Machine Kinematics
solved, then drawn
Concept

Pitch point — where it appears

The one point at which two bodies' material velocities are equal, which is the instantaneous centre of their relative motion. Every conjugate contact's common normal passes through it, and it is the only place where two meshing surfaces roll rather than slide.

Named by 13 essays across one field — each of them below, with the objects they name alongside it.

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.

meshing · Meshing A flank nobody would draw, and the partner it forces. On the left, a driving flank made up on purpose: a rising curve with a nine-cycle wobble on it, chosen to be nothing in particular. On the right, in the driven wheel's frame, the shape that has to mate with it. There was a contact at all 121 sampled positions and the worst residual of the meshing equation was 3.13e-10. Conjugate action does not single out the involute — every profile has a partner that holds the ratio exactly, and the reasons for preferring one tooth form over another are all somewhere else. Generating back the other way returns the original flank to 2.33e-5 mm over 61 points. positioned by solving, not by drawing.

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.

meshing · 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.

meshing · 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.

meshing · 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.

meshing · Meshing One member of the family is a pair of wheels. The same demand — one plus 0.4 sin φ — scaled by a constant, and for each the angle the driven wheel is out by after one turn of the input. It has to be zero, or the teeth do not line up with themselves and there is no wheel. Only the unscaled member closes; five per cent either way leaves the output 18° out, which is a third of a tooth on a thirty-tooth wheel and a mechanism that seizes on its second turn. Non-circular gearing is a search rather than a drawing for exactly this reason: the closure condition is one equation on a whole function, and almost no function satisfies it.

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.

meshing · 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.

meshing · 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.

meshing · 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.

meshing · 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.

meshing · 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.

meshing · 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.

meshing · 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.

meshing · Meshing

Named alongside it

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

Conjugate-actionEnvelopeContact normalMeshing equationCycloidal driveVelocity ratioInvoluteBacklashClearanceDescribing circleInstantaneous centreUndercut

All concepts