Machine Kinematics
solved, then drawn
More than one input

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.

Assumes One cross-ratio decides the compound split and The lever that is the gearset.

15 min read 6 figures Two routes to the same numberSomebody had a problemExact and approximate

One cross-ratio decides the compound split reduced a whole question to one number. A compound power split puts its variator between two shafts of a four-shaft gearset and its engine and output on the other two. Whether it can beat every split on a single planetary at a given variator share depends on the gearset only through the cross-ratio of the four shafts’ positions on its lever, Ω\Omega. A share of a third needs 2.52 and a share of a half needs 4.67. The two named families fell short of the second, or reached it only at the edge of what can be built. A Simpson stops at 3.87 inside a ring of 100 teeth. A Ravigneaux reaches 5.61 with planets of 13 teeth, the smallest the census allowed, and with planets of 17 it does not reach a half at all.

That essay ended by turning the question round. A cross-ratio is one number and it can be asked of any gearset. Simpson and Ravigneaux are two ways of building a four-shaft gearset, not the only two, and two simple planetaries can be joined into four shafts in several other ways. The question was whether one of them builds the compound split’s lever with ordinary teeth, and what it then costs inside.

Many do. And once many do, the cross-ratio stops deciding anything. The power that circulates inside the gearset decides it.

Two planetaries joined at two places make one four-shaft leverThe 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.sun₁carrier₁ring₁sun₂carrier₂ring₂outputenginevariatorvariatorsun–carrier, carrier–ringcross-ratio 4.73
Fig. 1 Two simple planetaries, each drawn on its own lever with sun, carrier and ring, laid so that their joined members coincide, and the four shafts of the gearset they make with the compound split’s engine, output and variator marked. The slider steps through every join that builds the lever a variator share of a half needs.

Eighteen ways to make four shafts

A simple planetary has three members, sun, carrier and ring. Two of them have six. Join two members of the first to two members of the second, each to one, and six members become four shafts: the two joints, and the member of each planetary left free. There are three ways to choose the first planetary’s two members, three to choose the second’s, and two ways to pair them, eighteen joins in all.

Some of the eighteen are the same gearset described twice. Swapping which planetary is called first turns a sun-to-carrier, carrier-to-ring join into a carrier-to-sun, ring-to-carrier one, which is the same machine. Six of the eighteen are their own mirror, so the eighteen make twelve distinct gearsets. The Simpson is one of them: sun joined to sun, and the first carrier to the second ring. The join of carrier to ring and ring to carrier is the one some automatic transmissions use under other names. Ravigneaux is not among them, since its planets mesh with each other and it is not two simple planetaries at all.

Each join is a two-freedom gearset, so it lies on a lever. A ratio is a null space found why: a gearset’s motions are the null space of its mesh equations, and a two-freedom gearset’s null space is a plane that always contains the motion in which everything turns together. The lever that is the gearset found that the lever is exact rather than a mnemonic. A simple planetary with ring-to-sun ratio KK lies on a lever with sun, carrier and ring at 0, KK and K+1K + 1. Joining two members of the second planetary to two of the first fixes the stretch and shift that lays its lever on the first’s. So each join’s four shaft positions, and their cross-ratio, are a closed function of the two ratios K1K_1 and K2K_2.

That construction was checked against the site’s general one, which builds the two planetaries as a gear train with their joined members as one link, finds the train’s motions in exact rational arithmetic and reads the lever from them. On every one of the eighteen joins and three pairs of tooth counts the two agree to 10−910^{-9}. Each join’s mirror, with the planetaries swapped, has the same cross-ratio. And the Simpson join with one sun shared by both planetaries, searched over every ring to 100, finds exactly the earlier essay’s best of 3.87.

What each join can reach

The census took every buildable simple planetary — a sun and planets of 13 teeth or more inside a ring of at most 100, ring-to-sun ratios from 1.35 to 7.6 — and every pair of them, 636,804 pairs for each join.

What each join can reach, and what it reaches with ordinary teeth. For each of the twelve distinct ways to join two simple planetaries, the largest cross-ratio of its four shafts over every pair of buildable planetaries (sun and planets of 13 teeth or more, rings to 100), as a bar, and the largest with both ring-to-sun ratios between 1.5 and 4.5, as a dot; the dashed lines are the 2.52 a third needs and the 4.67 a half needs. carrier–carrier, ring–ring: 7821, ordinary 7426; sun–sun, carrier–carrier: 5610, ordinary 4662; sun–sun, ring–ring: 3267, ordinary 2804; sun–carrier, carrier–sun: 74.22, ordinary 30.25; sun–ring, carrier–sun: 66.61, ordinary 25.75; sun–ring, ring–sun: 57.99, ordinary 20.25; sun–carrier, carrier–ring: 14.99, ordinary 9.17; sun–carrier, ring–ring: 14.25, ordinary 8.50; sun–ring, ring–carrier: 7.73, ordinary 4.68; sun–ring, carrier–carrier: 7.62, ordinary 4.50; sun–sun, carrier–ring: 7.38, ordinary 4.67; carrier–ring, ring–carrier: 4.57, ordinary 3.02.
Fig. 2 For each of the twelve distinct joins of two simple planetaries, the largest cross-ratio of its four shafts over every pair of buildable planetaries, as a bar on a logarithmic scale, and the largest with both ring-to-sun ratios between 1.5 and 4.5, as a dot, against the cross-ratios a third and a half need.

Eleven of the twelve can reach a half somewhere, and ten reach it with both ring-to-sun ratios between 1.5 and 4.5, which is the range taken here as ordinary proportions, though two of the ten only at the box’s corner. The Simpson’s own join is one of the eleven. Its limit of 3.87 came from the common sun: one sun gear meshing both planet sets ties the two ratios together, K1=r1/sK_1 = r_1/s and K2=r2/sK_2 = r_2/s. With two suns on one shaft, which is a joint like any other, the same join reaches 7.38. It reaches a half with ordinary teeth only at the corner of the box, ratios of 4.5 and 1.5, with a sun of 16 in a ring of 72 and a sun of 52 in a ring of 78.

The one join that cannot reach a half at all is carrier to ring and ring to carrier, which tops out at 4.57 over every buildable pair. One other cannot do it inside the ordinary box: sun to ring with carrier to carrier, whose best ordinary lever is 4.50. Sun to ring with ring to carrier reaches it in only eight ordinary pairs, all with both ratios at 4.5.

At the other extreme, three joins reach cross-ratios in the thousands. All three join two like members, carrier to carrier with ring to ring, or sun to sun with carrier or ring. When the two planetaries’ ratios are nearly equal, their free members land nearly on top of each other on the lever. A cross-ratio is large when two of its points are nearly together, so these joins can be given any cross-ratio at all by bringing the two ratios closer. It is the arithmetic of a hundred to one from a difference of one, where two nearly equal tooth counts subtract to give an enormous reduction, read on a lever instead of in a ratio. The earlier essay found the Ravigneaux’s cross-ratio growing without bound as one of its ratios approaches one, and this is the same mechanism.

The cost the cross-ratio does not show

Every power split is on one curve found that a split’s span against its variator’s share is a fixed curve for each kind of split on one planetary, and a second planetary lets the variator carry a difference found the compound split’s frontier beyond those curves. The split’s span, as a function of its variator share, depends only on the cross-ratio, which is why the earlier essay could reduce everything to it. What happens inside the gearset depends on more. Four points on a line have two numbers’ worth of shape once their ends are fixed, and the cross-ratio is only one of them. The power each member of each planetary carries also depends on the planetaries’ own ratios, not only on where the shafts land.

Each member’s power follows from the lever. A simple planetary carries torques on sun, carrier and ring in the fixed ratio 1:−(1+K):K1 : -(1 + K) : K, whatever it is doing. The two free shafts belong to one planetary each, so the external torques on them fix each planetary’s torque scale, and every member’s torque and power follow. That was checked by the route it does not use: at every operating point of every split computed, the joined members’ torques add up to the lever’s torque at their joint to 10−910^{-9}.

Some joins reach the lever by circulating power, and some do not need to. For each join that has a lever reaching a share of a half, the design that does it with the least power in any member of either planetary, at its best compound split with the variator carrying at most half: that largest member power over the engine's, as a bar, hollow where the split does not in fact beat one planetary because its travel would pass through a standstill. sun–carrier, carrier–ring: 0.80 at a span of 1.544; sun–ring, carrier–sun: 1.00 at a span of 1.545; sun–ring, ring–sun: 1.00 at a span of 1.538; sun–carrier, ring–ring: 1.00 at a span of 1.538; sun–sun, carrier–ring: 1.00 at a span of 1.538; sun–sun, carrier–carrier: 2.69 at a span of 1.538; carrier–carrier, ring–ring: 2.69 at a span of 1.538; sun–ring, ring–carrier: 1.16 at a span of 1.444; sun–ring, carrier–carrier: 1.41 at a span of 1.444; sun–sun, ring–ring: 4.23 at a span of 1.446. The line at one is the engine's own power.
Fig. 3 For each join that has a lever reaching a share of a half, the design that builds it with the least power in any member of either planetary, at its best compound split with the variator carrying at most half: the largest member power as a multiple of the engine’s, hollow where the split’s travel would pass through a standstill.

For each join, the designs whose cross-ratio just reaches the half share were collected, those in ordinary proportions first. Each was given its best compound split, over every arrangement of engine, output and variator and every starting point of the variator’s travel, with the variator’s share held to at most a half. The largest power any member of either planetary carries was then found, as a multiple of the engine’s power, and the design with the least was kept.

The three joins that reach any cross-ratio are the three that pay for it. At their cheapest half-share design, carrier-to-carrier with ring-to-ring and sun-to-sun with carrier-to-carrier both put 2.69 times the engine’s power through some member, and sun-to-sun with ring-to-ring 4.23. Their nearly coincident shafts turn at nearly the same speed, so moving the variator between them means large torques, and large torques at ordinary speeds are circulating power. The power that goes round twice measured exactly this in the single-planetary splits, and here it appears inside the gearset.

Five joins build the lever with no member carrying more than the engine’s own power. Four of them reach exactly 1.00, the engine’s shaft itself, among them the Simpson’s join with its suns separated. One does better. The sun-to-carrier, carrier-to-ring join, with planetaries of a sun of 26 in a ring of 52 and a sun of 41 in a ring of 71, reaches a cross-ratio of 4.732. Its best split spans 1.544 against one planetary’s 1.538, and no member of either planetary carries more than 0.80 of the engine’s power. Every design of this join near the half-share lever that was tried came out at 0.80, so the figure belongs to the join at that cross-ratio rather than to one lucky choice of teeth. The engine’s power is divided between the two planetaries before any of it reaches a shaft that carries it all.

Three joins fail differently. Sun to ring with ring to carrier, sun to ring with carrier to carrier, and sun to sun with ring to ring have levers with enough cross-ratio, but every travel that would reach the frontier passes the output through a standstill or reverses the variator’s power partway. The best splits they can actually make span 1.44 to 1.45, short of what one planetary does, and the last of them circulates 4.23 times the engine’s power doing it. A twelfth join, sun to carrier with carrier to sun, has a cross-ratio so large on every buildable pair that no travel keeps its variator at or under a half at all. A cross-ratio is necessary and, on these two, not sufficient.

Where in the ratios the lever lives

The winner’s cross-ratio can be drawn over its two ratios, to see how much of the space reaches a half and how much room the design has.

Where the sun–carrier, carrier–ring join reaches the half share. The cross-ratio of the sun–carrier, carrier–ring join's four shafts over the two planetaries' ring-to-sun ratios, from 1.35 to 7.6 on each axis: shaded darkest where it is 10 or more, next where it reaches the half share's 4.67, palest where it reaches a third's 2.52, and blank below. 65% of the square reaches a half. The box is ordinary proportions, both ratios from 1.5 to 4.5, and the dot is the design that builds the half-share lever there with the least circulating power: ratios 2.00 and 1.73.
Fig. 4 The cross-ratio of the sun–carrier, carrier–ring join over both planetaries’ ring-to-sun ratios from 1.35 to 7.6, shaded where it reaches a third, a half, and ten, with the box of ordinary proportions and the design that builds the half-share lever with the least circulating power.

The half-share region is a broad band, not a corner. In the census 222,741 ordinary pairs reach 4.67 on this join, 65% of the whole square of ratios. The cheapest sits well inside the box, with ratios of 2.00 and 1.73. Those are ordinary proportions for automotive planetaries: a sun half its ring’s size and one a little larger, with planets of 13 and 15 teeth. Nothing about the design is pressed against a limit. The Ravigneaux that reached a half did so with 13-tooth planets and suns of 69 and 74 in a ring of 100, every number at the edge of what the census allowed.

The law the cross-ratio obeys

The share a lever allows comes from the earlier essay’s law, and nothing in this essay changes it.

The share a compound split can take grows with one number. The largest variator share at which a compound split beats every split on a single planetary, against the cross-ratio of its gearset's four shaft positions: the law R(1 − 1/F)/(R − 1), F = ((√(RΩ) + 1)/(√Ω + √R))², with the variator's span R = 1/0.3 (line), and the same found by sweeping the travel's start along each cross-ratio's ray and comparing its best span with the single planetary's at every share (dots), which agree to the search's grid of 0.005. A third needs 2.52 and a half 4.67. The ticks mark the six gearsets of the hero figure: Simpson 34/74/74 2.46; Simpson 30/78/66 2.64; Ravigneaux 36/30/78 2.57; Ravigneaux 34/28/76 2.48; Simpson 13/99/39 3.87; Ravigneaux 69/74/100, planets 13 and 13 5.61.
Fig. 5 The largest variator share at which a compound split beats every split on a single planetary, against the cross-ratio of its gearset’s four shafts, as the closed-form law and as found by sweeping each cross-ratio’s travel, with the gearsets of the earlier essay marked.

The law depends on the variator’s own span, taken throughout as 1/0.3, and nothing else. What the joins add is a much larger supply of cross-ratios than the two named families had, and with it the freedom to choose among levers that are equal under the law by what they cost inside. This is the pattern four speeds from two numbers found in a Ravigneaux’s ratios, a gearset whose external behaviour is set by a couple of numbers while its internal loads are set by more.

Where the winner puts its power

The roles in the winner’s best split are worth reading off. Its engine is on the carrier-to-ring joint and its output on the sun-to-carrier joint, and the variator runs between the two free members, the first planetary’s ring and the second’s sun, which sit at the two ends of the lever. So the engine’s power arrives at a shaft that belongs to both planetaries and is divided between them at once. Neither planetary ever carries the whole of it, which is why the largest member power is below one, and why no single member of this gearset has to be sized for the engine. Two inputs and one output found a planetary dividing its output’s power between two inputs in fixed proportions; here two planetaries do it to each other.

What the lever does not show is why other joins circulate. The cheapest carrier-to-carrier, ring-to-ring design also has its variator on the lever’s two ends and its engine and output between them, and yet a member of it carries 2.69 times the engine’s power. The positions on the lever are the same kind of positions. The difference is in how each planetary’s fixed torque ratio, 1:−(1+K):K1 : -(1 + K) : K, divides the torques at the joints, and that depends on the two ratios themselves. The cheapest carrier-to-carrier, ring-to-ring design has ratios of 1.77 and 4.37, not two nearly equal ones, so the circulation is not simply the near-coincidence that gives these joins their large cross-ratios. A rule that reads circulation off a join’s name and its lever, without working through the planetaries’ torques, was not found, and the member powers here are computed rather than predicted.

The twelve, side by side

Twelve joins of two planetaries. Every distinct join of two simple planetaries, named by the members joined (S sun, C carrier, R ring, the first planetary's letter first): the largest cross-ratio of its four shafts over all buildable teeth, how many pairs with ordinary proportions reach the half share, the design that builds the half-share lever with the least circulating power, its compound split's span with the variator carrying at most half, and its largest member power over the engine's. CC·RR: 7821, 403390, 44/78, 19/83, 1.538, 2.69; SS·CC: 5610, 233998, 22/82, 21/57, 1.538, 2.69; SS·RR: 3267, 175034, 28/78, 22/78, 1.446, 4.23; SC·CS: 74.22, 422500, —, —, —; SR·CS: 66.61, 422500, 52/78, 52/78, 1.545, 1.00; SR·RS: 57.99, 276499, 35/61, 31/83, 1.538, 1.00; SC·CR: 14.99, 222741, 26/52, 41/71, 1.544, 0.80; SC·RR: 14.25, 167471, 37/83, 59/93, 1.538, 1.00; SR·RC: 7.73, 8, 16/72, 16/72, 1.444, 1.16; SR·CC: 7.62, 0, 17/93, 22/90, 1.444, 1.41; SS·CR: 7.38, 1, 16/72, 52/78, 1.538, 1.00; CR·RC: 4.57, 0, —, —, —. One planetary at a half spans 1.538. The Simpson is SS·CR, and its common sun, 13/99 with 13/39, reaches only 3.87.
Fig. 6 Every distinct join of two simple planetaries: its largest cross-ratio over buildable teeth, how many ordinary pairs reach the half share, the tooth counts of its cheapest half-share design, that design’s best span with the variator carrying at most half, and its largest member power over the engine’s.

Read across, the ledger sorts the joins into five groups. One join cannot reach a half, and one overshoots so far that its variator never carries as little as a half. Three reach it on paper and cannot use it, because their travels pass through a standstill. Two reach anything and pay with 2.69 times the engine’s power circulating inside. The remaining five build the lever honestly, four with no member carrying more than the engine’s shaft does and one with every member carrying less. The Simpson, with its common sun, is in none of the useful groups. With its suns separated it joins the honest five.

What the joins do not settle

Losses. Every gearset here is lossless. The reductions a planetary cannot give bounded a single planetary’s ratios, and a join’s internal ratios are bounded the same way; the bound is respected by the census, but the efficiency it implies is not computed. Circulating power matters because it is power that pays mesh losses without being delivered. A member carrying 2.68 times the engine’s power loses roughly that multiple of a mesh’s percentage, so the ranking by member power is a ranking by losses, but the losses themselves are not computed.

Physical layout. A join on a lever is not yet a gearset in a housing. Two shafts joined across two planetaries must be able to reach each other, a sun joined to a carrier has to pass through or around the planetaries, and a clutch or brake for each shaft has to fit. None of that packaging is checked. It is the kind of constraint that has made a handful of joins standard in automatic transmissions.

Planet spacing. Each planetary is taken as buildable when its sun and planets have at least 13 teeth. Equal spacing of several planets, which needs the sum of sun and ring to divide by the planet count, and clearance between neighbouring planets are not applied. Many of the 222,741 ordinary pairs would fail one or the other. The cheapest design’s planetaries, 26/52 and 41/71, happen to pass, taking three or six planets and four, seven or eight, but nothing in the census chose them for it.

Speeds. Member power is torque times speed, and a member can carry little power while spinning fast. Planet speeds relative to their carriers, which set bearing life, are not computed.

Still open: a split built on the winner

The sun-to-carrier, carrier-to-ring join builds the lever with ordinary teeth and modest internal power, and so far it exists only as a lever and a census entry.

The distinct argument would be that join built out as a gearbox. Choose its tooth counts with equal planet spacing applied, put clutches and brakes on its four shafts, and ask which fixed ratios it gives when the variator is locked out. Holding a member chooses the ratio found how a single planetary’s ratios come from which member is held, and a four-shaft gearset has more members to hold. The measurement would also price its mesh losses at the compound split’s operating points with a stated loss per mesh, against a Ravigneaux at the same span. If the join gives a useful set of fixed ratios as well as the compound split, it is a gearset for a transmission that switches between a power split and fixed gears. If its fixed ratios are poor, it is a gearset for the split alone, and the Ravigneaux’s place in transmissions is explained by what it does in fixed gears rather than in the split.

Where this goes next

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. 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. 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.

What this makes readable

Essays that name this one as a prerequisite.

About the same objects

Not linked from either essay — found by the objects both name.

What links here

Essays that link to this one from their own argument.

The objects this essay names

Each one links to every other essay that touches it.

Design ruleEnumerationEpicyclicGear trainRatioVariator