Machine Kinematics
solved, then drawn
Concept

Backlash — where it appears

The play between two meshing teeth, measured as the angle one wheel turns before the other moves. It is an allowance rather than an error: without it a pair cannot accommodate its own tolerances, and it is designed in by cutting the teeth thin.

Named by 19 essays across 6 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 Lost motion on a 24-tooth ratchet. A pawl can only drop into a tooth, so an input that has moved by less than one tooth pitch has moved the output by nothing at all. On 24 teeth the pitch is 15.0°, and that is the worst case with one pawl. Two pawls offset by half a pitch halve it and three thirds it, because whichever pawl is over a root drops first. None of this is a manufacturing question: the numbers are the same on a perfectly made ratchet, which is what separates them from the lost motion an as-built measurement would.

The resolution is the pitch

A ratchet's step and a ratchet's error are the same number. Nothing about how well it is made improves that, more pawls divide it by a whole number, and the obvious remedy — cut more teeth — runs into a wall that is geometric rather than practical: at a tooth depth of 0.16 radii the construction stops at twenty-nine.

timing · Intermittent 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 How far the crank turns before the rocker does. With a clearance of 0.01 at each pin, the crank must be turned this far on reversal before the rocker moves at all. Through most of the cycle it is about 2.8°, and at its best 1.44°. At the two positions where the rocker reverses — marked — it is unbounded: the output velocity passes through zero there, so no amount of crank rotation moves the rocker out of its clearance band. The peak in a plot like this is therefore a property of the sampling and not of the mechanism; it reads 402° at 96 samples and grows without limit as the sampling is refined. The number worth quoting is the plateau.

How far the crank turns first

Reverse the input of a four-bar with a hundredth of clearance at each pin and the output does not move for about 2.8° of crank rotation. At the two positions where the rocker reverses it does not move at all, however far the crank is turned — the lost motion is unbounded there, and the peak in any plot of it is a property of the sampling rather than of the mechanism.

practice · Clearance Eleven lobes from twelve pins, at 46° of eccentric. The pins are circles and the disc is their envelope. One turn of the eccentric moves the disc back by one lobe pitch — a reduction of 11 : 1 from a ring, a disc and an offset bearing, with no gear teeth anywhere — and the profile's 11 lobes are counted off the generated curve rather than put there. The roots sit at 50.00 mm and the tips at 60.00, which are R − r ∓ e: the eccentricity is the lobe height, twice over. positioned by solving, not by drawing.

Eleven lobes from twelve pins

A ring of round pins, a disc on an eccentric, and a reduction of eleven to one with no gear teeth anywhere. The disc's profile is not designed: one pin generates one lobe of it, the other ten lobes are the same curve, and the count that decides the ratio is a count of lobes on a shape nobody drew.

meshing · Meshing The transmission angle through one turn. μ is the angle at B between coupler and rocker, computed from each solved position rather than from a formula. It runs from 54.3° to 100.3° for these lengths. The shaded band is the usual design rule — keep μ between 40° and 140° — and this linkage stays inside it throughout. The rule is about geometry alone: nothing here knows about friction, and a mechanism with a comfortable μ can still be a poor machine.

What is still outside

Eight essays here end by saying that clearance, backlash, friction or wear are not modelled. This field took two of those four, because a tolerance is a set of geometries and a clearance is a short link, and both are questions about where a mechanism can be. The other two are not, and this is the page that says exactly where the line falls and why it is where it is.

practice · Practice 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 What preload takes away, and what it leaves. The same four-bar with 0.01 of clearance at each pin, driven so that the load through every joint keeps one sign. The upper line is the lost motion it had before: about 2.80° of crank rotation thrown away on every reversal, unrepeatable, and not removable by calibration. The lower curve is what is left once the pins are held against one side of their holes — a fixed offset of between -0.330° and 0.013°, which is a dimensional error rather than play, so its average of -0.136° comes out in a calibration and only the 0.343° of variation survives. A factor of 8.2, bought with a permanent parasitic load that is not modelled here.

Taking up the play

Preload does not make a clearance smaller. It takes away the clearance vector's direction, which is the property that made the error unrepeatable — so 2.80° of lost motion becomes a 0.343° offset, of which 0.136° is a constant that calibrates out. A factor of eight, bought with a permanent parasitic load that is not modelled here.

practice · Clearance The taut inner, and the walls it is actually held by. A sheath of two 8° bends on a radius of 40, its bore drawn at a clearance of 4, with the shortest path from ferrule to ferrule inside it. Nothing about a bend goes into that path: it is the shortest route the tube allows, found by tightening a funnel against every cross-section in turn, and the places it reaches the wall are where it is held. At this clearance the two bends hold 53% and 53% of their own turn, and the inner is short of the centreline by 0.854 against the closed form 1.117 — 76% of it. Widen the bore and the path lifts off.

Which walls a strand is held by

A Bowden inner is short of its sheath by the clearance times the total turning — a law with no bend radius in it and no route shape. It is exact while the inner touches the inside of every bend, and a bend shallower than the turn the inner spends crossing the bore is not touched at all. Past that the law is an over-estimate, and what the inner actually loses flattens onto a ceiling that has no clearance in it.

strands · Strand A disc cut undersize, turned until a pin touches. A twelve-pin cycloidal drive whose disc was cut as if its pins were 1 larger than they are, drawn at 10° of the eccentric with the disc turned a further 1.082° about its own centre — the turn at which the first pin touches. One pin touches, pin 0, marked in its own colour. The pins within half the clearance of touching are drawn in a second colour and the rest plainly: 5 of the twelve are within half the clearance and 7 further off. The clearance is exaggerated so that it can be seen; at every clearance measured, one pin touches first.

A clearance leaves one contact

A cycloidal drive's disc, exactly as its pins generate it, touches nine or ten of its twelve pins at once, and that is why it can drive where two identical rotors cannot. Cut the disc two hundredths undersize, as a real drive is cut, and a rigid disc turned until it touches meets one pin, at every angle of the eccentric and in both directions. The arm that pin gives is never worse than the exact disc's best. What a clearance takes away is load sharing, and deflection gives it back: contacts that may give by the whole clearance put six or seven pins within reach.

meshing · Meshing The disc, the wheel, and what has to be cut out of it. A 6-slot Geneva at 20° of driver, with the driver's locking disc of radius 13.5 drawn about its axis and the wheel drawn as the material it actually has — inside its rim, short of the 6 flat chords cut across it, one between each pair of slots, and clear of the 6 slots. The disc and the wheel share the region near the line of centres, so the disc has to be cut away wherever the wheel is ever there while the pin is driving. Swept over the whole index that cut-away spans 126.8°, against an index sweep of 120° — it is wider, because the rim is still swinging through the disc's circle after the pin has left the slot.

A flat holds the wheel to second order

A Geneva wheel is held between indexes by a disc on its driver sitting in a concave arc, and that disc is so large that one pin is all a wheel of four slots or more can carry. Hold the wheel on a flat chord instead and the disc must shrink to the crank times sin(π/n). Its cut-away falls from 236° to 127° on six slots, and a second pin fits at every count from four to twelve. The cost is the hold itself. An arc lets the locked wheel rock in proportion to the running clearance and a flat as its square root: 0.022° against 1.48° at a clearance of 0.02 on a crank of 30.

timing · Intermittent The same drive with a smaller eccentric: every pin touches. Twelve pins on a ring of 60, pin radius 5, and the disc exactly as those pins generate it, at 10° of the eccentric. Left, the eccentric at exactly R/N = 5, whose roots are round sockets: 9 pins touch (solid) and 3 stand clear (dashed), the pins near the roots. Right, the eccentric at 0.8 of that, 4.00, whose roots are smooth: 12 of twelve touch. Over a pin pitch the socketed disc touches 9 or 10, and every smaller eccentric measured touches all twelve.

A disc whose roots are not sockets

Put a cycloidal drive's eccentric at exactly the pin circle's radius over the pin count and every root of its disc is a round socket; take any smaller eccentric and the roots are smooth. The sockets looked like a benefit — a pin in one closes its gap both ways at up to the full pitch offset. Measured from a half of that eccentric to all of it, they are the opposite. The socketed disc touches nine or ten of its twelve pins, every smaller one touches all twelve, and its first contact keeps the smallest share of the pitch offset of the seven. What a smaller eccentric really costs is the pitch offset itself: the free turn per unit of clearance is 0.0365 divided by the eccentric's share.

meshing · Meshing Six balls, two grooves each, and a cage. A Rzeppa joint bent 20°, input shaft along z and output shaft turned about x. Each ball lies where the plane of its inner-race groove, through the input's axis, meets the plane of its outer-race groove, through the output's axis; at equal input and output angles every ball is in the plane bisecting the two shafts, and the cage holds them there. Left, from the side: the two shafts and the bisecting plane edge-on, the balls on it. Right, along the cage's normal: the six balls round their circle, the plane of the bend drawn across it. At an input angle of 0° a cage clearance of c lets the output lag or lead by up to 0.350 c radians, c in ball-circle radii.

A Rzeppa cage adds its clearances

A Rzeppa joint keeps constant velocity because its cage holds every ball in the plane that bisects the two shafts. Give the cage's windows a clearance c and the output can lag or lead the input. Six balls might be expected to limit that lag through the worst of them, or as a root sum of squares. They do neither. Opposite balls are one constraint, the cage's two tilts spend two of the three, and what is left adds the three pairs' allowances like clearances in series: 0.350 c at a 20° bend, twice what the worst ball alone would allow. The lag falls to nought each time a ball crosses the plane of the bend, grows with the bend itself, and a four-ball joint cannot hold it at all.

pairs · Pair The flank bends towards the disc, and the pin's radius is measured against that bend. One root of a twelve-pin cycloidal drive, pin radius 5 on a ring of 60, with the eccentric at 0.97 of R/N. The thin line is the curve the pin's centre traces in the disc's frame, its tip pointing at the disc's centre; the dashed line is that curve offset towards the disc by the pin's radius, which is the disc's edge wherever nothing trims it. Either side of the tip the flank bends the other way, towards the disc, most tightly 22.6° of the eccentric from the root, with a radius of 5.36 (the circle, centre marked). The pin's radius, 5, is measured along the same normal to the offset point. The offset point stops short of the centre of curvature, so the offset does not fold and the root is untrimmed. The two meet at 0.9740 of R/N.

The flank that trims the root

A cycloidal disc's roots are trimmed, and its pins start standing clear, above an eccentric that was known only for one drive: 0.974 of the pin circle's radius over the pin count, for twelve pins of radius 5 on a ring of 60. The trimming is a bend. Either side of each root the flank of the pin's path curves towards the disc, and where that bend is tighter than the pin the offset folds. The bend has a closed form, so the threshold does too, √(1 − ρ²(N + 1)³/27(N − 1)) for pins of radius ρ, and it falls well below one for fat pins or many of them. On every drive measured, the classical eccentric loses contacts and lever together.

meshing · Meshing Past the play, a caged joint shares its load and a grooved one leans on one ball. Two six-ball Rzeppa joints bent 20°, at an input angle of 15°, each twisted 0.03 of a window clearance past its free lag, with every contact a spring of the same stiffness and the cage and balls settled where the stored energy is least. Bars: each ball's share of the total contact load. Left, grooves offset 0.1, where the cage sets the play: 0.17, 0.20, 0.19, 0.16, 0.14, 0.15, 5.91 balls' worth. Right, grooves offset 0.7, where the grooves do: 0.00, 1.00, 0.00, 0.00, 0.00, 0.00, 1.00 balls' worth. Dotted: an even sixth.

A tight groove leans on one ball

A six-ball Rzeppa joint's free play is set by the tighter of its cage and its offset grooves, and the two limits look alike from outside. They are not alike once the play is taken up. With every contact made a spring of the same stiffness and the joint twisted past its free lag, a joint whose cage binds spreads the first load across four to six balls' worth and, a third of a window clearance on, gives every ball between 0.17 and 0.19 of it. A joint whose grooves bind puts the first load on one ball alone over most of a ball pitch, and two clearances on it still spreads it over fewer than six. At the same torque its hardest-pressed ball carries 1.5 to 3.0 times the caged joint's. Tightening the tracks to cut the play buys the play with uneven loading.

pairs · Pair Tighten the windows and the cage's limit drops below the grooves'. The six-ball Rzeppa joint bent 20° with its grooves offset 0.7 and a track play of one: across a ball pitch, the free lag the grooves alone allow (solid) and the cage alone allows with windows of clearance 0.5 (dashed). The joint's play is the smaller of the two. The cage is the limit wherever its curve is the lower, here at every input angle; it becomes the limit everywhere once the windows' clearance is below 0.559 of the track play. The largest play over the pitch is 0.163.

A tight cage spreads the load until the tracks join in

A Rzeppa joint whose offset grooves set its play loads one ball first, and at equal torque its hardest-pressed ball carries up to three times a joint whose cage sets the play. Tighten the cage's windows below 0.559 of the track play and the cage becomes the limit at every input angle: the first load spreads over 3.6 balls' worth or more everywhere, the play falls, and at low torque the worst ball carries no more than a caged joint's. The cage does not stay in charge. Once the twist reaches the gap between the two limits the tracks engage, unevenly, and the worst ball climbs back towards the untightened joint's. A tighter cage widens the gap and pushes that torque up, so the windows' clearance is a choice about the torque at which even loading should end.

pairs · Pair

Named alongside it

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

ClearanceToleranceBase circleEnvelopeInvoluteLost motionCentre distanceContact ratioCycloidal drivePressure angleConjugate-actionConstant velocity

All concepts