Machine Kinematics
solved, then drawn
Concept

Involute — where it appears

The curve traced by the end of a taut string unwound from a circle, and the tooth form of essentially every gear cut since about 1900. Its normal is always tangent to that circle, which is what makes a pair of them hold their ratio at any centre distance.

Named by 22 essays across 3 fields — each of them below, with the objects they name alongside it.

A ratio that is not a number. The output-to-input angular velocity of a four-bar, computed from the velocity solution at every position. It runs from -0.398 to 0.333 — it changes sign, because the rocker turns back — with a mean of 0.000 that no instant of the cycle actually exhibits. Quoting a single figure for a linkage's ratio is quoting the average of that curve. A gear pair is the case where the same phrase is honest: its ratio is 0.5000 and stays there, which is not a coincidence but the property the involute was invented to guarantee.

The ratio that is not a number

A four-bar's output-to-input speed ratio runs from −0.29 to 0.51 through one turn and changes sign on the way. Quoting a single figure for it quotes the average of that curve, which the mechanism never exhibits. A gear pair is the case where the same phrase is honest, and it is honest by construction.

wrong · Misconception 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 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 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 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 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 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 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.

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 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 The rotor is not a Reuleaux triangle. The generated rotor, with the Reuleaux triangle through the same three apexes drawn over it — three circular arcs, each centred on the opposite corner. They part company by 1.21 mm on a rotor of generating radius 100, and the flank is the one that is right: it is the envelope of the housing, and a Reuleaux flank would foul the wall. The measurement that cannot be argued with is the width. A Reuleaux triangle has the same width in every direction — that is what it is for — and this rotor's runs from 150.0 to 174.1, a spread of 16 per cent. positioned by solving, not by drawing.

Six things a shape is not

A rotor that is not a Reuleaux triangle, teeth that are not the same shape as each other, a fillet that is not an arc, a mesh that does not roll, a conjugate pair that cannot be built and a tooth form that was not deduced. Six claims in circulation, each with the number that kills it.

wrong · Misconception 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 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 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 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 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.

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

Named alongside it

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

Base circleUndercuttingConjugate-actionLine of actionPressure angleCentre distanceContact ratioMeshEnvelopeMeshing equationRatioBacklash

All concepts