The winning join is a gearbox too

The compound split has gearsets of its own searched every way to join two simple planetaries into four shafts for one that builds a compound power split’s lever. The split puts its variator between two shafts of a four-shaft gearset and its engine and output on the other two, which a second planetary lets the variator carry a difference found lets it beat every split on one planetary. At a variator share of a half it needs the cross-ratio of the shafts’ lever positions to reach 4.67. Ten of the twelve distinct joins reach it with ordinary teeth. They were then sorted by the largest power any member carries, and one stood out. The sun–carrier, carrier–ring join of a sun of 26 in a ring of 52 and a sun of 41 in a ring of 71 builds the lever with no member carrying more than 0.80 of the engine’s power.

It existed only as a lever and an entry in a census. The essay ended by asking for it to be built out: planets equally spaced, clutches and brakes on its four shafts, the fixed ratios it gives with the variator locked out, and its mesh losses priced against a Ravigneaux’s. If the join gives a useful set of fixed gears as well as the split, it is the gearset for a transmission that switches between the two. If not, the Ravigneaux’s place in transmissions belongs to its fixed gears rather than its split.

The answer is mostly the first, with one gap that every gearset here shares.

A fixed gear is a straight line through a brake #

The lever that is the gearset found that a two-freedom gearset’s shafts lie on a lever, with every shaft’s speed a linear function of its position. Two numbers fix every speed: the speed of one shaft and the speed of another. Brake one shaft and the line passes through nought there. Drive another at the engine’s speed and the line is fixed, and the output’s speed is read off where the output sits. So every fixed gear has a ratio that is a ratio of distances on the lever:

$$\frac{\omega_\text{engine}}{\omega_\text{output}} = \frac{x_e - x_b}{x_o - x_b}.$$

Holding a member chooses the ratio found this for one planetary’s three members. A four-shaft gearset has four. With the winning join’s lever normalised to run from 0 to 1, the second planetary’s sun is at 0. The joint of the first planetary’s sun with the second’s carrier is at 0.536, the joint of the first’s carrier with the second’s ring at 0.845, and the first’s ring at 1. The compound split puts its output on the joint at 0.536 and its engine on the joint at 0.845, with the variator between the two free members at the ends.

For a transmission that switches between the split and fixed gears, the output must stay where the split has it, since it is one shaft driving the wheels. The engine can be clutched to any of the other three shafts, and any shaft that is neither engine nor output can be braked. That gives six fixed gears, and direct drive when any two shafts are clutched together. Each ratio was found twice, once from the lever and once by solving both planetaries’ own kinematic equations with the braked member held, and the two agree to $10^{-9}$ in every gear of every join.

Five forward ratios below the split, one above #

The split itself covers engine-to-output ratios from 0.898 down to 0.581, a span of 1.544. On its own scale it is an overdrive range, which a final drive would move to wherever a vehicle needs it. The fixed gears fall around it. With the engine on the joint at 0.845, as in the split, braking the second planetary’s sun gives 1.58 and braking the first’s ring gives 0.333. With the engine clutched to the second planetary’s sun, braking the joint at 0.845 gives 2.73 and braking the first’s ring gives 2.15. With the engine on the first planetary’s ring, braking the second’s sun gives 1.87. Together with direct drive the set of ratios reads 2.73, 2.15, 1.87, 1.58, 1.00, then the split from 0.898 to 0.581, then 0.333.

That is a usable set of ratios. Its steps are 1.27, 1.15, 1.18 and 1.58 below direct drive, an 11% step from direct into the split’s lowest ratio, then the split’s continuous span and a final step of 1.74 to the overdrive. A vehicle could launch in the lowest fixed gear, move up through the others, and change to the split once it was moving. The split would then provide the continuously variable range a cruising engine wants, with the overdrive available for a light load at speed. The overall spread from 2.73 to 0.333 is 8.2. The one awkward step is 1.58, from the fourth gear to direct, and it lies just below the split’s range rather than inside it.

The variator’s own shafts, braked #

Two of the six gears are special, because they need no clutch beyond those the split already has. With the engine and output where the split puts them, braking either of the variator’s two shafts gives a gear: 1.58 with the second planetary’s sun held, 0.333 with the first’s ring held. Braking a variator shaft is the split run on past the end of its travel. The variator’s ratio has gone to nought, or to infinity, and the gearset delivers whatever the lever does at that limit.

So these two gears sit just beyond the split’s two ends, and the split’s own closed form says where. One cross-ratio decides the compound split wrote the split’s output ratio as $C(1 + Av)/(1 + Bv)$ in the variator’s ratio $v$. Run to $v = 0$ and $v \to \infty$ it gives the braked gears, and the steps from the travel’s ends to them are

$$\frac{1 + Av_0}{1 + Bv_0} \quad\text{and}\quad \frac{B}{A}\cdot\frac{1 + ARv_0}{1 + BRv_0},$$

with $v_0$ the travel’s start and $R$ the variator’s span. Both depend on the lever only through $B/A$, which is the cross-ratio. On the winning join the steps are 1.76 below the split and 1.74 above, and the formula and the braked lever agree to a millionth.

The figure’s other gearsets show what that dependence means. Simpson’s join with its suns apart and the carrier–carrier, ring–ring join both build the same half-share lever, with cross-ratios of 4.67, and both put their braked gears at 1.74 and 1.75 outside their splits. Their levers are quite different, and their other fixed gears are different too: the carrier-and-ring join’s lowest gear is 5.37. Only the braked variator gears are the same. The Ravigneaux builds a lever of larger cross-ratio and spans 1.612 at a share of 0.41, and its braked gears sit at 1.86 and 1.87. So a designer who chooses a compound split’s cross-ratio has also chosen, without asking, the two gears its variator’s brakes will give.

Planets that space evenly #

The census that found the winner checked only that every sun and planet had at least 13 teeth. A real planetary carries several planets, equally spaced, and they have to fit. Equal spacing needs the sun’s and ring’s tooth counts to sum to a multiple of the number of planets, so that each planet meets both in the same phase. Neighbouring planets also have to clear each other’s tips.

The winner’s planetaries pass easily. The first, a sun of 26 in a ring of 52 with planets of 13, sums to 78 and takes three or six planets with room between their tips. The second, a sun of 41 in a ring of 71 with planets of 15, sums to 112 and takes four, seven or eight. Simpson’s join with its suns apart takes four and five. The carrier–carrier, ring–ring join does worse: its first planetary, 44 in 78, sums to 122 and takes only two, which is a planetary that balances poorly and shares its load between two meshes. The earlier essay’s census had no way to see that, and among the half-share designs it is a mark against that join beyond its circulating power.

The Ravigneaux’s assembly condition is more involved, since its short and long planets must be placed together round the carrier, and it is not checked here.

What the meshes lose #

A mesh loses a share of the power it passes, and the power that matters is the power it passes as seen from its own carrier. A planetary whose members all turn together has meshes that pass no power at all, whatever its shafts carry. A planetary turning with its carrier held is an ordinary gear train, and its meshes pass everything. In between, a simple planetary’s two meshes both pass $|T_\text{sun}(\omega_\text{sun} - \omega_\text{carrier})|$, its rolling power. With 1% lost in each mesh, a figure typical of a good gear mesh, each planetary loses 2% of its rolling power. A Ravigneaux’s small sun drives the ring through two planets in series, three meshes, and its large sun through one, two meshes, with the long planet’s mesh with the ring carrying both. Its rolling powers were checked against the balance they must satisfy: in each carrier’s frame the members’ torques times their relative speeds sum to nought, to $10^{-9}$ at every operating point.

Along its split the winning join loses 0.41% of the engine’s power on average and 0.67% at worst. The Ravigneaux that builds the same lever, a small sun of 69 and a large one of 74 in a ring of 100 with 13-tooth short planets, loses 1.01% on average and 2.03% at worst. Its small sun’s ring-to-sun ratio is 1.45, close to one, and that is what gives it the cross-ratio. A double-planet set near a ratio of one carries nearly equal and opposite torques on its sun and ring for a small torque on its carrier, so its meshes pass a great deal of power without delivering it. The carrier–carrier, ring–ring join, which the power that goes round twice would recognise by the 2.69 times the engine’s power in one of its members, loses 0.70%.

The surprise is the Simpson’s join with its suns apart, which loses 0.42%, the winner’s figure to a hundredth, while one of its members carries the engine’s whole power. The earlier essay ranked the joins by the largest member power. That is the right measure of what a shaft or a bearing must be sized for. It is not a measure of loss. A member that carries the engine’s power while turning at nearly its carrier’s speed passes little through its meshes. The winner’s advantage of 0.80 against 1.00 is real for sizing and invisible in efficiency.

The figures are small against the variator’s. A variator of 90% efficiency carrying 37% of the power, the share at the winner’s best split, loses 3.7%. So the gearset’s choice moves the transmission’s total loss by about half a per cent out of four or five. That is worth having and it is not decisive, and it is the same proportion every power split is on one curve implied when it priced a split by its variator’s share: the variator’s loss is the large term, and the gearset’s is the correction.

In fixed gears #

In its four lower fixed gears the winning join loses between 0.73% and 1.43%, and the Ravigneaux between 0.85% and 1.55%, with no consistent winner. The two part at the extremes. In the overdrive the winner loses 1.33% and the Ravigneaux 6.68%. In reverse the winner loses 2.00% and the Ravigneaux 9.68%. Those are the gears that turn the Ravigneaux’s small sun hardest against its carrier. Direct drive loses nothing in either, since every member turns together and no mesh passes any power. That is why direct drive is the gear a fixed-ratio transmission spends most of its life in, and why the split’s own range sits just beyond it here.

The reverse no gearset gives #

The ledger’s one consistent failure is the reverse. With the output where the split needs it, the winner has one reverse gear. The engine drives the first planetary’s ring, at 1.0 on the lever, the joint at 0.845 is braked, and the output at 0.536 turns backwards at twice the engine’s speed. A reverse that fast is five and a half times faster than first gear, the opposite of what reversing a vehicle needs.

The lever says why, and it says it for every gearset at once. A reverse needs the brake between the engine and the output on the lever. Its ratio is the engine’s distance from the brake over the output’s, and the split has already put the output on an inner shaft with the engine beside it. The only shaft that can sit between engine and output is close to the engine, so every reverse is fast. Simpson’s join gives 0.56, the Ravigneaux 0.45, the carrier-and-ring join 1.46 against a first gear of 5.37. None comes near its own lowest forward gear.

Four speeds from two numbers found the Ravigneaux’s classic set of four forward gears and a reverse, with its output on its ring. Its own fixed gears here, with the output moved to its carrier for the split, are no better than the winner’s: a narrower spread from 2.35, and the same fast reverse. Moved the other way, with its output on the carrier-and-ring joint, the winner gives 6.46, 3.00, 1.50, 1.18 and 0.63 and a reverse of 1.73, as full a set as an automatic’s. That output is not the split’s. So the Ravigneaux’s place in transmissions is not explained by what it does in fixed gears under a split, where it is the weaker of the two. It belongs to its classic arrangement, which has nothing to do with the split. A transmission that both splits and shifts, on any of these gearsets, reverses some other way: a separate reverse gear, or, as hybrid drives do, by running the variator’s own machine backwards.

The ledger #

Read across, the winning join is the gearset the earlier essay hoped for. It splits with the lowest loss and with ordinary, evenly spaced planets, gives five speeds below its split and an overdrive above, and loses least in its extreme gears. Simpson’s join with its suns apart is close behind on every count but its extreme gears, where it loses 2.06% and 3.20%. Its planets space as well, and its member power is the engine’s, which matters for its bearings and not its efficiency. The carrier-and-ring join and the Ravigneaux each lose on two counts: the first on its planets and its circulation, the second on its losses at the split and in its extreme gears.

What this does not settle #

Clutch count and packaging. Six fixed gears need the engine clutched to three shafts and brakes on up to three, plus the split’s own variator connections. A real transmission would choose a subset, and whether the clutches fit round a sun-to-carrier joint and a carrier-to-ring joint is a question of housing, not of the lever. A ratio is a null space treats each held member as a constraint, and the clutches here are counted that way, not drawn.

Losses beyond the meshes. Bearing, churning and seal losses are not included, and the 1% a mesh is a stated round figure, not a measurement. The ranking depends on rolling power, so it survives any common figure per mesh; absolute efficiencies do not. The reductions a planetary cannot give bounded what one planetary’s teeth can do, and the bound on loss here is equally a statement about these teeth.

Shift quality. Changing from the fixed fourth gear to the split, or from the split to the overdrive, means handing the load from one clutch to another or to the variator. Whether that handover is smooth depends on the variator’s position at the change, and it is not modelled.

The Ravigneaux’s own planets. Its half-share design uses 13-tooth short planets with suns of 69 and 74, every number at the edge of the census. Its assembly with equally spaced planet pairs is not checked, and nor is the speed of those small planets.

Still open: a reverse that keeps the split’s output #

Every gearset here gives a reverse three to six times faster than its first gear when the output stays where the split needs it, and the lever explains why: the only shaft that can sit between engine and output is beside the engine.

The distinct argument would be a reverse that does not move the output. One way is a fifth shaft: a third simple planetary joined to two of the four, adding a node to the lever that can be placed between the output and the engine at any distance. Another is to reverse through the split itself, letting the variator’s travel run past one of its ends with its own direction reversed, as hybrid drives do with an electric machine. The measurement would take the winning join and find the third planetary, among ordinary teeth, whose added shaft gives a reverse equal to first gear while leaving the split’s lever and losses untouched. It would ask what the extra planetary costs in member power and mesh loss in the split, where it turns but carries no torque. If a reverse comes cheaply that way, the winning join is a complete gearbox with one more planetary. If the third planetary’s idle losses in the split eat the winner’s advantage over the Ravigneaux, the reverse is the price of the split.