given the housing and that motion, the rotor is not designed. It is computed, corners and all, by the routine that cuts a gear tooth.

part 5 · meshing

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

A rotor with three corners divides its housing into three chambers, and as the shaft turns they trade area: one grows exactly as fast as the other two shrink. The total does not move — and the fact that it does not is a measurement of whether the corners are actually touching the wall.

part 5 · meshing

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.

part 6 · meshing

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 is usually explained as a shape: a tooth with a waist in it. It is better understood as an event — the corner that leaves the fillet comes back through flank the straight edge has already generated — and seen that way the threshold at seventeen teeth is a comparison of two measured points that passes through zero.

part 6 · meshing

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

A conjugate pair is exact at every instant it has a contact. It does not have one for ever: a profile is an arc rather than a curve, and both ends of that arc are somebody's decision — which is why the useful question about a pair of shapes is not whether they mesh but for how long.

part 7 · meshing

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.

part 8 · meshing

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.

part 9 · meshing

A twist steadies what it cannot tighten

A helical rotor's sections are at different phases of the same mesh, so the clearance a machine has at one instant is a window along its own profile rather than a point on it. A wrap of exactly one lobe pitch holds the seal's open area constant through the turn — every harmonic at once, whatever the profile — and leaves its average, its tightest place and its widest place exactly where they were.

part 10 · meshing

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

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 nine or ten contacts at once instead of one, and never loses more than a fifth of the arm its geometry allows.

part 11 · meshing

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 cycloidal drive's disc, exactly as its pins generate it, touches nine or ten of its twelve pins at once, and that is why it can drive where two identical rotors cannot. Cut the disc two hundredths undersize, as a real drive is cut, and a rigid disc turned until it touches meets one pin, at every angle of the eccentric and in both directions. The arm that pin gives is never worse than the exact disc's best. What a clearance takes away is load sharing, and deflection gives it back: contacts that may give by the whole clearance put six or seven pins within reach.

part 12 · meshing

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

Put a cycloidal drive's eccentric at exactly the pin circle's radius over the pin count and every root of its disc is a round socket; take any smaller eccentric and the roots are smooth. The sockets looked like a benefit — a pin in one closes its gap both ways at up to the full pitch offset. Measured from a half of that eccentric to all of it, they are the opposite. The socketed disc touches nine or ten of its twelve pins, every smaller one touches all twelve, and its first contact keeps the smallest share of the pitch offset of the seven. What a smaller eccentric really costs is the pitch offset itself: the free turn per unit of clearance is 0.0365 divided by the eccentric's share.

part 13 · meshing

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

A cycloidal disc's roots are trimmed, and its pins start standing clear, above an eccentric that was known only for one drive: 0.974 of the pin circle's radius over the pin count, for twelve pins of radius 5 on a ring of 60. The trimming is a bend. Either side of each root the flank of the pin's path curves towards the disc, and where that bend is tighter than the pin the offset folds. The bend has a closed form, so the threshold does too, √(1 − ρ²(N + 1)³/27(N − 1)) for pins of radius ρ, and it falls well below one for fat pins or many of them. On every drive measured, the classical eccentric loses contacts and lever together.

part 14 · meshing

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

A cycloidal disc's roots stay untrimmed up to an eccentric where the pin path's flank bends with the pin's own radius, and it looked as though a disc cut exactly there would have its most conformal contact, the pin and the flank sharing a curvature. It has its least. The disc's edge is the path offset inwards by the pin's radius, so where the path bends round a centre on the disc's side the edge bulges towards the pin with the path's radius less the pin's: at the threshold, a point. A line contact's pressure per unit load there rises as the inverse square root of the distance to the threshold, and the pins passing it are carrying load. Each of four drives has an eccentric of least contact stress at 0.67 to 0.80 of its threshold; cut at 0.97, the peak pressure is 9% to 80% higher for a lever 21% to 46% longer.

part 15 · meshing

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 cycloidal disc cut near its trimming threshold has long levers and a point on each flank, where the pins carrying most of the load press hardest. Relieving the edge a little near the flanks can take most of that point away. On a twelve-pin drive at 0.97 of its threshold, a relief 0.01 deep spread across a radian either side of each flank lowers the peak contact pressure by a quarter and keeps every lever, to within 7% of the disc cut at its least-stress eccentric, whose levers are a fifth shorter. The relief does it by unloading the flank, not by reshaping it, so its depth must be the contact's own approach at the design torque. At a quarter of that torque it is worse than no relief at all, and a relief of hundredths of the pin's radius, which is what was expected to work, takes pins out of the load.

part 16 · meshing