Machine Kinematics
solved, then drawn
Series

Tooth — the series

23 essays on one idea, from the one that introduces it to the one that assumes the rest.
  1. 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.

    part 1 · gears
  2. 20 teeth driving 32. Both flanks generated from the involute, not approximated, at a pressure angle of 20°. The orange line is the line of action — tangent to both base circles, and the only place contact happens. Its length between the two tip circles divided by the base pitch is the contact ratio, 1.612 here, which means that for 61% of the cycle two tooth pairs are carrying the load and for the rest just one. The velocity ratio is 0.6250, and it is constant because the common normal never moves. The dashed extension runs between the two base tangency points, which are 8.89 mm apart; the heavy part is where contact actually happens.

    What happens in a mesh

    Contact between two gear teeth happens only along one straight line, and only over part of it. How much of that line lies between the two tip circles, divided by the base pitch, is the contact ratio — and if it drops below one the drive periodically stops being driven.

    part 2 · gears
  3. Undercutting, either side of 17.10 teeth. Five gears, drawn whole and scaled to a common pitch circle so the tooth counts can be compared by counting them. The dedendum sits a fixed 1.25 modules below the pitch circle and the base circle sits at r·cos α, so as the tooth count falls the base circle rises relative to the root. Below N = 2/sin²α = 17.097 it rises above it, and the part of the flank between them lies where no involute exists — the cutter removes it. The familiar rule says seventeen; the exact figure is 17.10, so seventeen undercuts slightly and eighteen is the smallest count that does not. The teeth are not to a common scale, because at a common module the small gears would be unreadable.

    Undercutting, and the seventeen-tooth rule

    Below a certain tooth count a standard cutter eats into the flank it is supposed to be forming. The count is quoted as seventeen. The formula gives 17.097, which means seventeen undercuts slightly and eighteen is the smallest that does not — the rounding went the convenient way.

    part 2 · gears
  4. Sun 24, ring 72, planet 24. An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. With the ring held, a sun input turns the carrier at 1 + 72/24 = 4.000 to one. Willis's equation and the tabular superposition method are independent derivations and both are computed here; the site requires them to agree, because planetary ratios are the most commonly mis-stated numbers in mechanism work and the sign errors are notorious.

    Epicyclic ratios, two ways

    An epicyclic train has three shafts and one equation relating them, so fixing any one gives a different ratio from the same gears. The sign errors are notorious, so every ratio here is computed by Willis's equation and by the tabular method, and the two are required to agree.

    part 2 · gears
  5. 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.

    part 3 · gears
  6. 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.

    part 4 · gears
  7. 53/11, instant by instant. The ratio a 53-tooth sprocket and a 11-tooth sprocket actually deliver, against the driver's rotation. The quoted 4.818 is the flat line, and it is exactly right as an average — the mean over a turn comes out at 4.8182, which the computation is never told. Within every tooth the ratio swings from 4.756 to 4.948, a fluctuation of 3.99%. The sharp corners are pins seating: the identity of the pin the strand runs from changes 53 times a turn on one sprocket and 11 times on the other, and the two do not coincide.

    The chain is a polygon

    A chain's pins sit on a polygon, so the radius that matters swings by 1 − cos(π/n) within every tooth: four per cent on an eleven-tooth sprocket. The quoted 53/11 is the mean of that, exactly — and how much of the fluctuation reaches the back wheel is decided by the fractional number of pitches in the taut strand, which nobody adjusts and which can take the drive from perfectly uniform to two per cent.

    part 5 · applied
  8. A reduction made out of a difference. A compound epicyclic's reduction is 1/(1 − z₁z₄/z₂z₃) — a ratio of two integers, so it is exactly what a catalogue says, at every position, forever. It is the only number in this field that survives being measured, and the reason is that it is a count. What it is not is insensitive: each row's denominator is a difference of two nearly equal products, and the last two rows differ by one tooth on one ring.

    A ratio that is a count

    A compound epicyclic's reduction is a ratio of two integers, because it is a ratio of tooth counts — 2176/106 for the drive here, exact at every position and on every unit ever made. It is the only quoted number in this field that survives being measured, and it is fragile in a way exactness does not protect against: one tooth on one ring moves it by forty per cent.

    part 6 · applied
  9. 404 ways to gear 60 to 1. Each dot is a pair of wheels and a pair of pinions whose ratios multiply to exactly 60, with wheels of 40 to 120 teeth and pinions of 6 to 16 leaves. The line is the hyperbola every exact answer must lie on; the dots are the ones that lie on it at whole numbers of teeth, and there are 404 of them. In a single pair there are 0 — a ratio of sixty needs a wheel of 360 teeth against a pinion of six, and nobody cuts that. Exactness is decided in integers here and never by comparing floating-point ratios, which is the one way a search like this quietly returns answers that are merely close.

    A clock is a factorisation

    A going train's job is a ratio and its parts are tooth counts, so whether a clock can be built is whether a number factorises inside the counts a wheel-cutting engine will cut. Sixty has 404 answers in two pairs and none at all in one. The ratio between a sidereal day and a mean one has none in any number of pairs, and the best two-pair train is out by a sixth of a second a day.

    part 7 · gears
  10. Which teeth one tooth ever meets: 20 on 40. Follow one tooth of the pinion round and mark every wheel tooth it touches. It does not touch them all. It touches 2 of 40, which is z₂ divided by the greatest common divisor of the two counts — 20 here — and it goes on touching the same ones for as long as the gears are in mesh. The pattern repeats after 2 turns of the pinion. Adding one tooth to the pinion makes the counts coprime and takes the count from 2 to 40. The marks are produced by walking the mesh, and the count they give is compared with the gcd rather than derived from it.

    Which tooth meets which

    A tooth on a pinion does not meet every tooth on its wheel. It meets z₂ divided by the greatest common divisor of the two counts, and it meets the same ones for the whole life of the drive. On the default planetary used throughout — sun 24, planets 24 — a planet tooth touches exactly one sun tooth and never touches another, and nothing in the drawing says so.

    part 8 · gears
  11. The string is the radius. An involute is generated by unwinding a taut string from the base circle, and the taut string is the flank's normal — so the point where it leaves the base circle is the centre of curvature and the string's length is the radius. At a flank radius r that length is √(r² − r_b²), and the dashed curve is that expression. The dots are the curvature of the polyline this site actually draws, measured by circumcircle through consecutive points, over 324 of them. Worst relative disagreement 1.9e-6. At the base circle the radius of curvature is zero, which is why a flank cut below it is not an involute and why undercutting removes exactly that part.

    Two flanks, one law

    At a gear mesh the two tooth flanks are conjugate profiles, the pitch point is the pole of their relative motion, and their radii of curvature are tied to each other rather than free. Each flank's radius varies fourfold across the mesh and the sum of the two does not vary at all.

    part 9 · gears
  12. 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.

    part 10 · gears
  13. 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.

    part 11 · gears
  14. One tooth count, three pressure angles. A 20-tooth gear cut at 14.5°, 20°, 25°, drawn at a common pitch radius and overlaid on its own pitch and base circles. The three teeth have the same thickness at the pitch circle — that is what the standard fixes — and differ everywhere else: a coarser pressure angle puts the base circle lower and leans the flank over, so the tooth gains metal where it is held and loses it where it reaches: thicknesses at the base circle run 1.628, 1.756, 1.967 modules and at the tip 0.866, 0.695, 0.510. Those two run opposite ways, and the second of them ends at a hard stop — a tooth whose tip thickness reaches zero has come to a point and cannot be cut.

    The angle the standard left free

    Involute geometry fixes the tooth curve and leaves one number open. Raising it buys smaller pinions and spends contact ratio, monotonically and in opposite directions, so there is no angle that is best at both — and the familiar twenty degrees is a choice with a date on it rather than an optimum.

    part 12 · gears
  15. Two cones, 20 teeth and 40, at 90°. The axial section, which is where every bevel quantity is read. Two cones share an apex and roll on one another along the element drawn heavy; their half-angles are 26.57° and 63.43°, adding to the shaft angle, and the ratio of their sines is the tooth-count ratio exactly. The dashed arc is the sphere of radius 22.36 on which a bevel tooth's profile actually lies. The two short lines perpendicular to the common element are the back cones; each is heading for its own axis at a distance r/cos δ from the pitch circle — 11.18 and 44.72 — and that distance is the pitch radius of the spur gear the tooth is really cut to. Both back cones lie on one line, because there is only one perpendicular to the pitch element at that point, and the two heavy stubs straddling it are the two teeth — each one addendum out from the pitch circle and 1.25 in.

    A tooth that lives on a sphere

    Every tooth in this field so far has been a curve in a plane, forced by the law of gearing and exact. A bevel tooth's profile lies on a sphere, no piece of a sphere flattens without stretching, and so the shape a bevel gear is actually cut to is an approximation — the only one in the field.

    part 12 · gears
  16. 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.

    part 13 · gears
  17. The field of action, straight and slanted. The rectangle a contact lives in: its width is the usable line of action, its height is the face width, and a new contact line enters every base pitch. On the left the teeth are straight, so the lines are vertical and each one arrives and leaves all at once — the total length in contact jumps by a whole face width, 40 mm, every base pitch. On the right the same pair with a 20° helix: the lines lean by the base helix angle, so a tooth enters at one end of the face before it has left at the other, and the total changes by 3.44 mm instead of 40. Nothing about the profile is different between the two panels.

    Contact that runs along the tooth

    A straight tooth engages along its whole face at once, so the amount of contact at a mesh is a square wave. Slant the tooth and a second contact ratio appears that has no tooth count in it and no pressure angle — bought with face width and helix angle alone, and able to reach a value at which the total length of contact stops varying at all.

    part 13 · gears
  18. 20 teeth driving 32. Both flanks generated from the involute, not approximated, at a pressure angle of 20°. The orange line is the line of action — tangent to both base circles, and the only place contact happens. Its length between the two tip circles divided by the base pitch is the contact ratio, 1.612 here, which means that for 61% of the cycle two tooth pairs are carrying the load and for the rest just one. The velocity ratio is 0.6250, and it is constant because the common normal never moves. The dashed extension runs between the two base tangency points, which are 8.89 mm apart; the heavy part is where contact actually happens.

    The module is a size, the ratio is a shape

    A spur pair has exactly one length in it. Every quantity it produces is either proportional to that length or completely independent of it — centre distance, base pitch and contact length scale exactly; ratio and contact ratio do not move at all, not even by a part in ten to the fifteen.

    part 14 · gears
  19. Where each flank slides worst, and a shift that makes them equal. A 17-tooth pinion driving a 51-tooth wheel, module 1 and 20°, with the pinion's cutter moved out by 0.3574 and the wheel's in by the same, so the centre distance does not change. The two curves are each flank's specific sliding along the path of contact — the sliding speed over that flank's own rolling speed — clipped at ±6. Each is worst at its own root: the pinion's at the start of contact, where it reads 1.962 in size, and the wheel's at the end, 1.961. Unshifted the pinion's is 11.09 and the wheel's 1.38. At a shift of 0.3574 the two are equal, at 1.961. The contact path — the part of the line of action drawn — slides along the line as the shift changes. Dragging changes the shift.

    The shift that evens the wear

    A pinion's flank slides hardest at its root, and on a small pinion much harder than anything on its wheel: a 17-tooth pinion on a 51-tooth wheel reads 11.1 at its root against the wheel's 1.38. Moving the pinion's cutter out and the wheel's in by the same amount leaves the centre distance alone and slides the contact path along the line of action, and at a shift of 0.357 the two roots slide equally, at 1.96. That shift is not the undercut cure — the same pinion needs 0.006 for that — and it grows with the ratio, so a pinion that needs no shift at all to be cut cleanly still needs a third of a module to wear evenly with its wheel.

    part 15 · gears
  20. Where contact runs on the line of action. A 17-tooth pinion driving a 51-tooth wheel, module one, drawn as its line of action from the pinion's base tangency (left end of each line) to the wheel's (right end), for four designs; the thick stretch is where the teeth are in contact. Specific sliding at a root grows without bound as contact nears that gear's own base tangency. Standard: operating pressure angle 20.00°, contact ratio 1.636, worst root 11.086; ±0.357 shift: operating pressure angle 20.00°, contact ratio 1.558, worst root 1.961; four knobs, shifts ≤ 1: operating pressure angle 26.18°, contact ratio 1.200, worst root 0.834; four knobs: operating pressure angle 27.94°, contact ratio 1.200, worst root 0.753. The four-knob designs push contact away from both ends at once, at a larger centre distance and pressure angle, until the contact ratio reaches its floor of 1.2.

    Four knobs for the root

    A shift of the cutter can balance the sliding at a gear pair's two roots, but on a 17-tooth pinion driving a 51-tooth wheel both then slide at 1.96. Free both shifts and both addenda, let the centre distance go where zero backlash puts it, and keep a contact ratio of 1.2, tips a quarter of a module thick and a tenth of a module of clearance: the worse root comes down to 0.83 with shifts no larger than a module, and 0.75 without that limit — two and a half times less. Every optimum has its two roots exactly equal, runs at an operating pressure angle of 24° to 29°, and sits on its contact-ratio floor, which is what the gain is paid with.

    part 16 · gears
  21. A taller, narrower field carries the same line. The field of action of two 17–51 pairs, drawn to one scale with the path of contact stretched 7 times: across, the transverse path of contact; up, the face width; the dark lines are the contact lines in it at one instant, a new one entering every base pitch. Left: the spur pair whose four knobs give the least worst-root sliding at a transverse contact ratio of 1.2, 0.834, on a face of 10 modules. Right: a 30° helical pair whose four knobs are chosen for a transverse contact ratio of 0.4, worst root 0.209, on a face of 26.5 modules, which is what carries the same mean length of line, 12.00 against 12.00 modules. Its overlap ratio is 4.21 and its total 4.61. At this instant 10.00 and 12.10 modules of line are in contact.

    A helix pays for the root with face width

    A spur pair's least worst-root sliding, 0.834 on a 17-tooth pinion driving a 51-tooth wheel, was bought by spending contact ratio down to the floor of 1.2 that continuity needs. A helical pair gets its continuity from the overlap along the face, so its transverse contact ratio can fall below that floor, and the root's sliding falls with it, nearly in proportion: 0.209 at a third of the contact and 0.102 at a sixth, on the pitch point's law π εα (1/z₁ + 1/z₂) / tan αw. The helix angle itself buys under a tenth. What pays is the face: the mean length of line in contact is the transverse ratio times the face, so carrying the spur pair's load on as much line at a third of the contact takes a face 2.65 times as wide.

    part 17 · gears
  22. The line in contact swings, and the least of it is what a tooth carries. A helical pair at a transverse contact ratio of 0.4 and an overlap ratio of 3.5. Left: the field of action, 0.4 of a base pitch across and the face up, with the contact lines at the moment the total length in contact is least; each line crosses the face while advancing 3.5 base pitches, so only a short piece of each lies in the field. Right: the total length in contact over one base pitch, as a share of its mean, 0.857 at its least and 1.143 at its most; the closed form b/(εβ cos βb)·[εα εβ − min(nα nβ, (1 − nα)(1 − nβ))] gives the least exactly. Dashed: the spur optimum at εα 1.2, which carries its whole face for four fifths of a pitch and twice it for one fifth, 0.833 and 1.667 of its mean.

    The face carries the least line

    A helical pair that gives up transverse contact to cut its root's sliding was priced by the face it needs to carry the spur optimum's mean length of contact line: 2.65 times as wide, at a third of the contact on a 30° helix. But the length in contact swings through every base pitch and the peak load stands on its least value. That least has a closed form in the fractional parts of the two contact ratios, exact against the swept field to 10⁻¹³. The spur optimum's own least is its face, a sixth below its mean, so least against least the factor is 2.21 when the overlap ratio is a whole number. Real faces land between whole numbers and need 2.21 to 2.87 times the spur's, 2.39 on a spur face of ten modules, and above 2.65 only on faces under ten. And a steeper helix is not always more line: at a fixed face, raising the overlap ratio from 1 to 1.5 takes a third of the least line away.

    part 18 · gears
  23. Where on the field the root band sits, and when it is loaded most. A helical pair at a transverse contact ratio of 0.4 and an overlap ratio of 1.5. Left: the field of action, the path across and the face up, with the bands a tenth of the path wide at each end shaded, where the sliding is worst, and the contact lines at the moment the share of the load in those bands is largest, 1.36 times its even value. Right: over one base pitch, the share of the load in the root bands over its even value (solid), and the total length in contact over its greatest (dashed), least 0.667 of its mean.

    The least line carries the root

    A helical pair's length of contact line swings through every base pitch, and the peak load stands on its least value. The roots, where the sliding is worst, sit at the two ends of the path of contact. The share of the load carried there swings too, and in the limit of a thin root band it reaches its peak while the line in contact is least: half as much again as an even share at overlap ratios just above one, a quarter above two, a sixth above three. A whole-number overlap ratio removes both swings at once. So the overlap that makes the least line largest is also the one that spreads the root's load evenly, and a face sized on the least line alone understates the root's worst moment by the root's own swing.

    part 19 · gears

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