Machine Kinematics
solved, then drawn
Concept

Contact ratio — where it appears

The length of a gear pair's contact path divided by the base pitch, counting how many tooth pairs are engaged at once. Below one a pair of teeth lets go before the next arrives, so the drive stops being continuous and becomes a series of impacts.

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 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.

gears · Tooth 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 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.

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 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 One shaft angle, five places along a twisted rotor. A mismatched pair — a cycloidal rotor against a circular-tipped one of the same height, which are not each other's conjugates — with the rotors twisted by 1 lobe pitch along their length. The section at a given place along the shaft is the flat pair at an input angle shifted by the twist so far, so at one instant the five sections shown are at five different phases of the same mesh: the two bodies are into each other by 2.42 at the tightest section and 2.34 apart at the widest, with a mean of -0.408 across the whole seal. All five are drawn at one scale, and nothing here is meshed twice: one planar profile is read along a window.

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.

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 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 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.

Line of actionBase circleInvoluteUndercuttingDesign rulePitchPressure angleProfile shiftOverlap ratioBacklashCentre distanceConjugate-action

All concepts