Machine Kinematics
solved, then drawn
Concept

Centre distance — where it appears

The separation of two meshing gears' axes, half the sum of their pitch diameters when both are cut to standard proportions. For an involute pair it can be wrong without changing the ratio at all, which is the property that made the involute the tooth form of the whole industry.

Named by 14 essays across 6 fields — each of them below, with the objects they name alongside it.

The involute, unwound. Hold a string taut against a circle and unwind it: the end traces this curve. Two properties follow immediately and between them they are the whole of gear geometry. The string is always tangent to the base circle, and it is always perpendicular to the curve. So the normal to an involute at any point is a tangent to its base circle — measured on the generated points here, to 2.3e-5 — and when two involutes touch, the common normal is a line tangent to both base circles, which does not move as the gears turn.

Why a tooth is an involute

A gear tooth is not a shape anybody chose for its looks. It is what one requirement forces — that the ratio of the two shafts' speeds stays exactly constant while the contact point slides along the flank. Impose that and the curve is essentially determined.

gears · Tooth A 6-slot Geneva wheel. The driving pin enters a radial slot, carries the wheel through 60°, and leaves. The centre distance is not free: it must be crank ÷ sin(180°/6) = 60.00 so that the pin enters along the slot, with the crank and slot perpendicular. That is the mechanism's one design requirement and it is what makes the driven wheel start and stop from rest — measured here at 1.5e-3 against a peak of 1.000. What it does not fix is the acceleration, which peaks at 1.35 and is why film sprocket holes tear.

Stopping thirty times a second

A Geneva wheel turns continuous rotation into steps, and its one design requirement is that the pin enters the slot along the slot so the driven wheel starts and stops from rest. That fixes every dimension from the slot count. What it does not fix is the acceleration, which is why film sprocket holes tear.

cams · Cam 11 teeth, cut with three different shifts. A 11-tooth gear cannot be cut with a standard rack without the cutter eating into the flank near the root — the tooth is undercut, and what it loses is exactly the part that does the work. The fix is to hold the cutter further out by a fraction x of the module. Here the threshold is x = 1 − z sin²α / 2 = 0.3566, measured rather than quoted: at 0.347 the gear still undercuts and at 0.3566 it does not. What the shift costs is at the other end of the tooth. The tip thickness falls from 0.606 to 0.240 of a module, and a tooth shifted far enough comes to a point and breaks — so the technique has a ceiling as well as a floor.

Moving the cutter out

A gear with too few teeth is undercut by the tool that generates it, and the fix is to hold the tool further out. What that does to the tooth is easy to say. What it does to the pair is not what most readers expect — the two gears no longer mesh at the centre distance the sum of their radii would give, and the pressure angle they run at is no longer the one they were cut with.

gears · Tooth 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 Backlash is what the centre distance buys. A 20-and-40-tooth pair, module 1, run at centre distances either side of the one at which the two teeth exactly fill the circular pitch. Backlash is measured from the drawn tooth thicknesses — no involute equation appears in the calculation — and plotted against the textbook linearisation j = 2 Δa tan α_w. Left of zero the teeth interfere and the pair cannot be assembled at all. So backlash is not slop and it is not wear: it is a quantity a designer buys with a centre distance, and buying none of it means specifying a centre distance that has to be exact at every temperature.

Backlash is an allowance

A gear pair with no backlash cannot be run, and a pair with the wrong amount cannot be assembled. It is bought with a centre distance — 0.03 too far apart on a 30 mm centre buys 0.022 of it, which is a quarter of a degree at the pinion — and the textbook formula that says so is right about the slope and drifts 4.5% at a centre-distance error nobody would accept anyway.

gears · Tooth Six gearsets, three conditions. Every one of these can be drawn, and five of the six are drawn in some textbook or other. The columns are the three conditions a planetary has to satisfy: that a whole planet fits between the sun and the ring, that the sun and ring teeth add to a multiple of the planet count, and that the planets clear each other. The last column is how far out of mesh the worst planet station is, in teeth — a quantity that is zero or is not, and that no drawing shows, because a drawing of a planetary at this scale draws circles.

The gearset that could not be assembled

A planetary drawing shows a sun, a ring and three or four planets between them, and if the circles are the right sizes at the right stations it looks right. The condition that decides whether the second planet can actually be dropped in is arithmetic — the sun and ring teeth must add to a multiple of the planet count — and it appears in no drawing, at no scale, in any style.

wrong · Misconception A chain is a strand with a smallest piece. 53 teeth driving 11 at a pitch of 12.7 mm. The taut strand is drawn as this field computes it; inside each pitch circle is the polygon the pins actually sit on. The two agree on the straight spans and not on the wraps, and the gap is what the applied field's chordal action is: the effective radius swings between R cos(π/n) and R within every tooth, which is 4.05% on the small sprocket and 0.176% on the large one. A strand has no such number, because a strand has no pitch. positioned by solving, not by drawing.

A chain is not a strand

A chain has a smallest piece, and two things follow that no continuous model can have. Its pins sit on a polygon, so the radius that matters swings by 4.05% within every tooth of an eleven-tooth sprocket — and its loop must contain a whole number of pitches, so the centre distance that closes it comes in steps of 6.4834 mm.

strands · Strand One planet shaft, two centre distances. A compound epicyclic gets its enormous reduction from two meshes whose tooth counts are nearly in the same proportion. The two planet gears are on one shaft, so their axes are at one radius — and at a common module the two rings ask for radii that differ by 0.50 of a tooth. The exactness the reduction is famous for is bought with a pair of meshes running away from the centre distance they were cut at, and the difference is made up by profile shift — the same correction the teeth field applies for a different reason. It is not a rounding: it is the mechanism's own condition, and it is the reason a catalogue reduction of this kind comes in a short list of tooth counts rather than in any combination.

A hundred to one from a difference of one

A harmonic drive reduces by a hundred to one in a single stage with two gears in it, and the hundred is the flexspline's tooth count divided by the two teeth the circular spline has more than it. The same null space that answers a planetary answers it. What each of the three single-stage reductions pays for that arithmetic is different, and the compound epicyclic's price is a pair of meshes whose centre distances differ by half a tooth.

transmission · Transmission One cutter, five wheels. The same rack — one module, one pressure angle, one straight flank — rolled on five different pitch circles. Each wheel gets a different involute, of its own base circle, and the worst departure anywhere is 4.26e-14 mm. This is why gears are interchangeable: the tool defines the tooth, so any two wheels cut by the same rack mesh with each other whatever their tooth counts, and a workshop needs one cutter per module rather than one per pair. It is also why the standard is written as a rack — the rack is the definition and the wheels are consequences of it.

One rack and every wheel

Two gears mesh if they were cut by the same tool. That is not a manufacturing convenience laid on top of the geometry — it is the geometry, and it is why a gear standard is written as a description of a cutter rather than as a family of tooth curves.

gears · Tooth Reverted trains of 12 : 1. A reverted train has its input and output shafts in line, which means the two stages share one centre distance — so the tooth counts must satisfy z₁ + z₂ = z₃ + z₄ and give the ratio asked for. Two equations in four integers, and there is no reason for a solution to exist. For 12 : 1 there is none at all until the wheels are allowed to reach 63 teeth; below that the coaxial condition and the ratio simply cannot both be met. This is the same kind of arithmetic as the clock trains of the timing field, with one extra equation, and the extra equation is what a shaft position costs.

Two shafts that must be in line

Asking a two-stage gear train for a ratio is easy. Asking it for a ratio and for its input and output shafts to be coaxial is asking for a solution of two equations in four integers, and there is no reason for one to exist. A twelve-to-one reverted train needs a sixty-three-tooth wheel before it has any solution at all — while sixteen to one, a larger ratio, manages with fifty-six.

transmission · Transmission 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.

meshing · Meshing A tooth flank is the end of an unwound strand. A strand wrapped on a circle of radius 45.105 mm, unwound while kept taut. Its free end traces the involute — the same curve the gears field builds from its own parameterisation, agreeing to 1.5e-14 mm over the whole flank. The strand is the important part of that sentence and not the curve: the taut portion is a tangent to the base circle, its length is the arc it has left, and both of those are statements about a strand rather than about a tooth. At this position the free length is 27.965 mm. positioned by solving, not by drawing.

A tooth flank is an unwound strand

Unwind a taut string from a circle and its free end traces the involute, to 1.5·10⁻¹⁴ mm of the curve the gears field draws. That is not a curiosity: the line of action of an involute pair is a crossed strand on the two base circles, so the property the involute is chosen for — a ratio that does not care where the shafts are — is a belt's property rather than a curve's.

gears · Tooth 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 24 teeth inside 72. An internal pair, drawn from the same involute the external pairs are drawn from. Three things are different and they are one difference. The centre distance is 24.0 — the difference of the pitch radii rather than their sum. Both base tangency points lie on the same side of the line of action, so the pitch point falls outside the segment between them rather than inside it. And the contact ratio is 1.931 against the external pair's 1.707, because the annulus's tip circle cuts the line on the far side of its own tangency and lengthens the contact path instead of shortening it. The pinion turns the same way as the annulus, which no external pair ever does.

The mesh with one curvature reversed

Turn an annulus's teeth inward and the same involute law produces a different machine: a centre distance that is a difference, two base tangencies on one side of the line of action, more contact than an external pair carries, and three separate floors on the tooth counts, all of them the same statement about where an involute stops existing.

gears · Tooth

Named alongside it

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

Base circleInvoluteConjugate-actionDesign ruleUndercuttingBacklashLine of actionPressure angleProfile shiftContact ratiocycloidal motionEpicyclic

All concepts