Machine Kinematics
solved, then drawn
Concept

Epicyclic — where it appears

A gear train in which some axes are carried round another, so its members satisfy one relation rather than a fixed proportion. It is a mechanism with two freedoms rather than one, so it can add, subtract or hold, and every ratio it is sold with names something being held.

Named by 22 essays across 5 fields — each of them below, with the objects they name alongside it.

A bevel differential, turning. cage in, hold left, with every member's speed taken from the train's null space and every angular position that speed integrated. The teeth are marked at the pitch points rather than cut as involutes — the flank is the teeth field's subject — but the count is the tooth count and the positions are the solved ones, so what turns and how fast is real. Drag it and watch which way each member goes: left 0.000 · right 2.000 · cage 1.000.

Two inputs and one output

Every mechanism in this collection so far has had one input, and its output has been a function of that input. A differential does not. Its cage turns at the mean of two wheels, so knowing one of them tells you nothing at all about where the third shaft is going — and that is not a complication of the mechanism, it is a different kind of object.

transmission · Transmission A simple planetary, as a graph. Members are vertices and meshes are edges. Each edge carries the two tooth counts and the body the two axes are stationary in — its carrier — and that third label is the whole of what makes an epicyclic different from an ordinary train. Write the mesh relation relative to the carrier and one formula covers both: an ordinary train is the case where every carrier is the frame. This train has 2 meshes across 5 members and 2 freedoms.

A ratio is a null space

Write a gear train as a graph — bodies for vertices, meshes for edges, and on every edge the body the two axes are stationary in — and one formula covers a countershaft gearbox, a planetary, a harmonic drive and a car's differential. The ratio is the null space of a matrix whose entries are tooth counts, so it comes out as a fraction and not as a number that is nearly one.

transmission · Transmission The lever of a simple planetary. Each member sits at a position on the lever fixed by the tooth counts alone, and its speed is the height of one straight line over that position. The line here is drawn through sun and ring; every other member is plotted where the train's null space puts it, and lands on the line exactly — the residual is zero in rationals, and 2.2e-16 once the coordinates have been rounded to doubles for the drawing. Where the line crosses the axis is the member that is standing still, and that is what a brake does: it pins the line to the axis at one position and leaves it free to pivot there. The planets are on the lever too, off the end of it, which is where they belong — they are members of the train and are not shafts anybody can reach.

The lever that is the gearset

The lever diagram of an epicyclic is usually offered as a mnemonic. It is exact, and the reason is a fact about the null space: a gearset whose frame carries no teeth can turn as a block, and that one motion supplies the coordinate every member is plotted at. Where the line crosses the axis is the member standing still, and the ordering of the members on the lever settles which gears are reductions and which run backwards, without a formula anywhere.

transmission · Transmission 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 What a Simpson gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 4 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.

Holding a member chooses the ratio

A gearset offers a plane of motions and a shift element is one linear condition, so a gear is a line in that plane. Enumerate every brake and every clutch and a Ravigneaux's twenty-seven combinations collapse to seven ratios — eighteen of them being the same gear, because locking any two members at all locks the whole gearset solid. Neutral is not one of the seven, because neutral is not a gear.

transmission · Transmission What a Ravigneaux gearset can be made to do. Every distinct ratio the gearset offers, as a reduction. Each one is the same mechanism with one more constraint imposed — a brake holding a member to the case or a clutch locking two members together — and the input is a choice as well, which is why a four-speed needs an input clutch rather than four brakes. The shaded rows are the 5 a real transmission on this gearset is sold with; the rest are ratios the mechanism has and the gearbox does not buy the elements to reach. Every value is exact: the reductions are ratios of integers and are printed as such.

Four speeds from two numbers

A Ravigneaux gearset has five tooth counts and gives seven exact ratios. Two of the counts do not appear in any of them — the short planet is an idler and its size is free — and the remaining three enter only through two dimensionless numbers, so the whole set of shift ratios of a four-speed automatic is a function of ring-over-sun and ring-over-the-other-sun. A Simpson three-speed is a function of one number.

transmission · Transmission Six gearsets, three conditions. Every one of these can be drawn, and five of the six are drawn in some textbook or other. The columns are the three conditions a planetary has to satisfy: that a whole planet fits between the sun and the ring, that the sun and ring teeth add to a multiple of the planet count, and that the planets clear each other. The last column is how far out of mesh the worst planet station is, in teeth — a quantity that is zero or is not, and that no drawing shows, because a drawing of a planetary at this scale draws circles.

The gearset that could not be assembled

A planetary drawing shows a sun, a ring and three or four planets between them, and if the circles are the right sizes at the right stations it looks right. The condition that decides whether the second planet can actually be dropped in is arithmetic — the sun and ring teeth must add to a multiple of the planet count — and it appears in no drawing, at no scale, in any style.

wrong · Misconception The steps, against the ones the rule asks for. The design rule everybody quotes is that the steps between gears should be equal in ratio, so that the engine returns to the same speed after every shift. That makes the sequence geometric, and the ideal step for this spread over this many gears is 1.5324, marked. The steps a gearset actually gives are not free: the whole sequence is a function of the tooth counts, so once the top and bottom are chosen there is nothing left to spend on the middle. The worst step here is off the ideal by 10.8%.

The steps are not free

A gearbox is supposed to have equal steps between its gears, so that the engine returns to the same speed after every shift. A gearset has one or two numbers to spend on three or four gears, so from the third one the steps are a consequence rather than a choice — and asking for them to be equal turns out to be a quadratic whose root is the golden ratio, realised in tooth counts by consecutive Fibonacci numbers.

transmission · Transmission One planet shaft, two centre distances. A compound epicyclic gets its enormous reduction from two meshes whose tooth counts are nearly in the same proportion. The two planet gears are on one shaft, so their axes are at one radius — and at a common module the two rings ask for radii that differ by 0.50 of a tooth. The exactness the reduction is famous for is bought with a pair of meshes running away from the centre distance they were cut at, and the difference is made up by profile shift — the same correction the teeth field applies for a different reason. It is not a rounding: it is the mechanism's own condition, and it is the reason a catalogue reduction of this kind comes in a short list of tooth counts rather than in any combination.

A hundred to one from a difference of one

A harmonic drive reduces by a hundred to one in a single stage with two gears in it, and the hundred is the flexspline's tooth count divided by the two teeth the circular spline has more than it. The same null space that answers a planetary answers it. What each of the three single-stage reductions pays for that arithmetic is different, and the compound epicyclic's price is a pair of meshes whose centre distances differ by half a tooth.

transmission · Transmission Everything one planetary can do, and the gap in the middle. A single epicyclic has three shafts, so there are six ways of choosing which is held, which is driven and which comes out. Each gives a band of reductions as the tooth counts run over every design that can be cut, assembled with three planets and kept clear of undercutting. The bands above 1 are drawn; between them is a gap running from 1.6304 to 2.5862 that no single planetary reaches in any configuration — and a reduction of exactly 2, which is the most ordinary thing anybody asks a gearbox for, is inside it. The gap's width as a factor is exactly the smallest achievable ring-over-sun ratio, 1.5862, which is a statement about how small a planet may be and how large a sun may be.

The reductions a planetary cannot give

One epicyclic offers six ratios, and the formula for each of them suggests the whole positive line is available. Sweep every design that can actually be cut and assembled and the reachable set has a hole in it running from 1.630 to 2.586 — the width of which has a closed form — and a reduction of exactly 2, the most ordinary thing anybody asks a gearbox for, sits in the middle of it.

transmission · Transmission A reduction made out of a difference. A compound epicyclic's reduction is 1/(1 − z₁z₄/z₂z₃) — a ratio of two integers, so it is exactly what a catalogue says, at every position, forever. It is the only number in this field that survives being measured, and the reason is that it is a count. What it is not is insensitive: each row's denominator is a difference of two nearly equal products, and the last two rows differ by one tooth on one ring.

A ratio that is a count

A compound epicyclic's reduction is a ratio of two integers, because it is a ratio of tooth counts — 2176/106 for the drive here, exact at every position and on every unit ever made. It is the only quoted number in this field that survives being measured, and it is fragile in a way exactness does not protect against: one tooth on one ring moves it by forty per cent.

applied · Tooth A variator that spans under four to one, driving a machine that spans everything. The whole machine's ratio against the variator's own, for an engine split between a variator and a straight path and summed by a planetary with 50 sun teeth and 70 ring teeth. The variator's travel runs from 0.544 to 2.045, a span of 3.76 to one, and the output's ratio is (1 + K) / (K − v) with K = 1.4: a hyperbola with a pole at v = 1.4. Left of the pole the output turns one way, right of it the other, and the vertical scale is cut at ±12 because nothing else would fit. The measured extremes over the travel are 2.80 and -3.72, and between them the ratio passes through every value there is.

A bounded ratio made unbounded

Split an engine between a variator and a straight path and add the two with a planetary, and a variator that spans 3.76 to one becomes a machine whose ratio passes through infinity. The setting at which the output stands still is the planetary's own tooth ratio and nothing the belt does moves it. The price is a sensitivity that grows as the reciprocal square of the distance to that setting, and the variator's one and a half per cent becomes a hundred and ten before the setting is reached.

transmission · Transmission Sliding the variator's travel across the pole. The variator of the power split — travel 0.544 to 2.045 — geared by a fixed ratio k ahead of a planetary with K = 1.4, so the planetary sees k times the variator's ratio. At each k the travel is trimmed wherever the output's ratio is uncertain by more than 10%, and what is left gives a forward span and a reverse span. Solid lines use the variator's own tolerance, which grows from 0.98% at one end of its travel to 2.24% at the other; dashed lines use one tolerance of 1.6% everywhere. With one tolerance the forward span peaks sharply, at 5.59 where the travel's top meets the trim, and falls as the pole moves into the travel. With the variator's own it peaks at 4.81 at k = 1.00 and stays within a tenth of that from k = 0.6 to 1.2, while the reverse span climbs from one, meeting the forward span at k = 1.26.

Sliding the travel across the pole

A power split's ratio has a pole the variator's own tolerance makes unusable, so the question is where to put the variator's travel relative to it. With a tolerance that is one number, the best forward span comes where the travel's top just meets the trim — 1 + τ(1 − r)/p, with no gearset in it. With the variator's real tolerance, which grows along its travel, that peak flattens into a plateau: the pole can be moved well inside the travel, buying reverse, for under a tenth of the forward span.

transmission · Transmission 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 A ball bearing turned by its inner race. A bearing of pitch diameter 40 with 8 balls of diameter 8, its outer race held and its inner race turned 60°. Rolling without slipping at both contacts leaves the cage — the ring that keeps the balls apart, marked by the dark tick — turned 24.00°, a share of 0.4000 of the race, and each ball spun 120.0° backwards about its own centre, marked by its own tick. The share is (1 − d/D)/2, less than a half by d/2D; the gear-train solver, handed a planetary with a sun of 32 teeth and a ring of 48, returns 2/5 for its carrier. Dragging turns the inner race.

A bearing is a planetary with no teeth

Roll a ball between two races and nothing but the two rolling constraints decides how fast its centre goes round. Solved, they put the cage at (1 − d/D)/2 of the inner race's speed — always less than half — and the gear field's train solver, handed a planetary with a sun of D − d teeth and a ring of D + d, returns the same fraction exactly. And a tapered roller rolls without slipping along its whole line only if its axis meets the bearing's at the apex, the wheel field's concurrency one dimension up.

rolling · Rolling What each branch carries, against what the engine delivers. The power in the variator's branch and in the straight path, as multiples of the engine's own, for a planetary of K = 1.40. Both are ratios of powers, so the load cancels and nothing here is a force: the split comes from requiring the gearset to be lossless at every admissible set of speeds, which fixes the torques at 1 : K : −(1+K). The variator carries v/(K − v) and the straight path K/(K − v), and both run away at the pole. The variator first carries the engine's whole power at v = 0.700, which is exactly half the way to the pole — and the straight path is already carrying more than the engine everywhere past nought, flowing the other way through the planetary. That excess is the circulation.

The power that goes round twice

Sliding a power split's travel towards its pole buys ratio span for nothing, on the kinematics. It is not for nothing. The variator's own branch carries v/(K−v) of the engine's power and overtakes it at exactly half the way to the pole, and the tolerance trim the span was computed from is not reached until the variator is rated for six times the engine — which no machine is.

transmission · Transmission Every power split, on one curve. 1,994 power splits: all six ways of putting an engine, a variator and an output on a simple planetary's three shafts, with the variator coupled at the input or at the output, turning either way, on nine gearsets, with its travel placed at eleven positions. Every variator spans 3.333, as the variator of the previous essay does. Each dot is one split: the largest share of the engine's power its variator carries anywhere on the travel, and the span its output covers. They fall on three curves and nowhere else, to 7e-16: with no circulation, span R/(R(1 − S) + S), never more than the variator's own; with the variator's branch running backwards, 1 + S(1 − 1/R); with the variator carrying more than the engine, 1 + S(R − 1). 1188, 394 and 412 splits lie on the three. Which curve a split is on is decided by where its travel sits, not by which shafts it uses: every one of the twelve attachments appears on all three.

Every power split is on one curve

There are twelve ways to attach an engine, a variator and an output to a planetary's three shafts, and asking which is best turns out to be the wrong question. On 1,994 splits across all twelve and nine gearsets, the output's span is (1 + R·a)/(1 + a) to 5 × 10⁻¹⁵, where R is the variator's own span and a is the power the variator carries over the power the gears carry. Every split lies on one of three curves of span against the variator's largest share. The previous essay's 1.70 is 2 − 1/R on the worst of them, and a split beats its variator's own span only by making the variator carry more than the engine.

transmission · Transmission A second planetary buys span at every variator share, if its lever can be built. For a variator spanning 3.333, the most output span a power split reaches when its variator carries at most a given share of the engine's power. On one planetary it is R/(R(1 − S) + S) (dashed). On a four-shaft gearset with the variator between two of its shafts — the compound split — it is the square of (√R + k)/(1 + k√R), k = (1 − S)/(1 + S) (solid): 1.124 against 1.075 at a share of 0.1, 1.802 against 1.538 at 0.5, and both reach R at 1. The dots are the best compound split on four real Simpson and Ravigneaux gearsets at each share: ahead up to 0.3 (1.320 against 1.266), behind from 0.4 (1.423 at 0.5).

A second planetary lets the variator carry a difference

On a single planetary every power split lies on one of three curves of span against the variator's share of the engine's power. Build the split on a four-shaft Simpson or Ravigneaux instead and nothing changes while the fourth shaft is free: 3,588 splits, all on the curves. The one new arrangement is the variator between two of the gearset's shafts. That compound split is an input-coupled split and an output-coupled split in series, and its variator carries only the difference of their shares — so at every share below one it can span more than any single planetary, the square of (√R + k)/(1 + k√R). It needs a lever with the variator's shafts exactly its share apart, which real gearsets do not have beyond a share of about a third.

transmission · Transmission The best split for a given variator rating. The most span a single-planetary power split can have when its variator may carry no more than the share on the axis, for variators spanning 2, 3.333 and 6. Up to a share of one the best is a split with no circulation, reaching the variator's own span exactly at one; beyond it, a split whose variator carries more than the engine, gaining R − 1 of span per unit of share. The dashed curves are the third branch, with the variator's power running backwards, which is below the best everywhere. The ring marks the previous essay's split at a unit rating: 1.700, which is 2 − 1/R. The same variator in a split with no circulation, its share never passing 0.9, spans 2.703, and it approaches 3.333 as that share approaches one.

Six things a power split is not

A multiplier of its variator's span, a machine whose neutral the belt decides, a source of any ratio near that neutral, a travel that slides towards its pole for free, a design whose attachment is the choice that matters, and a machine a second planetary always improves. Six claims about continuously variable transmissions built as power splits, each of them what a careful person would say, each answered with a measured number.

wrong · Misconception A compound split sees its gearset through one cross-ratio. The four shafts of the Ravigneaux 69/74/100, planets 13 and 13 on its lever, each at the position that makes every shaft's speed a straight-line function of it: sun2 at 0.000 (engine), carrier at 0.483 (variator), ring at 0.840 (variator), sun1 at 1.000 (output). The best compound split puts the variator between the two shafts shaded together and the engine and output on the other two. What the split can do depends only on the cross-ratio of the four positions, here 5.613, and a split with that cross-ratio beats every split on a single planetary up to a variator share of 0.542. A share of a third needs a cross-ratio of 2.52 and a half needs 4.67.

One cross-ratio decides the compound split

A compound power split, its variator between two shafts of a four-shaft gearset, can beat every split on a single planetary, but four real gearsets let it do so only up to a variator share of about a third. Whether teeth can do better turns on one number: the cross-ratio of the gearset's four shaft positions on its lever. A third needs 2.52 and a half 4.67. Every Simpson inside a ring of 100 teeth stops at 3.87, a share of 0.454. A Ravigneaux with large suns and 13-tooth planets reaches 5.61, a share of 0.542, and solved on its lever it passes a half. With planets of 17 teeth, neither family does.

transmission · Transmission Two planetaries joined at two places make one four-shaft lever. The sun–carrier, carrier–ring join: planetary 1, a sun of 26 in a ring of 52 (ratio 2.00), and planetary 2, a sun of 41 in a ring of 71 (ratio 1.73), each on its own lever with sun, carrier and ring at 0, K and K + 1, laid so that the joined members coincide. The four shafts below are the two joints and the free member of each. Their cross-ratio is 4.732, enough for a compound split to beat every split on one planetary up to a variator share of a half; its best split there spans 1.544, with the engine, output and variator where marked, and no member of either planetary carries more than 0.80 of the engine's power.

The compound split has gearsets of its own

A compound power split depends on its gearset through one number, the cross-ratio of the four shafts on its lever, and a variator share of a half needs 4.67. Simpsons stop at 3.87 inside a ring of 100 and Ravigneaux reach it only with 13-tooth planets. Two simple planetaries can be joined into four shafts in eighteen ways, twelve of them distinct gearsets, and ten of the twelve reach a half with ordinary ring-to-sun ratios. The cross-ratio is then no longer the question. Three joins reach any cross-ratio by setting two shafts nearly together and pay with 2.7 to 4.2 times the engine's power inside, and one, carrier to ring both ways, never reaches a half. A sun-to-carrier, carrier-to-ring join builds the lever with planetaries of 26/52 and 41/71, and nothing inside it carries more than 0.80 of the engine's power.

transmission · Transmission Each fixed gear is a straight speed line through a braked shaft. The sun–carrier, carrier–ring join of a 26-tooth sun in a 52-tooth ring and a 41-tooth sun in a 71-tooth ring, its four shafts at their lever positions: sun₂ at 0.000, sun₁ + carrier₂ at 0.536, carrier₁ + ring₂ at 0.845, ring₁ at 1.000. The output stays where the compound split has it, on sun₁ + carrier₂. In the gear shown the engine drives carrier₁ + ring₂ and sun₂ is braked, so the shafts' speeds lie on a straight line through nought at the brake and one at the engine; the output turns at 0.634 of the engine's speed, a ratio of 1.58.

The winning join is a gearbox too

The sun–carrier, carrier–ring join of two simple planetaries builds a compound power split's half-share lever with ordinary teeth. Locked out of the split and given clutches and brakes, the same four shafts give five forward ratios below the split's range, from 2.73 to direct drive, and an overdrive of 3 above it. The two gears that brake a variator shaft sit 1.76 and 1.74 outside the split's ends, and three gearsets with the same cross-ratio put them at the same steps. Its planets space evenly, three and four to a set, and its meshes lose 0.41% of the engine's power along the split where the Ravigneaux that builds the same lever loses 1.01%. What none of these gearsets gives, with the output where the split needs it, is a usable reverse.

transmission · Transmission

Named alongside it

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

Design ruleRatioGear trainVelocity ratioTransmission relationVariatorContinuously variableMobilitySensitivityEnumerationMeshNull space

All concepts