Machine Kinematics
solved, then drawn
Concept

Base circle — where it appears

The circle an involute tooth flank is unwound from, of radius equal to the pitch radius times the cosine of the pressure angle. Every normal to the flank is tangent to it, and it belongs to the wheel rather than to the pair — which is why an involute pair does not mind where the shafts are.

Named by 26 essays across 5 fields — each of them below, with the objects they name alongside it.

A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

Prescribing motion

A linkage gives the motion its geometry allows. A cam gives the motion it was asked for, which sounds like an improvement and is a trade — the displacement becomes free and the derivatives stop being.

cams · Cam 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 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 The curves that do not move. A rotation about a point carries a curve into itself if and only if the curve is an arc of a circle centred on that point, and that one sentence is every exact dwell a machine has. Each bar is how far a curve moves when it is turned two degrees about the axis its mechanism turns about, as a fraction of its own radius. The two arcs about their own centres — a cam's dwell and a deadbeat's locking face — sit at the sampling floor, which is the sagitta of the polyline they are measured as and not a property of the geometry; the number is quoted with the floor beside it because an agreement quoted without its resolution is a mistake that is easy to make. Everything else is orders of magnitude above it, including the near-circular stretch of a coupler curve that a six-bar builds its approximate dwell out of.

The arc that is concentric with the pivot

A rotation carries a curve into itself exactly when the curve is an arc of a circle about the centre of rotation. Every exact dwell in a machine is that one sentence applied — a cam's dwell, a Geneva's locking disc, a deadbeat escapement's locking face — and the six-bar dwell that is merely very good is what happens when the curve is nearly one.

timing · Intermittent A 34-unit roller on a 20-unit base: the cam cannot be made. The outer curve is the pitch curve — where the roller's centre has to go, which is just the base radius plus the follower's displacement, plotted in polar coordinates. The inner curve is what has to be cut: the pitch curve offset inward by the roller's radius along its own normal. Where the pitch curve bends more tightly than the roller is wide, that offset crosses itself, and the metal the profile needs has already been removed. The tightest radius of curvature here is 19.43 at 85°, against a roller of 34 — so the cam is undercut by 14.57 and no amount of care in manufacture recovers it. The pressure angle, meanwhile, peaks at a perfectly comfortable 21.2°: this cam fails a test it never looked like failing.

The cam that cannot be cut

A cam has to be big enough for two reasons, and they are not the same reason. One is that the follower will jam in its guide if the pressure angle is steep. The other is that the roller will gouge the profile if the curvature is tight — and there is a combination where the pressure angle is comfortable and the cam still cannot be manufactured at all.

cams · Cam 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 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 A cycloidal cam at 60°. A 20-unit rise over 120°, a dwell, a return and a dwell. The dashed curve is the pitch curve — where the roller's centre travels — and the solid one is the surface that has to be cut, which is the pitch curve offset inward by the roller radius along its own normal. The pressure angle at this instant is 25.5°: the angle between the follower's direction of travel and the normal to the surface, and the number that decides whether the follower jams in its guide rather than sliding.

A lift is a size and a law is a shape

A cam's motion law is a dimensionless function of a dimensionless argument, so everything the cam field says about laws — which has the least acceleration, which has an impulsive jerk, which one costs least — transfers between cams of any size. What does not transfer is anything with the lift or the base circle in it.

cams · Cam 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 A flat face on a base circle of 30, where the profile bends tightest. The cycloidal programme — rise 10 over 120°, return over 120° — cut for a flat-faced follower on a base circle of 30, turned to 212.4°, where the profile's radius of curvature is smallest. The contact sits −5.37 from the follower's axis, which is ds/dθ, and the dashed circle is the one that fits the profile there: R₀ + s + s″ = 24.668, and 24.668 fitted through the drawn curve. The face's pressure angle is zero at every angle. What it needs instead is for that radius to stay positive, and it does so on every base circle above 5.332.

A flat face asks for a convex cam

A flat-faced follower is pushed at right angles to its face, so its pressure angle is zero at every angle and the constraint that sizes a roller's cam disappears. What replaces it is convexity: the profile's radius at the contact is R₀ + s + s″, and a base circle below 5.332 leaves a cycloidal cam with a hollow the face cannot reach — held 0.446 high on a base of 2.

cams · Cam 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 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 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 A flat face on a swinging arm, and the cam it asks for. The cam a flat-faced follower needs when its face is carried on an arm pivoted 300 from the cam's centre, with a base circle of 20 and a lift of 20, drawn in the cam's own frame at three cam angles. The heavy line is the face, the dashed line is the perpendicular from the cam's centre to it, and the dot is the contact. Its distance from the foot of that perpendicular is the offset, which for a sliding face would be the programme's own velocity and here is a quantity with the arm in it. Over the turn it runs from -20.41 to 17.95, so the face must be 38.4 wide.

A flat face on an arm is worse

Carry a cam's roller on a swinging arm instead of a slide and the pressure angle improves — a long arm beats an offset. Carry a flat face on one and the opposite happens. The cam sees the sine of the follower's rotation rather than the rotation, and the distortion costs convexity: the smallest workable base circle rises from 10.66 on a slide to 28.68 at a pivot three base circles out, and below a certain arm length no base circle works at all.

cams · Cam 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 same arm with its face through the pivot and stood off it, each at its smallest convex cam. A flat-faced follower on an arm pivoted 90 from the cam's centre, delivering the symmetric cycloidal programme with a lift of 20, drawn at 60° of cam rotation. On the left the face is a line through the pivot, and the smallest base circle that keeps the cam convex is 28.679. On the right the face stands 33.56 off the pivot on a bracket, parallel to where it was, and the smallest convex base is 26.136 — 8.9% smaller. A sliding face needs 10.664, and neither arm reaches it. The dot is the contact; the small circle at the right is the pivot, and on both sides it clears the cam.

Where on the sine the arm swings

A flat face on a swinging arm delivers its programme through d sin ψ. Differentiating that twice brings in a square of the follower's speed, multiplied by sin ψ, and sin ψ has a sign. Standing the face off its pivot moves the swing down the sine until it straddles ψ = 0, where the square helps on one half of the stroke. On an arm ninety out that lowers the smallest convex base from 28.68 to 26.14, and it lets an arm of 61 have a cam where the face through the pivot needs 80. It never reaches the sliding face's 10.66.

cams · Cam 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 roller offset by 5.88, at 42.3° of the rise. The cycloidal programme — rise 20 over 90°, return over 170° — on a prime circle of 40 with a roller of 10, drawn with the follower fixed above and the cam turned anticlockwise to 42.3°. The follower's line of travel is 5.88 to the right of the cam's centre. The contact normal leans 21.81° from that line, which is the pressure angle, and measured off the drawn pitch curve it is 21.81°. At this offset the peak on the rise is 21.81° and on the return 21.81°; centred, they are 27.37° and 15.33°.

Six things a cam drive is not

A follower that is better off offset, a flat face with no size limit, a swinging arm that improves any follower, a groove that keeps its roller on one wall, a smooth law that is gentle everywhere, and a camshaft that delivers the law it was cut to. Six things people say about cam drives, each the natural reading of something true, each answered by a number the cam essays measured. They fail the same way: each reads the cam at first order, in positions and slopes, and the drive is decided at second order, in accelerations and curvatures.

wrong · Misconception A tilted face and a shorter parallel bracket hold the same line. A flat face on an arm pivoted 70 from a cam's centre, at 70° of cam rotation, the symmetric cycloidal programme with a lift of 20, base circle 33.725. Dark bracket: the face held 43 off the pivot and tilted 30° to the bracket's normal. Light bracket: the face held parallel, 37.24 off the pivot, which is 43 cos 30°. Both put the face on the same line at every angle of the cam — the support from the cam's centre agrees to 2e-14 and the normal's direction to 9e-16 radians over a turn — so the cam, the contact and the force on the pivot are the same; the tilted arm only swings 30° further round.

An arm that lifts against the cam can beat a slide

A flat face on a swinging arm seemed to offer three choices: how far its pivot is from the cam, how far the face stands off the pivot, and the angle the face makes with its bracket. The angle is not a choice. A line fixed to a body turning about a pivot is fixed, up to the turning, by its distance from the pivot, so a face tilted β on a bracket c long is the face on a parallel bracket c cos β long, the same line at every instant. The real third choice is which way the arm lifts. On a symmetric programme the two ways are mirror images. On a quick rise, an arm lifting against the cam's turning needs a base circle of 6.6 to 13 where a sliding face needs 32.85, and one lifting with it needs 41 to 59, or has no convex cam at all.

cams · Cam A 45° rise: the slide's cam and the arm's, each as small as it can be. The quick programme with its rise of 20 shortened to 45°, cut for two flat-faced followers at the smallest base that keeps each cam convex, drawn to one scale. Left: a sliding face, base 185.56. Right: a face on an arm pivoted 90 from the cam's centre, lifting against the cam's turning and stood off −44.9, base 23.26, 8.0 times smaller. At that base the arm's cam stays convex for bases up to 1.0% larger, and no further.

A steep rise spends the arm's room

A flat face on an arm that lifts against the cam's turning needed a fifth of a sliding face's base on a 90° rise. Steepen the rise to 45° and the slide's base grows 5.6 times, to 185.6, while the arm's least base grows only 3.2 times, to 23.3. But the arm's least base sits where its cam is about to go concave on the return as well as the rise: the base can be raised by only a hundredth before the return fails. Ask for a tenth's room above the base and the arm's base grows 5.1 times, nearly the slide's rate, and at 35° no stand-off gives that room at all.

cams · Cam

Named alongside it

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

InvolutePressure angleUndercuttingDesign ruleContact ratioLine of actionEnvelopeCentre distanceProfile shiftPitchBacklashConjugate-action

All concepts