Machine Kinematics
solved, then drawn
Concept

Variator — where it appears

The pair of split pulleys and the belt between them whose running radii set a stepless drive's proportion. Its two running radii are not independent: the belt has a fixed length, so one radius decides the other, and the ratio range follows from the geometry alone.

Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.

A ratio with no steps in it. Two pulleys whose sheaves slide on their shafts, and one belt. Pushing the primary's sheaves together makes the belt ride further out; the secondary's radius is then not a choice, because the belt is a fixed length and its length over two pulleys at a fixed centre distance is a function of both radii. So the secondary's radius here is the root of that equation, solved rather than assumed, and the drawn belt is the length it is supposed to be to 0.0e+0 mm. Ratio 1.000, with the two radii adding to 110.00 — a number the received rule of thumb says should not change and which changes by 2.6% across the travel.

A ratio with no steps in it

Push a variable pulley's sheaves together and the belt rides further out. The other pulley's radius is then not a choice — the belt has a fixed length — so it is the root of an equation, solved rather than set. The rule of thumb that says the two radii add to a constant is true to first order and wrong by 7.4% of the ratio at full shift, and the departure has a closed form.

transmission · Transmission What a belt ratio's tolerance is made of. The three ways a variator's ratio can be wrong when the parts are wrong, at a nominal ratio of 1.848: a belt 0.3% long, a centre distance 0.2 mm out, and a sheave 0.15 mm from where it was commanded. Worst case they add to 2.98% and combined in quadrature they are 2.04% — the two conventions the practice field already argues about. The last row is a gear train's ratio under the same treatment, and it is empty: there is no length in a tooth count for a tolerance to be on.

The ratio that has a tolerance

Every dimension in this collection has been given a range at some point, and the ratios never were, because a gear ratio is a count and a count has no tolerance. A belt ratio is a quotient of two solved lengths, so every length in the mechanism is in it — and the amplification from belt length to ratio runs from 2.7 to 6.4 across the travel, which puts a whole per cent on a mechanism whose gearbox equivalent has none at all.

practice · Tolerance 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 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 ruleEpicyclicRatioContinuously variableGear trainEfficiencySensitivityToleranceBelt driveEnumerationVelocity ratioApproximation

All concepts