Machine Kinematics
solved, then drawn
Concept

Line of action — where it appears

The common tangent to two gears' base circles, along which every contact between two involute flanks takes place. It is where every contact of an involute pair happens, and this site recovers it by fitting a line to solved contacts rather than by constructing it.

Named by 15 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

gears · Tooth 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 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 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.

meshing · Meshing 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.

gears · Tooth 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 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.

gears · Tooth 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 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.

gears · Tooth 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.

gears · Tooth 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.

gears · Tooth 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.

gears · Tooth 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.

gears · Tooth 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.

gears · Tooth

Named alongside it

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

InvoluteContact ratioBase circleUndercuttingDesign rulePressure angleProfile shiftCentre distanceEnvelopeOverlap ratioPitchBacklash

All concepts