If you’re familiar with transmissions like the Chrysler TorqueFlite and GM Turbo Hydra-Matic (among others), you may have heard of the “Simpson gearset.” In this installment of Ate Up With Motor, we look at the origins and function of the Simpson gearset and briefly introduce you to its inventor, the late Howard W. Simpson.

HOWARD W. SIMPSON

Born in 1892 in Kalamazoo, Michigan, Howard W. Simpson studied engineering at the University of Michigan and after World War I went to work for Henry Ford’s farm tractor company, the Henry Ford & Son (Fordson) Tractor Plant. Fordson became a division of the Ford Motor Company in 1920, but Simpson remained with Ford for almost two decades, working principally on tractors and farm equipment.

Although not all of his work was transmission-related, Simpson was a noted expert in planetary (or epicyclic) gears. Henry Ford, of course, had long been a committed supporter of planetary transmissions, which he considered simpler to use than gearboxes with sliding spur gears. By World War I, most of the auto industry abandoned planetary gearsets in favor of dual-shaft gearboxes, but Ford held out until the advent of the Model A in 1927. After the war, epicyclic gears had seen something of a renaissance in preselector transmissions and later an emerging breed of semiautomatic and automatic transmissions, the latter best exemplified by the original Hydra-Matic.

Before World War II, planetary gearsets remained the exception rather than the norm for automotive use, but, like Henry Ford, Simpson felt that their drawbacks could be overcome through careful design. Throughout the mid-thirties, he developed both two- and three-speed planetary transmissions for passenger cars, the three-speed intended for rear-engine/rear-drive applications. Neither saw production.

Simpson left Ford Motor Company in 1938 and the following year went to work for Detroit Harvester, where he served as chief engineer until 1943. He subsequently became a freelance engineering consultant, again working primarily in transmission and drivetrain design. In 1944, he developed and applied for a patent on a new hydraulically operated six-speed planetary transmission for tractors, followed in 1946 by a closely related three-speed passenger car unit.

At first, it didn’t appear that Simpson would live long enough to exploit these patents, which were not granted until 1950 (as U.S. Patent Nos. 2,518,824 and 2,518,825 respectively). In 1947, he was diagnosed with what appeared to be terminal cancer and had to set aside his work to focus on treatment. The initial prognosis turned out to be overly pessimistic — Simpson survived 17 years after his original diagnosis — but his convalescence was protracted and led him and his family to escape Detroit winters in favor of milder climates in California and the American Southwest. Nonetheless, he did continue to work at least sporadically, applying (ultimately unsuccessfully) for an additional transmission patent in 1948.

By 1950, Simpson had recovered enough to rededicate himself to his work on planetary transmissions, filing a series of new patent applications late that year. Intended to address certain limitations of his earlier designs, his latest disclosures covered several additional improvements on the three-speed/dual-planetary layout, now using two multi-disc (rather than cone) clutches and multiple brake bands, sometimes supplemented with overrunning clutches.

Simpson quickly discovered that despite their virtues, his inventions were not an easy sell in Detroit. While he was a well-respected auto industry veteran, that wasn’t enough to overcome major manufacturers’ pronounced disdain for “NIH” (“Not Invented Here”) ideas or technology. Moreover, most of the inventors who had previously broken through that barrier — like Earl Thompson, designer of GM’s “Silent Synchro-Mesh” gear synchronization system and later one of the principal architects of the first Hydra-Matic — had been able to present working models of their designs, something Simpson wasn’t in a financial position to do.

If Simpson wasn’t independently wealthy, he was nonetheless very thorough. Having conceived what he deemed a viable layout, he also considered a long list of refinements and variations of that basic concept, many of which he also patented. That left automakers who wanted to take advantage of related ideas no legal alternative but to seek a licensing agreement with Simpson; he left little room for patent workarounds.

In June 1953, Ford Motor Company became Simpson’s first major licensee, although Ford didn’t actually adopt Simpson’s designs for any production transmission until the advent of the C4 and C6 Cruise-O-Matic in the mid-sixties. (Ford continued to also use variations of its older Ravigneaux gearset automatics into the nineties.)

Ford was followed in 1955 by Chrysler, whose new TorqueFlite design sufficiently overlapped Simpson’s patents that the company was obliged to license them. General Motors followed suit in the early sixties, eventually followed by Daimler-Benz. All of GM’s three-speed Turbo Hydra-Matic transmissions would use Simpson gearsets, as would the three-speed torque converter automatics introduced by Mercedes-Benz, Borg-Warner, JATCO, and ZF in the early seventies.

This is not to say these transmissions were identical — each had unique features of its own — or that Simpson himself designed them, which he did not. (He did naturally take a keen interest in production applications, but as an outside observer, not a participant.) Rather, their designs shared common elements covered by one or more of Simpson’s patents. For instance, Chrysler’s patent for the original TorqueFlite, granted in 1960 and credited to Bert W. Cartwright, Teno Iavelli, and Ervin R. Miller, cited no fewer than four Simpson patents.

Simpson remained busy with his engineering work and in the Dearborn community into the early 1960s, continuing to invent and patent new planetary transmission designs. By the time he died in November 1963 at the age of 71, he had a total of 40 patents in his name (including ones assigned to Ford or Detroit Harvester during his employment there), 23 of those in transmission design. A 41st and final patent was filed a month and a half after his death and was granted in 1967.

THE SIMPSON GEARSET

Having dispensed with the biography, you’re probably still wondering, “So, what is a Simpson gearset?”

The short answer: What we now call a “Simpson gearset” is a compound planetary gear train consisting of two simple epicyclic gearsets with a common sun gear, taking input through one ring gear and output through the second ring gear. (Simpson’s patents specify that the sun gear doesn’t necessarily have to be one piece; it can be two gears splined or otherwise linked together so that they act as a single unit.) Use of a common sun gear immediately distinguishes a Simpson gearset from the rival Ravigneaux gearset, which has two sun gears of different sizes, using short and long planet gears on a common planet carrier to mesh with a single ring gear, and which usually outputs through the planet carrier.

In a Simpson gearset, the annulus (ring gear) of the second gearset — let’s call it B — is connected to the output shaft. Typically, the planet carrier of the first gearset — let’s call it A — is as well, although in some variations of this layout, carrier A is only connected to the output shaft in the forward gears. This description is necessarily general because there are many different variations of the basic concept, many of them essayed by Simpson himself. The different permutations use different types or arrangements of clutches and brakes and arrange the various components in different ways. For instance, some variants actually place gearset A behind gearset B, requiring an additional intermediate shaft.

(We should also emphasize that during his long professional life, Howard Simpson designed a lot of different transmissions, not all of which follow this model. Therefore, not all Simpson-designed gear trains are Simpson gearsets!)

A basic Simpson gearset provides three forward speeds and one reverse. The indirect ratios are obtained by holding the planet carrier of gearset B stationary in first and reverse and holding the sun gear stationary in second. In many, though not all, Simpson gearsets, the gears in gearsets A and B are identical. (This isn’t actually specified in Simpson’s patents, but it’s very common.) The commonality reduces production costs — it’s usually cheaper to make multiple sets of an identical part than several different ones — and simplifies assembly, but it imposes certain limitations on the range of possible ratios, an issue we’ll discuss in more detail below.

Simpson gear trains have a variety of advantages. They’re relatively compact and mechanically straightforward. They allow their various radial and thrust forces to be balanced with a minimum number of bearings, reducing internal friction as well as production costs. Using a common set of gears for both forward and reverse also represented a useful advance over the original Hydra-Matic, which had an additional planetary gearset used only in reverse.

By modern standards, of course, any three-speed automatic is hopelessly anachronistic. However, at the time this layout was designed and for many years afterward, three forward speeds was the norm for U.S. manual transmissions. Transmissions with integral overdrives were very rare and many four-speed transmissions (including Hydra-Matic) were essentially three-speeds with an additional extra-low gear, a function that could be performed well enough by a torque converter.

Simpson designed a number of transmissions that added a third planetary unit behind a conventional dual-planetary Simpson gearset, giving six forward speeds, but those designs were intended for tractors, which typically have their engines governed to a single fixed rpm; he didn’t consider the additional complexity necessary for most passenger cars. Interestingly, in the mid-fifties, he did design some dual-planetary four-speed transmissions with overdrive top gears, although they used a different gear train layout with separate sun gears.

More technical details can be found on the following pages.

THE SIMPSON GEARSET IN OPERATION

Presenting a detailed history and description of each production transmission to use a Simpson gearset is well beyond the scope of this article. However, we will explain the basic principles of its operation, which will help you understand the production applications and their relatively minor variations on the theme.

A simple planetary gearset has three elements: a sun gear, a set of planet pinions journaled on a planet carrier, and an annulus (ring gear). All the gears are in constant mesh: The planet gears mesh with the sun gear and the annulus meshes with the planets. Obviously, the planet carrier is not a gear; rather, it’s the armature holding the pins on which the planet gears rotate, which may itself rotate at a different speed than the planets, the sun gear, or the annulus.

A Simpson gearset is a compound planetary gear train with five elements: the annulus of gearset A (which we’ll abbreviate AA); a set of planet pinions on the planet carrier of gearset A (which we’ll abbreviate CA); the sun gear (S); the annulus of gearset B (AB); and another set of planet pinions on the planet carrier of gearset B (CB). Annulus B is permanently affixed to the output shaft.

(To be clear, what we’re describing as “gearset B” is the gearset whose annulus is connected to the output shaft, regardless of actual physical position. Depending on the specific layout, gearset A may be located behind gearset B or vice versa; this description and all the calculations below are based on each gearset’s function, not its position.)

Any Simpson gear train requires at least two brakes and two clutches:

• A forward clutch that connects the input shaft to annulus A in the forward gears. (In some variants, annulus A is permanently connected to the input shaft and the forward clutch instead connects planet carrier A to the output shaft.)
• A direct/reverse clutch that connects the input shaft to the sun gear in third and reverse. (Simpson also designed some split torque variants where this clutch instead connects the sun gear directly to the engine, usually through the torus cover of the fluid clutch.)
• A low/reverse brake that can hold planet carrier B stationary in certain gears.
• An intermediate brake that can hold the sun gear stationary in certain gears.

We say “at least” because some transmissions use multiple brakes of different types, a point we’ll discuss in more detail below.

BASIC OPERATION

Simpson gearsets obtain three forward and one reverse speed by combining the above elements in the following combinations:

• First (low): The forward clutch engages, allowing the transmission input shaft to drive annulus A (or connecting planet carrier A to the output shaft). A low/reverse brake is applied to planet carrier B. As annulus A rotates, reaction torque created by the inertia of carrier A causes the sun gear to turn backward. The sun gear’s reverse rotation attempts to turn planet carrier B backward as well, but the low/reverse brake holds the second carrier stationary instead. This causes annulus B to rotate around the stationary carrier in the opposite direction — which in this case is forward — at reduced speed. Since carrier A is also affixed to the output shaft, it must also rotate forward at the same speed.
• Second (intermediate): The forward clutch remains engaged, so the input shaft continues to drive annulus A forward. The low/reverse brake disengages, allowing planet carrier B to rotate freely. An intermediate brake is applied to hold the sun gear. As annulus A rotates, reaction torque attempts to turn the sun gear backward, but the brake holds the sun gear stationary instead. This forces planet carrier A — and the output shaft — to rotate forward around the now-fixed sun gear at reduced speed. The rotation of the output shaft drives annulus B forward at the same speed, causing planet carrier B to rotate idly forward around the stationary sun gear.
• Third (high): The forward clutch is still engaged, allowing the input shaft to drive annulus A. Both the low/reverse and intermediate brakes are released, allowing planet carrier B and the sun gear to rotate freely. The direct/reverse clutch engages, locking the sun gear to either (depending on the version) planet carrier A or annulus A. This forces all elements of both gearsets to turn at the same speed, putting the transmission in direct drive. (In split torque variants, the sun gear turns at engine speed, annulus A turns at input shaft speed, and carrier A resolves the difference.)
• Reverse: The direct/reverse clutch engages, allowing the input shaft (or, in split torque versions, the torus cover) to drive the sun gear forward. A low/reverse brake is applied to hold planet carrier B. The sun gear’s rotation attempts to turn carrier B forward, but the brake prevents the carrier from moving, forcing annulus B to rotate backward at reduced speed. The forward clutch is released, disconnecting annulus A from the input shaft (or, if annulus A is permanently connected to the input shaft, disconnecting carrier A from the output shaft) so that gearset A can’t transmit any torque. (If annulus A is released, it spins idly backward; if the planet carrier A is disconnected, it turns idly forward at engine speed or close to engine speed.)
• Neutral: All clutches and brakes are released. No element is connected to the input shaft (or the torus cover) and no element is held.

BRAKES AND CLUTCHES

In principle, a Simpson gearset can use any type of brake or clutch; Simpson’s earliest versions of this layout used cone and dog clutches along with band brakes. Most production applications use a mix of multi-disc clutches, band brakes, and one-way clutches.

Unlike other types of brake, which keep the elements to which they’re attached from turning in either direction, a one-way clutch (or overrunning clutch) allows a gear element to rotate in one direction, but not the other. Many Simpson gear trains (and other automatic transmission layouts) use sprag-type or cam-and-roller one-way clutches to simplify the mechanics of shifting.

For example, you’ll notice in the above summary that in first gear, the sun gear of a Simpson gearset rotates backward (i.e., opposite the direction of engine rotation and attempts to rotate planet carrier B backward as well. In second gear, carrier B rotates idly forward. If carrier B is connected to a one-way clutch that allows it to turn forward but not backward, that clutch will hold carrier B stationary in first and automatically releases upon the shift to second, or vice versa. The one-way clutch requires no external mechanism or synchronization.

There are two drawbacks to using a one-way clutch in this way. First, the one-way clutch isn’t effective in reverse, when the forward rotation of the sun gear attempts to rotate carrier B forward. Second, if the output shaft overruns the engine (for example, when descending a steep hill in first gear), the rotation of annulus B will also rotate carrier B forward, causing the one-way clutch to immediately unlock and leaving the transmission effectively in neutral. Consequently, the one-way brake must be supplemented by a band or clutch-type brake, which is used in reverse or when the driver manually selects “Low.”

Some transmissions with Simpson gearsets, such as the Chrysler TorqueFlite, use both a one-way clutch and a separate brake for first gear, but only a single band- or clutch-type intermediate band for second. To use a one-way clutch to hold the sun gear in second, there must be some means of neutralizing that clutch in first gear, when the sun gear has to turn backward. Simpson proposed doing that by attaching the outer race of the one-way clutch to a brake drum and band brake. Reverse rotation of the sun gear would always lock the inner race against the outer race, but with the brake released, the whole drum would simply rotate backward. Some production transmissions, including the GM Turbo Hydra-Matic, accomplished the same effect by using a multi-disc clutch to lock the one-way clutch’s outer race to the transmission case.

As with a one-way clutch on carrier B, a separate overrun or coast brake is still needed to keep the sun gear locked when coasting. Turbo Hydra-Matic included an overrun band for this purpose, functional only when manually selecting a lower speed range and disengaged in Drive. This gave the driver greater ability to keep the transmission in second gear for engine braking in hilly terrain.

While this arrangement is obviously more complex than a simple two-brake/two-clutch Simpson gearset, it allows each shift in Drive to be accomplished by engaging or disengaging one element. In a Turbo Hydra-Matic, for example, engaging the forward clutch from a neutral condition would give first gear; also engaging the intermediate clutch would give second; and engaging the forward, intermediate and direct/reverse clutch gave third. Downshifts were just as straightforward; from third, releasing the direct/reverse clutch would produce an immediate downshift to second and then releasing the intermediate clutch put the transmission back in first. The additional mechanical complexity was repaid with simpler (and usually smoother) shift action between all forward speeds.

On the following page, we’ll explain how a Simpson gearset’s ratios are calculated.

SIMPSON GEARSET RATIOS

As with any gear train, the gear ratios of a Simpson gearset are the number of rotations of the input shaft required for each rotation of the output shaft in a particular gear. The output shaft of an automotive transmission is connected to the driveshaft and differential; in an automatic transmission, the input shaft is usually driven by the turbine of the fluid coupling or torque converter. If we call the velocity of the input shaft V_I and the velocity of the output shaft V_O, the gear ratio therefore equals:

V_I / V_O

(For these calculations, it’s important to specify velocity rather than speed because the direction of rotation is also significant. A positive velocity signifies rotation in the same direction as the input shaft (i.e., forward) while a negative velocity signifies reverse rotation (i.e., backward.))

The ratio of any gear train depends on the relative velocities and numbers of teeth of the various gear elements. For two meshed gears, the ratio is the teeth of the driven gear divided by the teeth of the driving gear. For example, if a gear with 25 teeth drives a gear with 40 teeth, the gear ratio will be 40 / 25, or 1.6:1, meaning that the driving gear must turn 1.6 times for each complete rotation of the driven gear.

If the planet carrier is stationary while the annulus drives, the rotation of the annulus drives the planet gears forward at a ratio equal to planet teeth divided by annulus teeth. The planet gears then drive the sun gear backward at a ratio equal to sun gear teeth divided by planet teeth. The net ratio is the product of those two ratios:

Annulus Velocity * (Planet Teeth / Annulus Teeth) * -(Sun Gear Teeth / Planet Teeth) = Sun Gear Velocity

As mentioned above, the negative sign indicates a reversal of direction. You’ll also notice that the two planet teeth values cancel out; planet teeth divided by planet teeth equals 1. Therefore, we can drop those values and simply write:

Annulus Velocity * -(Annulus Teeth / Sun Gear Teeth) = Sun Gear Velocity

Conversely, if the sun gear drives while the carrier is stationary:

Sun Gear Velocity * -(Sun Gear Teeth / Annulus Teeth) = Annulus Velocity

For both of the above equations to be simultaneously true, the velocity of the planet carrier must equal:

(Annulus Velocity * Annulus Teeth + Sun Gear Velocity * Sun Gear Teeth) / (Annulus Teeth + Sun Gear Teeth)

That’s for a simple planetary gearset, but this equation still applies to compound planetary gearsets. In a Simpson gearset, the velocity of carrier A (which we’ll abbreviate V_CA) must equal:

V_CA = (Annulus A Teeth * Annulus A Velocity + Sun Gear Teeth * Sun Gear Velocity) / (Annulus A Teeth + Sun Gear Teeth)

At the same time, the velocity of carrier B (which we’ll abbreviate V_CB) must equal:

V_CB = (Annulus B Teeth * Annulus B Velocity + Sun Gear Teeth * Sun Gear Velocity) / (Annulus B Teeth + Sun Gear Teeth)

For the sake of brevity, we’ll abbreviate “sun gear teeth” as ST, “annulus A teeth” as AAT, and “annulus B teeth” as ABT.

In a Simpson gearset, annulus B is always affixed to the output shaft. Therefore, the velocity of annulus B (which we can abbreviate V_AB) must always equal V_O. Typically, carrier A is also affixed to the output shaft or else affixed to annulus B. Either way, the velocity of carrier A (V_CA) must also equal V_O. However, in variants where annulus A is permanently connected to the input shaft and carrier A is connected to the output shaft through the forward clutch, V_CA only equals V_O when the forward clutch is engaged.

FIRST GEAR

In any iteration of the Simpson gearset, both the forward clutch and the low/reverse brake(s) are engaged in first. Therefore, in first gear, the velocity of annulus A (which we’ll call V_AA) equals V_I and the velocity of carrier A (V_CA) must equal the velocity of annulus B and the output shaft (V_O). Since carrier B is held by a brake in first, the velocity of carrier B (V_CB) equals 0.

We don’t yet know the velocity of the sun gear (which we’ll abbreviate as V_S), but we do know that:

V_CA = (AAT * V_AA + ST * V_S) / (AAT + ST)

Since the velocity of carrier A is the velocity of the output shaft:

V_O = V_CA = (AAT * V_AA + ST * V_S) / (AAT + ST)

We also know that:

V_CB = (ABT * V_AB + ST * V_S) / (ABT + ST)

The velocity of annulus B also equals the velocity of the output shaft, so we can substitute V_O for V_AB as follows:

V_CB = (ABT * V_O + ST * V_S) / (ABT + ST)

The low/reverse brake is engaged in first gear, so carrier B cannot move and V_CB is 0. Therefore:

(ABT * V_O + ST * V_S) / (ABT + ST) = 0

Solving for V_S gives us:

V_S = -ABT / ST * V_O

Because gearsets A and B share a common sun gear, we can substitute this result for V_S in the first equation above. If:

V_O = (AAT * V_AA + ST * V_S) / (AAT + ST)

… and:

V_S = -ABT / ST * V_O

… then:

V_O = (AAT * V_AA + ST * (-ABT / ST) * V_O) / (AAT + ST)

We can then simplify and solve for V_S:

V_S = V_AA * -(ABT / ST) / (1 + ST / AAT + ABT / AAT)

With the forward clutch engaged, the velocity of annulus A is the velocity of the transmission input shaft, so we can substitute V_I for V_AA:

V_S = V_I * -(ABT / ST) / (1 + ST / AAT + ABT / AAT)

Note the negative sign, indicating a reversal of direction. Since the velocity of the input shaft is positive (forward), the sun gear’s velocity must be negative (backward).

Since we determined earlier that V_S equals -ABT / ST times V_{O, then:}

V_O = V_S * -ST / ABT

The sun gear’s velocity is already negative, so the two negative signs cancel each other out, making the velocity of annulus B positive. In other words, the sun gear’s backward rotation causes annulus B, carrier A, and the output shaft to rotate forward.

We can again substitute this result into the earlier equation, giving us:

V_O = (AAT * V_I – ABT * V_O) / (AAT + ST)

… which simplifies to:

V_O = V_I / (1 + ST / AAT + ABT / AAT)

Therefore, the gear ratio in first gear is:

V_I / V_O = 1 + ST / AAT + ABT / AAT

(We could also simplify this 1 + (ST + ABT) / AAT, but the above will help to illustrate an important point later.)

SECOND GEAR

In second gear, the forward clutch remains engaged, but the low/reverse brake releases and the intermediate brake engages, which brings the sun gear to a halt and prevents it from turning backward.

As before, the velocity of carrier A — and thus the velocity of the output shaft — must equal:

V_CA = (AAT * V_AA + ST * V_S) / (AAT + ST)

However, with the intermediate brake engaged, V_S = 0. Therefore:

V_CA = AAT * V_AA / (AAT + ST)

… which simplifies to:

V_CA = V_AA / (1 + ST / AAT)

Since the forward clutch is engaged in second, V_AA equals V_I and V_CA equals V_O. Therefore:

V_O = V_I / (1 + ST / AAT)

… and the gear ratio in second gear is:

V_I / V_O = 1 + ST / AAT

You’ll notice that we haven’t said anything about gearset B. However, we know that the velocity of annulus B (V_AB) equals the velocity of the output shaft and that the sun gear is stationary. Therefore:

V_CB = ABT * V_AB / (ABT + ST)

… and:

V_CB = V_AB / (1 + ST / ABT)

Since annulus B rotates forward in second gear, V_AB is a positive number. Therefore, V_CB is also positive, meaning the rotation of annulus B attempts to rotate carrier B forward at reduced speed. If carrier B is held by a one-way clutch in first, the one-way clutch will allow carrier B to rotate idly forward. If carrier B is held instead by some other type of brake, that brake must be released for the shift to second.

THIRD GEAR

In third gear, the forward clutch remains engaged, but the intermediate brake is released and the direct/reverse clutch engages.

Once again, the velocity of carrier A equals:

V_CA = (AAT * V_AA + ST * V_S) / (AAT + ST)

… and the velocity of carrier B equals:

V_CB = (ABT + V_AB + ST * V_S) / (ABT + ST)

With the forward clutch still engaged, V_AA continues to equal V_I. However, with the direct/reverse clutch also engaged, the sun gear is also linked to the input shaft, so V_S must also equal V_I. Therefore:

V_CA = (AAT * V_I + ST * V_I) / (AAT + ST)

… and so:

V_CA = V_I

Since the velocity of annulus B is the same as the velocity of carrier A:

V_CB = (ABT * V_I + ST * V_I) / (ABT + ST)

Therefore, V_CB also equals V_I. Since both carrier B and the sun gear each have a positive velocity (rotating forward), any one-way clutches attached to those elements will unlock and remain unlocked in third.

Both carrier A and annulus B are connected to the output shaft, so their velocity equals V_O. Consequently, in third gear:

V_I = V_O

The gear ratio in third is therefore 1:1 — direct drive.

In Simpson’s split torque variants, the direct/reverse clutch does not connect the sun gear to the input shaft, but rather to the torus housing of the fluid coupling or torque converter. This causes the sun gear to rotate at engine speed, which is always somewhat faster than input shaft speed due to hydraulic slippage in the fluid clutch. The velocity of carrier A and the output shaft will therefore equal:

V_O = (AAT * V_I + ST * Engine Speed) / (AAT + ST)

For example, if the sun gear has 39 teeth and annulus A has 81 teeth, output shaft velocity will be:

V_O = (81 * V_I + 39 * Engine Speed) / 120

… or:

V_O = 0.675 * V_I + 0.325 * Engine Speed

Therefore, the sun gear receives 32.5% of engine torque, effectively demultiplying any slippage in the fluid coupling or torque converter by that amount. For instance, if the engine is turning 2,000 rpm and there’s 5% slip in the converter, the input shaft will turn at 1,900 rpm. The output shaft will therefore turn at about 1,933 rpm. (Our earlier article provides more explanation of the split torque concept than you probably wanted.)

What about gearset B? The sun gear rotates at engine speed and annulus B rotates at output shaft speed. The velocity of carrier B is:

V_CB = (ABT * V_AB + ST * V_S) / (ABT + ST)

Since V_AB equals V_O:

V_CB = (ABT * V_O) + ST * engine speed) / (ABT + ST) = (1 + ABT / ST) * V_O + (1 + ST / ABT) * engine speed

We could simplify this further, but we’ll spare you and just say that carrier B will rotate idly forward, turning somewhat slower than the sun gear, but faster than the output shaft.

REVERSE

For reverse, the forward clutch releases and both the direct/high clutch and low/reverse brake engage. With the direct/high clutch engaged, the sun gear now rotates with the input shaft, so V_S equals V_I. As in first, carrier B is stationary, so V_CB equals 0.

As we noted above:

V_CB = (ABT * V_AB + ST * V_S) / (ABT + ST)

Therefore, since V_CB is 0, then:

V_AB = -(ST / ABT) * V_S

… and:

V_O = -(ST / ABT) * V_I

The gear ratio in reverse is therefore:

V_I / V_O = -ABT / ST

Note the negative sign, indicating a reversal of direction. As in first gear, holding carrier B stationary while the sun gear turns causes annulus B to rotate in the opposite direction. Since the sun gear is rotating forward, annulus B must turn backward.

The behavior of gearset A in reverse depends on the layout:

• In Simpson gearset variants where annulus A is permanently connected to the input shaft, annulus A will naturally continue to rotate forward at a velocity equal to V_I. However, the release of the forward clutch disconnects carrier A from the output shaft. Since V_AA and V_S both equal V_I, just as in third gear, carrier A will rotate idly forward at input shaft speed.
• In variants where disengaging the forward clutch disconnects annulus A from the input shaft, carrier A will rotate at the same velocity as annulus B. In this case, that means carrier A rotates backward as the sun gear rotates forward. Annulus A resolves that difference by rotating idly backward at a velocity equal to V_S * (ST² / AAT² – 2 * ST / AAT).

NEUTRAL

Most Simpson gearsets achieve neutral by releasing all clutches and brakes. With none of the elements driving, no torque can be transmitted to the output shaft.

In variants where annulus A is permanently connected to the input shaft, the forward clutch is disengaged in neutral to disconnect carrier A from the output shaft. Gearset A therefore tends to rotate idly forward as a unit. The rotation of the sun gear and the inertia of annulus B (which is connected to the mass of the output shaft) tend to rotate carrier B forward, so if there is a one-way clutch on carrier B, it remains unlocked, allowing carrier B to idle.

GEARING SUMMARY

If your eyes rolled back in your head at all the math above, here’s a quick summary of a Simpson gearset’s ratios in each gear:

• First: 1 + ST / AAT + ABT / AAT
• Second: 1 + ST / AAT
• Third: 1 (direct drive)
• Reverse: -ABT / ST

… where ST means “sun gear teeth,” “AAT” means “annulus A teeth,” and “ABT” means “annulus B teeth.”

GEARING LIMITATIONS

We mentioned earlier in this article that it’s very common for both gearsets in production Simpson gear trains to use identical gears for reasons of production economy. This imposes certain mathematical limits on the possible ratio spread.

In a Simpson gearset, if annulus A and annulus B have the same number of teeth, then ABT divided by AAT must obviously equal exactly 1. The gear ratio in first gear would therefore become 2 + ST / AAT. As you can see, that will always be the ratio in second gear plus 1.

For example, early GM Turbo Hydra-Matic 400 (TH400) transmissions had a sun gear with 39 teeth, identical planet pinions with 21 teeth, and identical ring gears with 81 teeth each. Since 39 divided by 81 is 0.481, a TH400’s forward ratios are therefore 2.481, 1.481, and 1.000. Reverse is 81 divided by 39, or -2.077. The early Chrysler TorqueFlite had a sun gear with 28 teeth, identical planet pinions with 17 teeth, and identical ring gears with 62 teeth each, giving ratios of 2.452, 1.452, 1.000, and -2.214.

Ratios like these were well-suited to American V-8s or big six-cylinder engines, especially with a torque converter to provide extra multiplication under load. Even a “tight” torque converter with a modest 2:1 stall ratio gave these transmissions a robust maximum starting ratio of almost 5:1. Second gear, meanwhile, provided a useful passing gear for typical U.S. highway speeds (45 to 75 mph, 72–121 km/h).

However, gearing like this was less than ideal for smaller engines with less torque, which needed a shorter (higher numerical) first gear for decent acceleration off the line. A Simpson gearset with identical gears can’t obtain a first gear ratio shorter than about 2.6:1 without using tiny planets on an enormous sun gear, and doing so makes second gear too short to be an effective highway passing gear.

For this reason, some later transmissions with Simpson gear trains (and some aftermarket replacement gearsets) sacrifice the cost advantages of identical gears in order to provide a shorter (numerically higher) first gear. For example, keeping the TH400’s stock gears for gearset B while replacing the gears of gearset A with smaller planet pinions of 15 teeth each and a smaller annulus of 69 teeth would make first gear:

1 + 39 / 69 + 81 / 69 = 2.739

… while second gear would become:

1 + 39 / 69 = 1.565

Reverse would remain -2.077:1 and of course third gear would still be 1:1. (These ratios, incidentally, are very close to the published gear ratios of GM’s light-duty TH200 and TH200C, for which we don’t have gear teeth counts.)

Some aftermarket gearsets go the opposite direction, using larger gears for gearset A to obtain a lower numerical first gear. The object is to provide closer-spaced gears to keep a highly tuned engine in its power band. For example, if the sun gear has 36 teeth, annulus A has 90 teeth, and annulus B has 66 teeth, first gear will be:

1 + 36 / 90 + 66 / 90 = 2.133

Second will be:

1 + 36 / 90 = 1.400

… and reverse becomes:

-66 / 36 = -1.833

With these ratios, engine speed drops only 34.4% on the 1–2 shift and 28.6% on the 2–3. By comparison, the stock gears of a Turbo Hydra-Matic 400 give an rpm drop of 40.3% from first to second and 32.5% from second to third. However, such a tall first gear would make for sluggish launches unless accompanied by a very short axle ratio and/or a torque converter with a high numerical stall ratio. A gearset like this is therefore intended mostly for racing.

FOUR OR MORE

By the late seventies, the limitations of three-speed automatics were already becoming apparent. With an axle ratio tall enough to provide relaxed, economical cruising, smaller emission-controlled engines needed wide-ratio gearsets for decent off-the-line punch, which left painful gaps between the ratios and made it harder to keep the torque converter locked up for better fuel efficiency. Compromise gearing often made for an unhappy combination of lackluster acceleration, sub-par fuel economy, and irritatingly buzzy highway rpm.

The immediate solution, other than to stick with a manual gearbox, was automatic transmissions with an overdrive top gear. Overdrive allowed a shorter axle ratio for better acceleration while still keeping cruising rpm lower than would generally be practical with a three-speed automatic.

Many four-speed overdrive automatics were essentially revisions of existing three-speed units. Some obtained the additional ratio by rearranging how the existing elements interconnected; Ford’s four-speed AOD transmission, which we’ve described elsewhere, was a revamp of the Ravigneaux gearset MX/FMX transmissions. Others, like the GM 200-R4 and the ZF 4HP22, were Simpson gearset automatics with an additional planetary gearset to provide an overdrive ratio after the the shift to the direct drive third gear.

Although three-speed automatics remained common throughout the eighties and persisted on some inexpensive models into the 21st century, four-speeds became the norm by the early nineties. These eventually gave way to automatics with five or more speeds. Some of these later multi-speed transmissions still retained some recognizable elements of older Simpson gearset designs, although calling them that strains the definition.

However, in this regard, Simpson himself was ahead of the curve. Back in 1959, he designed a dual-planetary transmission using three brakes and six clutches to provide eight forward speeds, including direct drive and two overdrive ratios. (A patent was granted for this design in May 1962, U.S. Patent No. 3,031,901.) Simpson intended this transmission for tractor duty — as evidenced by the four reverse gears — but with today’s proliferation of eight-, nine-, and even 10-speed passenger car automatics, it looks surprisingly prescient.

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AUTHOR’S NOTE

I’d like to make a special shout-out to Matthias Wandel’s Woodgears.ca website. In writing both this article and the earlier split torque article, I looked at a lot of different references to try to get my head around how to calculate planetary gearset ratios. Although not specifically about automotive transmissions, Wandel’s straightforward explanation was enormously helpful for your math-challenged author to grasp the basic principles well enough to extrapolate them to different situations. (For the record, I have no affiliation with that site and have never spoken with Wandel; I merely found his published work very helpful!)

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NOTES ON SOURCES

Our sources on Howard Simpson and his designs included R. August, R. Kasuba, J.L. Frater, and A. Pintz, “Dynamics of Planetary Gear Trains,” NASA Contractor Report 3793 (Grant NAG3-186), June 1984; Cernil Bagci, “Efficient Methods for the Synthesis of Compound Planetary Differential Gear Trains for Multiple Speed Ratio Generation,” Gear Technology July/August 1990: 14–35; John Barach, “Cadillac History,” Motor Era, n.d., www.motorera. com/ cadillac/index.htm, last accessed 21 October 2017; Roy Beardmore, “Epicyclic Gears,” Roymech, 20 January 2013, www.roymech. co.uk/Useful_Tables/Drive/ Epi_cyclic_gears.html, accessed 20 August 2017; Ford R. Bryan, Henry’s Lieutenants (Chicago: Wayne State University Press: 1993); Joseph M. Callahan, “Design of Dying Engineer Sweeping Auto Industry,” Automotive News 6 July 1964; Bert W. Cartwright, Teno Iavelli, and Ervin R. Miller, assignors to Chrysler Corporation, U.S. Patent No. 2,932,990A, “Transmission,” filed 14 October 1954, issued 19 April 1960; Charlie Tranny, “C6,” n.d., www.charlietranny. com/ c6.htm, 2 October 2017, and “Ford C4 transmission,” n.d., www.charlietranny. com/ c4.htm, accessed 2 October 2017; Chrysler Division, Chrysler Corporation, “Imperial” [brochure CS-375], September 1956; “Chrysler’s TorqueFlite automatic transmission,” Allpar.com, n.d., www.allpar. com, accessed 17 October 2017; David E. Davis, Jr., “Chrysler Imperial,” Car and Driver Vol. 26, No. 7 (January 1981): 69–75; Kevin Elliott, “Automatic for the People,” Hot Rod 14 September 2009, www.hotrod. com/ articles/ 0911rc-ford-automatic-transmission/, accessed 2 October 2017; Ford Division of Ford Motor Company, “The New Ford Mustang” [press kit], April 1964; Michael Galimi, “Ford C6 Transmission Upgrades – Built Tough,” Modified Mustangs & Fords 7 February 2012, www.mustangandfords. com/ how-to/ drivetrain/ mdmp-1203-ford-c6-transmission-upgrades/, accessed 2 October 2017; Philip G. Gott, Changing Gears: The Development of the Automotive Transmission (SAE Historical Series) (Warrendale, PA: Society of American Engineers, 1991); T. Grace, Automatic Transmission Service Guide (Union, NJ: Lincoln Technical Institute, September 1966); Tom Hand, “The TorqueFlite Automatic Transmission,” WPC News Vol. 15, No. 5 (January 1984): 4–22; Ari Holopainen, “Planetary Gears,” LUGNET News, 2005, www.lugnet. com/~3813/epicyclic, accessed 4 September 2017; Roger Huntington, “The Great Transmission Controversy: Coupling vs. Converter,” Car Life Vol. 10, No. 2 (March 1963): 18–21; Jim Kaekel, Jr., “GM TH-200: From the Court Room to the Race Track,” Motor State Distributing, May-June 2013, www.motorstate. com/GMTH-200.htm, accessed 2 October 2017; Alexander L. Kapelevich, “Analysis and Optimization of Asymmetric Epicyclic Gears,” Gear Solutions Vol. 61, No. 7 (August 2016): 50–55; Claus-Peter Köth, “50 Jahre Automatgetriebe von ZF,” Automobil Industrie, 5 August 2015, www.automobil-industrie.vogel. de, accessed 21 October 2017; Michael Lamm, “Driving the 1980 Dodge, Plymouth, and Chrysler Models,” Popular Mechanics Vol. 152, No. 4 (October 1979): 102–103, 246–248; “Man with a Pencil: Engineering Genius of the Modern Automatic Transmission,” Motor Trend Vol. 16, No. 10 (October 1964): 82–85; Le Roy F. Maurer, assignor to Automatic Turbine Drive Company, Inc., U.S. Patent No. 2,329,724, “Transmission,” filed 7 December 1937, granted 21 September 1943; Karim Nice, “How Gears Work,” HowStuffWorks.com, 16 November 2000 science.howstuffworks. com/ transport/ engines-equipment/gear.htm, accessed 11 August 2016; the Old Car Brochures website (oldcarbrochures.org); Nissan Motor Co.., Ltd., Export Engineering Dept., Service Manual: Nissan Full Automatic Transmission Model 3N71A (Publication No. 29811), ca. 1972; the Old Car Manual Project (www.oldcarmanualproject.com); the Online Imperial Club website (www.imperialclub.com); George Reid, “How to Assemble Ford C4 Transmissions: Cruise-O-Matic / Select Shift,” DIY Ford, n.d., www.diyford.com/ assemble-ford-c4-transmissions-cruise-o-matic-select-shift/, accessed 2 October 2017, and “How to Build a Ford C6 Select Shift Transmission: Step by Step,” DIY Ford, n.d., www.diyford. com/ build-ford-c6-select-shift-transmission-step-step/, accessed 2 October 2017; Rick O. Rittenberg, Challenger and Barracuda Powertrain Databook (Totowa, NJ: Lightning Press, 2021); Howard W. Simpson, assignor to Ford Motor Company, U.S. Patent No. 1,488,136A, “Fender and Safeguard for Tractors,” filed 7 November 1921, granted 25 March 1924; U.S. Patent No. 1,875,767A, “Tractor Pulley,” filed 8 September 1930, granted 6 September 1932; U.S. Patent No. 1,963,686A, “Tractor Wheel Spade,” filed 10 June 1931, granted 19 June 1934; U.S. Patent No. 2,088,782A, “Transmission,” filed 28 September 1935, granted 3 August 1937; U.S. Patent No. 2,132,728A, “Transmission,” filed 2 January 1936, granted 11 October 1938; U.S. Patent No. 2,177,951A, “Transmission,” filed 24 February 1936, granted 31 October 1939; Howard W. Simpson, “Discussions: Finds TorqueFlite Design Gives Satisfactory Performance,” SAE Transactions Vol. 66 (1958): 165; U.S. Patent No. 2,518,824A, “Transmission,” filed 16 September 1944, granted 15 August 1950; U.S. Patent No. 2,518,825A, “Transmission,” filed 27 June 1946, granted 15 August 1950; U.S. Patent No. 2,749,773A, “Hydrodynamically Driven Planetary Transmission,” filed 15 December 1951, granted 12 June 1956; U.S. Patent No. 2,749,775A, “Planetary Transmission for Self-Propelled Vehicle,” filed 27 December 1951, granted 12 June 1956; U.S. Patent No. 2,749,777A, “Planetary Transmission for Self-Propelled Vehicle,” filed 15 December 1951, granted 12 June 1956; U.S. Patent No. 2,786,369A, “Planetary Transmission,” filed 6 February 1953, granted 26 March 1957; U.S. Patent No. 2,826,936A, “Variable Speed Transmission,” filed 3 August 1953, granted 18 March 1958; U.S. Patent No. 2,856,794A, “Planetary Transmission for Self-Propelled Vehicle,” filed 13 December 1955, granted 21 October 1958; U.S. Patent No. 2,856,795A, “Planetary Transmission for Self-Propelled Vehicle,” filed 11 June 1956, granted 21 October 1958; U.S. Patent No. 2,873,625A, “Planetary Transmission for Self-Propelled Vehicles,” filed 4 April 1957, granted 17 February 1959; U.S. Patent No. 3,031,901A, “Twelve Speed Power Shift Planetary Transmission,” filed 23 September 1959, granted 1 May 1962; and U.S. Patent No. 3,319,491A, “Heavy Duty Planetary Transmission,” filed 24 December 1963, granted 16 May 1967; Howard W. Simpson, assigner of one-third to General Motors Corporation, U.S. Patent No. 2,914,967A, “Planetary Transmission,” filed 20 August 1956, granted 1 December 1959; Jim Smart, “How to Build a Better C4 Transmission,” Modified Mustangs and Fords, 1 August 2009, www.mustangandfords. com/how-to/ drivetrain/ mdmp-0404-ford-automatic-transmission-c4/, accessed 2 October 2017; “Test Update: The Cat Strikes Back,” Autocar 7 October 1987, reprinted in Jaguar XJS Gold Portfolio 1975–1988, ed. R.M. Clarke (Cobham, Surrey: Brooklands Books Ltd., ca 1989): 147–149; William K. Toboldt and Larry Johnson, Goodheart-Willcox Automotive Encyclopedia (South Holland, IL: The Goodheart-Willcox Company, Inc., 1975); “Transmission Gearing Ratios,” TCI Automotive, n.d., www.tciauto. com/tc/ gear-ratios/, accessed 2 October 2017; Matthias Wandel, “Planetary gear ratio calculations,” Woodgears, woodgears.ca/ gear/ planetary.html, last accessed 12 October 2017; and ZF AG, “50 Years of Automatic Transmissions,” ZF.com, 5 July 2015, www.zf. com/corporate/en_de/ magazine/magazin_artikel_viewpage_22099816.html, accessed 22 October 2017.

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