Machine Kinematics
solved, then drawn
Concept

Continuously variable — where it appears

A drive whose output-to-input proportion is a smooth function of one geometric setting rather than a member of a finite set. It replaces a finite set of ratios with an interval, and what it costs is a contact that has to transmit by friction rather than by teeth.

Named by 8 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 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 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

Named alongside it

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

Design ruleVariatorEpicyclicRatioGear trainSensitivityToleranceBelt driveEfficiencyVelocity ratioApproximationConditioning

All concepts