The lever that is the gearset

There is a diagram every automatic-transmission course draws. The members of a planetary gearset are marked as points on a horizontal line, spaced according to their tooth counts; a second, sloping line is drawn across them; and the height of the sloping line above each point is that member’s speed. Hold a member and the sloping line is pinned to the axis there. Drive another and the line tilts about the pin. Read the third off the picture.

It is presented, almost always, as a mnemonic — a way of getting the signs right without the algebra, useful because planetary signs are notoriously easy to get wrong. It is not a mnemonic. It is exact, it is exact for a reason that can be stated in two sentences, and the reason also says which gear trains have such a diagram and which do not.

The motion that makes it work #

Take a gearset and give every member the same speed, the frame excepted. Ask each mesh whether it is happy.

It is. Every mesh relation on this site is written as a statement about differences — $z_a(\omega_a - \omega_c) + \sigma z_b(\omega_b - \omega_c) = 0$ — so if all three of $\omega_a$, $\omega_b$ and $\omega_c$ are equal, both terms vanish. Nothing is straining. The whole assembly rotates bodily inside its case, teeth in mesh, nothing sliding.

That is a real motion of the mechanism, and it is what a gearbox is doing in direct drive. It is also the key to the whole diagram, because a gearset has a two-dimensional space of permitted motions and that block rotation is one particular direction in it. Pick any second motion independent of that one and every motion is a combination:

$$\omega_m \;=\; a\cdot 1 \;+\; b\cdot x_m$$

where $x_m$ is the $m$-th member’s coordinate in the second motion and $a$, $b$ range over the two freedoms. That expression is the equation of a straight line of height $a + bx$ over a position $x$. Take the member’s lever position to be $x_m$ and the diagram is not an analogy at all: it is the statement that the motion space is two-dimensional and that one of its directions is the rigid one.

Where the positions come from #

The coordinates are not chosen; they fall out. For a simple planetary with sun $z_S$ and ring $z_R$, take the second motion to be “sun held, ring turning at 1”. Then the constraint

$$z_S\omega_S + z_R\omega_R = (z_S + z_R)\,\omega_C$$

gives $\omega_C = z_R/(z_S+z_R)$, so with the sun at 0 and the ring at 1 the carrier sits at

$$x_C \;=\; \frac{z_R}{z_S + z_R}.$$

For 24 and 72 that is 0.75. The lever’s proportions are the tooth counts and nothing else, which is why the diagram can be drawn from the parts list before anything is assembled.

The planet gets a coordinate too, and it is 1.5 — outside the segment. That is correct and worth keeping in the picture rather than tidying away: a planet is a member of the train and is not a shaft anybody can reach, so it lives off the end of the lever, spinning faster than anything else and connected to nothing outside the case. The library distinguishes the two — a train declares which of its members are shafts — and the lever is normalised between the extreme shafts, so where the planets land is a result rather than a convention.

What the ordering settles #

Here is the part that turns a picture into a theorem. On the simple planetary’s lever the carrier is between the sun and the ring, always, because $0 < z_R/(z_S+z_R) < 1$ for any positive tooth counts. Three consequences follow with no algebra at all:

• Hold the ring and drive the sun. The line is pinned to zero at $x=1$ and lifted at $x=0$. The carrier is between them, so its height is between 0 and the sun’s, and the same sign. A planetary driven at its sun with its ring held is therefore always a reduction, and never a reversal. For 24/72 it is exactly 4.
• Hold the sun and drive the ring. Mirror image: the carrier is again between, again the same sign, again slower. Reduction 4/3.
• Hold the carrier. Now the pin is between the other two, so the sun and the ring are on opposite sides of it and have opposite signs. That is the reason the carrier-held cases run backwards, and it is a statement about betweenness rather than about a minus sign in a formula. For 24/72 it is −3 one way and −1/3 the other.

The sign rule that gets memorised — carrier held means reverse — is the observation that a point strictly inside a segment has the two ends on opposite sides of it. Nothing else is going on.

Bigger levers #

The construction does not care how many members there are. A Simpson gearset — two planetaries sharing one sun, with the front carrier joined to the rear ring — has four shafts, and its lever has four stations:

and a Ravigneaux, with two suns of different sizes, has its four in a different order:

Read the four-speed off it. Pin the line at the ring and lift it at the large sun: the carrier comes out at a fraction of the sun, and the reduction is 2.48. Pin it at the small sun instead and the reduction is 1.46. Lock two members and the line is horizontal — direct drive, 1.00. Pin it at the small sun and lift the ring: now the carrier is outside the segment between them, so it moves faster than the input, and the ratio is 0.689, an overdrive. Four gears, one line, four places to put the pin.

And the reverse: pin the line at the ring and lift the small sun, which sits at 1 while the carrier sits at 0.3151 and the ring at 0.5280. The carrier is now on the far side of the pin from the input, so it runs backwards. −2.2174, and again the sign is a betweenness fact.

Reading a gearbox off one line #

The lever is at its most useful when a gearset has more members than anybody can keep in their head, and the four-station levers above are already at that point. So it is worth doing one gearbox completely, as a reading exercise rather than a calculation.

Take the Simpson gearset — sun at 0, rear carrier at 0.4695, output at 0.6852, front ring at 1 — and ask for its three forward gears and its reverse.

First. Drive the front ring, at position 1. Brake the rear carrier, at 0.4695. The line is pinned to zero there and lifted at 1, so the output at 0.6852 has height proportional to $0.6852 - 0.4695 = 0.2157$ while the input’s is $1 - 0.4695 = 0.5305$. Reduction $0.5305/0.2157 = 2.4595$.

Second. Same input, brake the sun at 0. Now the output’s height is proportional to 0.6852 and the input’s to 1, so the reduction is $1/0.6852 = 1.4595$ — and the two gears differ by exactly one, which is the identity the closed forms make explicit.

Third. Any clutch. Horizontal line, everything at the same speed, 1.0000.

Reverse. Drive the sun at 0, brake the rear carrier at 0.4695. Now the output at 0.6852 is on the far side of the pin from the input, so its height has the opposite sign: $-(0.6852 - 0.4695)/0.4695 = -0.4595$, a reduction of −2.1765.

Four gears, four readings of one line, no algebra at any point. The reason this works — and the reason it is worth teaching — is that the two things a gearbox does, choosing a pin and choosing a lift, are exactly the two things a straight line has freedom to do.

The trains that have no lever #

A statement that applied to everything would say nothing about the things it applies to, so the second half of the check is the more important one.

An ordinary train — two pairs of spur gears on three shafts in the case — does not admit the block rotation. Give every shaft a speed of 1 and the first mesh’s relation reads $z_a + z_b \ne 0$: the gears are being asked to roll on each other while turning the same way, which they cannot. Its mesh residual is 60 rather than 0, and it has no lever. That is not a defect; it is that an ordinary train has mobility 1, one ratio, and nothing for a two-dimensional diagram to describe.

A compound epicyclic is the interesting refusal, because it looks like a gearset. It has a carrier, planets that orbit, two rings — and one of those rings is bolted to the case. That single bolt means the frame carries teeth, the block rotation is refused, and the mobility is 1. There is no lever for a Wolfrom drive, and the reason is not that it is too complicated: it is that fixing a ring to the case is already one of the two constraints, so there is no plane left to draw.

The check assertTheLeverExistsExactlyForGearsets asserts both halves — a lever for the four gearsets and no lever for the two trains — and the second half is what stops the first from being a tautology.

Exact, and what “exact” is being claimed about #

The residual quoted at the top of these figures is zero, and the arithmetic behind it is rational: the members’ coordinates are fractions, the motions are fractions, and the difference between a member’s null-space speed and the line’s height at its position is $0/1$.

That distinction was worth building into the figures rather than glossing. The first version read the residual off the drawn coordinates, which are doubles, and reported $2.2\times10^{-16}$. Which is true, and is a statement about rounding a fraction to a double for the purpose of putting a dot on a canvas, and would have appeared in the caption looking exactly like a measurement of the gearset. It is not: the gearset’s residual is zero and has no unit of doubt attached. The figures now compute the residual in rationals and report the double’s error separately, as a fact about the drawing.

This is a small instance of something this site keeps finding. A number that is small has to be traced to what it is small about before it goes in a caption, and “1e−16” is the most convincing wrong answer available, because it looks like a machine-precision confirmation of whatever claim it is put next to.

Why the positions are what they are #

One question the diagram invites and rarely answers: why is the carrier at $z_R/(z_S + z_R)$ rather than at the midpoint, or at the planet’s orbit radius, or anywhere else that would be geometrically natural?

Because the lever’s coordinate is not a length. It is a coefficient in a motion, and its resemblance to a physical position is a coincidence of the simple planetary being nearly symmetric. The Ravigneaux makes that plain: its two planet species land at $-0.210$ and $+1.628$, one of them at a negative coordinate, and neither is anywhere near where the planet physically sits. A negative coordinate means only that the member turns the other way when the datum member is held, which is exactly what a short planet driven through a long one does.

So a lever diagram is a picture of the null space, drawn on a line because the null space is two-dimensional and one of its directions has been used up by the rigid motion. The line has no units on it and the spacing has no length in it, and treating it as a layout drawing — as though a member nearer the sun were physically nearer the sun — is the one way to be misled by it.

The members outside the segment #

The planet’s coordinate came out at 1.5, outside the span from the sun at 0 to the ring at 1, and that is worth more than a parenthesis because it is a statement about the mechanism rather than about the drawing.

The lever’s rule is that every member’s speed is the value of one straight line at that member’s coordinate. A member inside the span between two others therefore always has a speed between theirs. A member outside the span never does: whatever the line’s slope, its value at 1.5 lies beyond its values at 0 and 1, on one side or the other.

So the planet is always at an extreme. It is never the intermediate speed of the set; in every gear, in every configuration, it is turning either faster than every other member or slower — and since the line’s slope changes sign between gears, it swaps ends. The single exception is direct drive, where the line is horizontal and everything including the planet turns together, which is the block rotation the lever is built out of.

That is the kinematic reason planet bearings are the hard part of an epicyclic. Not that planets are small, though they are; that the geometry guarantees the planet carries the extreme speed of the whole gearset in every gear but one. A designer reading the lever can see it in the position of the coordinate, without computing a single speed, and the same reading applies to any member whose coordinate falls outside the span of the ones that carry shafts.

It also gives a clean way to see the result the enumeration reaches by counting. Lock any two members and the line is pinned to zero at two distinct coordinates; a straight line through two zeros is the zero line; every member’s speed is nought, including members outside the span. The gearset is solid. That is the eighteen-of-twenty-seven collapse read off a picture in one sentence, and it needed no null space at all.

What the lever is not #

Two boundaries, since the diagram is seductive.

It is not a free-body diagram. The lever’s positions are kinematic, but the analogy is often extended to torques — a member’s torque as a force applied at its station, with the whole thing balancing like a beam. That extension is true and it is dynamics; it needs the assumption that the gearset is lossless, which is a statement about friction, and it is outside this site entirely. Nothing here balances any moments and nothing needs to.

It does not say which gears the gearbox has. The lever shows every line that could be drawn. Which of them a transmission is built to reach depends on how many brakes and clutches are fitted and where, which is a separate enumeration and is the next essay’s subject. A gearset’s lever with four stations offers more gears than any four-speed uses, and the leftovers are exact ratios that nobody bought the parts for.

The lever’s real value is that it makes the field’s premise visible in one picture. There is no ratio in a gearset. There is a line with two degrees of freedom, and every gear anybody has ever driven in is that line held down somewhere.

Reading the coordinate as a sensitivity #

There is one more thing the positions carry, and it is the reason a designer looks at the lever rather than at the ratio table when deciding what to change.

The line’s value at a member is that member’s speed, so the spacing between two members is how much a change in one moves the other. Two members close together on the lever are nearly locked to one another: pinning one and lifting the other requires a steep line, and everything far away swings wildly. Two members far apart are loosely coupled, and a large change at one end produces a modest change at the other.

That makes the coordinates a picture of the gearset’s sensitivities as well as its ratios, and it explains why a Simpson’s output at 0.6852 and its front ring at 1 give the ladder they do. The gears whose input and held member are close together on the lever are the extreme ratios; the ones whose input and held member are far apart are the moderate ones. A designer wanting to move one gear without moving the others is asking for a coordinate to shift, and a coordinate is a function of tooth counts — so the lever turns which gear needs to change into which member needs to move, and in which direction, which is a question about two integers rather than about a ladder.

None of that is available from the ratio table, where the seven expressions all contain both design numbers and nothing says which is doing what. It is the ordinary advantage of a geometric representation over an algebraic one: the algebra is exact and the picture is where the derivatives live.