mailto:email@example.comMore bicycle articles by David S. Lawyer
Three-speed gearing for bicycles isn't too hard to understand but it's a lot more complex than the derailleur (aka derailer) bicycles which greatly outnumber 3-speeds today. Anyone can easily see how the derailleur works by just looking at the chain and the toothed wheels (aka sprockets = chainrings) at the front and back of the chain. But the internal workings of the 3-speed are hidden from view.
Although the 3-speed was invented shortly after 1900, it didn't become popular in the United States until the 1950's. And then by about 1970 it was surpassed by the derailleur. New 3-speeds became nearly extinct by the 1980's, but later on it was reintroduced on some so called "Cruiser" and "Comfort" Bicycles". But 3-speeds today likely account for only a few percent of the bicycle market.
There are also internal gear hubs with more than just 3 speeds, such as 5-speeds, 7-speeds, etc. They use some of the same principle as the 3-speed but are more complex, such as having sun gears that rotate. The simplest such hub is the 5-speed hub which is really nothing more than 2@ 3-speed systems combined inside the same hub. Thus learning how a 3-speed works can help understanding the hubs with more speeds. It might seem that combining 2@ 3-speeds in a hub would result in 6-speeds. But for every 3-speed sub-system, one of the speeds is just direct drive. So adding on an additional 3-speed mechanism only provides 2 more additional speeds in this case but if the sun gear is allowed to rotate even more speeds can be provided.
This article is for readers who are mechanically inclined. You should be familiar with common hand tools like a ratchet, socket, screwdriver, and saw. You should be able to identify the basic components of a bicycle, like the chain, hubs, spokes, hand-grips, pedals, axle, etc. You should have seen a toothed gear wheel engaging with and turning with another such gear wheel, etc. But it's assumed that you know nothing about what's inside a 3-speed hub (except for ball bearings and an axle). Hopefully, you have also looked over a 3-speed bicycle and perhaps actually ridden one and shifted its gears.
So what is a 3-speed? If you look at a 3-speed bicycle, you'll see only one sprocket on each end of the chain. Derailleur bicycles have multiple sprockets (or a cluster) on each end of the chain and the rider can use derailleurs to move the chain from one sprocket to an adjacent sprocket in order to change gears.
A sprocket is a circular metal disk with teeth on the circumference of the circle, and these teeth engage the chain. Sprockets often have large holes in their sides to reduce weight. The front sprocket(s) (often called a chainring) is pumped by the pedals via crank arms which rotate around a bearing housing known as the bottom bracket. On a 3-speed, the front sprocket (chainring) is larger than the rear one so that if the chainring is twice the diameter of the wheel sprocket, then for each turn of the crank (and chainring) by your feet, the rear sprocket spins 2 turns.
The 3-speed has only one gear ratio, yet the result is 3 gear ratios. Let's explain. Imagine one gear ratio, say 1 to 2. This could be obtained with just two gear wheels that engage each other, with one gear having twice as many teeth (and twice the diameter) of the other. This means that when one gear turns one revolution, the other gear turns two revolutions. Now if you want to increase the rotating speed of your rear tire-wheel (rpm or revolutions per minute), you put the input from the pedals-crank into the large gear and take the output to the rear wheel from the small gear. But if we connect up this gear set so that the input and output are interchanged, then the rpm of the output is cut in half rather than doubled. Thus with a 1 to 2 gear ratio, interchanging the input and output gives us a 2 to 1 ratio. In general, if a gear ratio is x, we can make this interchange, resulting in a ratio of 1/x. This is in fact what a 3-speed does. It gets two gear ratios from one set of fixed gears where one ratio is just the inverse (1/x) of the other. Where's the 3rd gear ratio in this? Well, its just direct drive (a 1 to 1 ratio) which doesn't require any gears at all.
Judging from the above, a 3 speed is simple, except for two things. First, it's nice to have the rotation of the input and output on concentric shafts: the input rear sprocket and the output rear bike tired-wheel rotate around the same center. Thus, the two gear-wheel example presented above will not work easily since the input and output shafts are not on the same axis. To get the input and output on the same axis a 3-speed uses what are known as "planetary gears", also known as "epicyclic gears". Secondly, it requires various clutches (or the like) to interchange the input with the output.
Before explaining the mechanical complications of interchanging the input and output with clutches let's first discuss planetary gears that just create one gear ratio but do so with a concentric geometry.
Planetary gears (aka epicyclic gears) for a bicycle work like this. A planetary gear-system consists of 3 types of gears: sun, planets, and ring. To assembly one on a table top place an ordinary gear, called the sun gear flat on the table. Fix it to the table with glue, etc. It's actually rigidly attached to the rear axle and concentric with it. Then around the sun you uniformly place 3 or 4 planet gears, about the same size as the sun and also called "pinions" or "pinion gears" or "planet pinions". The planet pinions mesh with the sun. The planets are all at the same distance from the sun and all the same size. Then around this whole "solar system" you place a ring gear, which is a metal ring with internal teeth that engages the teeth of every planet. The ring gear is several times the diameter of a ring you might wear on your finger. We now have 6 gears in all: 1 sun gear, 4 planet gears, 1 ring gear. But there is still one major element missing: the planet carrier which we'll describe shortly.
Rotating the ring gear around the fixed sun gear causes all the planets to rotate, both around their own axes and around the sun too. Kind of a double rotation like the earth rotating both around its own axis and also rotating all the way around the sun once a year. But how do we capture the rotation of the planetary gears around the sun gear? After all, the individual axis of any planet gear is obviously not concentric with the sun.
To obtain another object rotating concentric with the sun and containing the planets, we first make an axle pin for each planet gear by cutting a common nail into say 2 cm. lengths. Then, after ensuring that the 4 planets are symmetrically placed around the sun, each 2 cm. long pin is inserted into a center hole of each planet pinion with a loose fit so that the pinion can rotate around its pin. Then we take a large metal flat washer, lay it down concentric with the sun gear so that it contacts all the pins and then weld all the pins to it.
Now when the ring gear is rotated, the planet pinions circle the sun just as before, only the flat washer rotates concentric with the sun since the flat washer is linked to the planets via the 4 axles (pins). This flat washer (and the pins welded to it) rotates at the same rpm around the sun as the planets rotate around the sun. Of course, the planets, in addition to rotating about the sun, also spin around their own axes (the pins).
To make the construction stronger, we'll weld on another flat washer at the other end of the pinion pins so that now the planetary pinions pins are supported from both sides. So that we can replace the pinions when they wear out we'll undo all the above mentioned welding of pins and instead drill holes in the flat washers for the pinion pins so that both the pins and pinions can be removed and replaced when worn. Then we'll weld some rods (or the like) to more rigidly connect the two flat washers to each other in their separated position with the pinions positioned on pins between the two washers. This is called the planet carrier (or planet cage): it carries the planets. It's not actually made from flat washers but is cast as one piece. It has holes for the pins and space for the pinion gears which rotate on these pins. One catalog lists a "planet carrier, complete" as including the pins and pinions. There also needs to be some kind of retention to keep the pins from slipping out of their holes.
The planet carrier (aka "planet wheel carrier") must do more than just carry the planets (or pinions). It must engage with clutches so that it can lock itself either to the bicycle hub (to drive the rear spoked wheel) or to the sprocket (so the chain and your feet can drive it). Similarly, the ring gear also needs two clutches. There are various types of clutches and the various makes and models of 3-speed hubs differ mainly in the types and layout of the clutches. The clutches are cleverly designed and if you take apart a hub you may have a hard time identifying them. But before going into the details of the clutches, let's determine the gear ratio of planetary gears.
If we place the planetary gear system flat on a table top (and unglue the sun gear so that it too can rotate), we can turn both the sun gear and the planet carrier and observe how far the ring gear has turned. Let's say we first note the initial positions (with respect to rotation) of the 3 concentric "wheels": the sun gear, the planet carrier, and the ring gear. Then we rotate the sun x turns (fractional turns are allowed) from its initial position and also rotate the planet carrier y turns from it's initial position. We then observe z, the turns the ring gear has rotated from its initial position. We claim that for given values of x and y, the result z will always be the same regardless of the sequence of inputing x and y rotations. For example, if we first rotate the sun by x (holding the carrier fixed at y=0) and then rotate the carrier by y (holding the sun fixed at x) then the value of z (the rotation of the ring gear) is the same as if we did the rotations in the opposite sequence: first rotate the carrier by y and then the sun by x. And it's also the same if we simultaneously turned the sun by x and the carrier by y even if we turned them in jerks and not necessarily simultaneously. Expressed as a mathematical function we assert that z = f(x,y). Once you convince yourself of this, calculating the gear ratio when the sun is held fixed (as it is in a 3-speed bicycle) is easy.
We are going to turn the carrier one turn clockwise (with the sun fixed) and determine how many turns the ring gear turned . Expressed mathematically, we want to find z1 = f(0,1) where z1 will be the turns the ring gear has made and is also the gear ratio of the planetary system. This result will be achieved in two steps: 1. Turn both the sun and carrier one turn. 2. Holding the carrier fixed, turn the sun gear one turn back to its original position 0. The result will be f(0,1) since the first step made x=1 and the second step changes x by -1 bringing the resulting number of turns of x back to zero. This proceeds as follows:
Lock all the 6 gears together so that the pinion gears can't rotate around their axles. You could just glue all the gear teeth together where the teeth are making contacts. All the gears are now just like one solid mass. Now we rotate this mass clockwise one turn thus turning the carrier, ring gear and sun gear thru one turn. Note that we really didn't need to lock the gears together if we turn the sun gear and the carrier simultaneously at exactly the same speed (in revolutions per minute --rpm). Then we unlock (unglue) the pinions so they can rotate on their axles and holding the carrier fixed to the table (at one turn), rotate the sun one turn counter-clockwise back to where it was originally. Now, based on the reasoning of the last paragraph, the amount of displacement of the ring gear is exactly the same as if we had initially held the sun gear fixed and turned the carrier one turn. Note that moving the sun gear counter-clockwise actually moves the ring gear further in the clockwise direction. So we now need to determine how much further the ring gear turned due to the one-turn rotation of the sun gear back to it's original position.
In this step 2, the sun gear circumference has moved thru distance Cs where Cs is the circumference of the sun (recall it was turned one revolution). The pinion gears just translates this motion to the ring gear, but reverses the direction. This is because the circumferential velocity of a rotating planet-pinion gear is the same at any position on its circumference and the positions of interest here are the pinion-sun contact point and the pinion-ring contact point. So the ring gear in step 2 has moved distance Cs along its periphery in a clockwise direction. Since in step 1 the ring gear moved one revolution clockwise (or distance Cr where Cr is the circumference of the ring gear) the total ring gear circumference movement is Cs + Cr.
So the number of revolutions the ring gear has rotated is obtained by dividing how far a point on its circumference has moved by its circumference. It's ( Cs + Cr ) / Cr. = 1 + Cs/Cr = 1 + Rs/Rr where the Rs is the radius of the sun gear and Rr is the radius of the ring gear. The last equality is true since C = 2 pi R where C is the circumference of a circle and R is its radius. Thus the gear ratio for the planetary system is just 1 + Rs/Rr (one plus the radius ratio of the sun to the ring gear). In other words it just 1 + R where R is the radius ratio. Since Rs < Rr, R < 1, and the gear ratio is always less than 2. Also note that this gear ratio doesn't depend at all on the diameter of the planet pinion gears.
Take the planetary gear system and place it on a table top with the table resting on the earth. Suppose we fix the sun gear to the table so it can't move or rotate and then rotate (drive) the planet carrier clockwise. The planet pinions are thus forced to circle the sun since their pins are attached to the planet carrier, but due to their engagement with the sun, they also rotate around their own axles. The outer edges of the pinions are moving clockwise and push the ring gear clockwise too. It may seem complicated but it's somewhat simplified because everything (the planets, carrier, and ring gear) are all rotating clockwise.
Now ask yourself the question of just how fast the ring gear rotates as compared to the carrier. What's the gear ratio? One way to calculate this is to rotate the carrier 1 turn and then observe how much the ring gear has turned. If the ring gear were to turn 1.5 turns then the gear ratio would be 1:1.5 or 2 to 3 (2:3). How much the ring gear rotates with respect to the earth (the table) is the sum of the rotation of the carrier with respect to the earth plus the rotation the ring gear with respect to the carrier. So if we turn the carrier one turn then all we need to determine is the number of revolutions that the ring gear turns with respect to the carrier. By adding this number and 1, we obtain the number of revolutions that the ring gear has turned with respect to the table (and this number is also the gear ratio). Now imagine that there was an observer on the carrier to observe how far the ring gear turns with respect to the carrier. This is similar to a rider (observer) on a merry-go-round. For the planetary gear set, that observer would see both the ring gear and the sun gear rotate with respect to the carrier.
The observer on the carrier sees the sun gear, which is actually fixed, rotate counter-clockwise thru the same angle (and at the same angular speed of rotation, such as rpm) as a stationary observer on the earth sees the carrier rotate clockwise. For example, as the carrier rotates one turn, an observer on the carrier sees the stationary sun gear rotate one turn. It's similar to a rider on a merry-go-round who senses that the world outside goes around one turn when it's actually the merry-go-round that makes one turn. But the outside world (including the stationary equipment at the center of the merry-go-round) appears to the rider to rotate in a direction opposite to the merry-go-round's rotation.
To our observer on the carrier, the apparent counter-clockwise rotation of the sun gear, forces the pinions to rotate clockwise. We have the teeth of the sun, pinions, and ring all moving in arcs around various centers at various radii but these movements all have one thing in common: the linear speed of all the teeth of all the gears is the same (to the observer on the carrier). The speed is measured at the circumference of any gear wheels where the circumference passes through approximately the center of each tooth.
For example, with a pen draw a black straight radial line on the carrier body, passing thru the center of one of the planetary gears and extending out to the ring gear. Since it's a radial line, an extension of this line would also hit the center of the sun gear. This straight line is rigidly attached to the planet carrier and rotates with the carrier. Then take a red marker pen and put 4 red dots along this line, but put the dots on the gear teeth and not on the carrier: One dot on a ring gear tooth, a dot on the teeth on each side of the planet gear (2 dots on the planet gear), and a final dot on the sun gear tooth. Thus we have one line and 4 red dots on gear teeth along the black line. When the gears start to rotate all the red dots will move off the black line. In reality, the line may not exactly intersect the tip of a particular gear tooth and the dot may have to be put in between two teeth. But imagine that each dot can be put in it's correct location and that it rotates with the gear that it's drawn on.
Now rotate the carrier and see what happens. The black line remains fixed to the carrier, but the red dots, which were on the line, move off the line. When the red dot on the sun gear is 1 cm. (in arc length) away from the line (measured along the circumference of the sun gear with a flexible tape measure) where are the other dots? It's easy to see that all the other dots are also 1 cm. in arc length away from the line since each dot has moved by the same number of teeth (such as 6.43 teeth). So the observer on the carrier, sees the black line remaining stationary and the red dots all moving the same arc length away from the line (at the same circumferential velocity).
Since the circumferential movement of the sun and ring gear are equal. It's just like the sun gear directly drove the ring gear with the pinion gear merely serving as a means to change the direction of motion. Remember that all the above assumes that the observer is on the carrier.
So what is the gear ratio to this observer? Let Cr be the circumference of the ring gear and Cs be the circumference of the sun gear. In one revolution, the dot on the sun gear has moved Cs and since dots move the same arc length, the dot on the ring gear gear has also moved arc length Cs. If do on the ring gear had moved Cr it would have moved thru an entire revolution. But it's only moved Cs which amounts to Cs/Cr of a revolution. Note that Cs/Cr is the ratio of the diameters of the sun and ring gears which we'll call R. One could also determine this gear ratio by dividing the number of teeth on the sun gear by the number of teeth on the ring gear.
So for one turn of the carrier the ring turns 1 + R turns with respect to the earth since the carrier has turned one turn and the ring gear has turned R turns with respect to the carrier. This results in a gear ratio of 1 + R, the same as found by the other explanation. Since the diameter of the sun is always less than the ring, R is always less than one and thus the gear ratio always lies between 1 and 2. It's also the inverse of this if the input and output shafts are interchanged by means of clutches. Since the pinion gears can't be too tiny, the ring gear must always have a diameter significantly greater than the sun gear, resulting in actual ratios of say 4:3. In the next section, we'll discuss one type of clutch using pawls and dogs. It's also like a ratchet or "free-wheeling" device.
One type of clutch used for 3-speed hubs is a one-way clutch which is free-wheeling and is constructed using pawls and dog. Dogs? A dog is something like a ring-gear tooth, but it's more like the shape of a saw tooth than a gear tooth. It doesn't engage with any gear but provides a bumps (or notches) for a pawl to push on it. Pawls pushing against dogs are the main elements of a ratchet, like the socket wrenches used by mechanics. They also provide for freewheeling where a bicycle wheel can spin, even when the pedals aren't being turned. All modern bicycles (including derailleurs) have freewheeling. But during the "golden age" of the bicycle in the 1890's, bicycles were not freewheeling and one had to pump (rotate ones feet) even when coasting. Free-wheeling principles are essential to the operation of the 3-speed hub. These mechanisms not only allow the 3-speed to freewheel, but also serve as clutches.
A freewheel or ratchet is a clutch that has a rotary input and a rotary output and is free to slip in one direction of rotation. For a ratchet socket wrench, the handle is the input and socket (which fits over a nut or bolt) is the output. If you are tightening a nut and apply torque to the input in say the clockwise direction, then the output becomes temporarily locked to the input and turns at the same speed in the same direction (clockwise). However, if you hold the input fixed, the output may still freely rotate in one direction. More generally, if you rotate the input at a given speed (rpm) then if someone turns on the output at a higher rpm, the output shaft will freely turn at this faster output and the input and output are no longer locked together. Try this if you have mechanic's ratchet handy. In a bicycle, there's no need to reverse the direction of rotation for transmitting power like there is for a mechanic's ratchet.
If you don't understand the ratcheting operation, get a hand saw (or possibly just a saw blade) and a common screwdriver (for slotted screws). Set the saw horizontally on a table with the saw blade up with its teeth pointing to the ceiling. Look at the teeth. Each tooth has one side nearly vertical and the other side sloped, forming a triangle. Take the screwdriver, place it over the saw's blade (near the middle of the blade) and point it towards the saw handle. Now push the flat edge of a sawtooth with the tip of the screwdriver. Notice that you can push the whole saw this way and slide it along the table. Now pull back on the screwdriver in the opposite direction but gently keep the screwdriver tip in contact with the saw teeth. Notice that as the screwdriver blade slips along the teeth, the saw stays in the same position and doesn't slide along the table. Here the saw teeth are the "dogs" and the screwdriver blade is the "pawl". This is just like a ratchet. If you push on the screwdriver, it drives the saw. But if you pull on the saw in the same direction with the screwdriver kept stationary, the saw will just "freewheel" in that direction and the screwdriver and saw are not locked together anymore.
This experiment is for linear motion, but the same sort of thing happens for rotary motion. For a bicycle, the dog teeth go around in a circle on the inside of the rear hub shell so it's called a "dog ring". It's something like a ring gear but the teeth are a lot different and the exterior of dog ring is rigidly attached to the wheel and it's spokes. Inside the hub shell there are two dog rings, each about 1/2 inch or so wide. So when you look inside a rear hub on a 3-speed, you'll see dog rings as part of the inside surface of the hub shell. If pawls attached to the ring gear or planet carrier turn the dog ring, the hub, bike wheel, and tires must turn too.
One of the two dog rings in the hub is for the pawls on the planet carrier and one for the pawls on the gear ring. Some dog rings can be unscrewed and replaced using special tools but some can't and are part of the hub shell. In some cases, the dog rings may have been made separately, but were pressed into the shell when it was made.
What do pawls look like. They only bear a little resemblance to a screwdriver blade (used in the example above) since they must be attached to the periphery of a rotating object and engage with a dog by spring action. So if you had to mount a screwdriver blade on a rotary part, you would likely make the blade and shaft rectangular, put a hole in it to mount it, and let it pivot abound a pin inserted into the hole. You would also provide a small spring that pushes the tip of the pawl gently against the dog ring so that it will rub against the dog ring. There might even be some mechanism to retract the pawls so that they disengage from the dog ring, but more on this later.
Pawls are attached to both the planet carrier and the ring gear. They often are mounted on a metal ring (pawl carrier) that is effectively locked to the planet carrier or to the ring gear. The pawl carriers may be cast as part of the ring gear or planet carrier or they may be attached by pins. They drive the hub via dog rings which can be considered to be part of the hub. If the hub is rotating faster than both the carrier and ring gear, then both sets of pawls will click and the bicycle is freewheeling: the wheel is spinning freely just like the front wheel.
Now suppose that the bicycle is not freewheeling and both the planet carrier and ring gear are being driven by your feet. The ring gear rotates 1 + R times faster than the planet carrier. So the pawls of the ring gear drives the hub via the dogs. What happens at the carrier pawls? The hub is turning at the speed of the ring gear which is faster than the planet carrier. So by the freewheeling principle, if the dog ring rotates faster than pawl carrier, the pawls just click and you have freewheeling here (turns freely). Thus there is no engagement between the pawls on the planet carrier and the hub. The pawls just click as they slip over the dogs. Thus the output just comes from the ring gear as originally assumed.
But suppose we want the take the output from the planet carrier via the pawls? This would rotate the hub (via the dog ring) at the speed of the carrier. Since the ring gear is turning faster than the carrier (and faster than the hub) hub, the ring gear's pawls would engage the dogs and force the hub to turn at the speed of the ring gear as described above. The only way to have the carrier drive the hub is to somehow disable the pawls on the ring gear. This may be done by retracting these pawls so that they engage nothing. It can also be done by sliding the pawl carrier to one side, so that the pawls are no longer contacting the dog ring. Instead, the pawls slide freely on a smooth interior hub surface with no dogs contacting them.
We have described about how the one-way clutches (consisting of pawls and dogs) operate. But if you look at a parts list for a 3-speed hub you are not likely to see the name "one-way clutch". The "clutch" name that you will find in the parts list is not for a one-way clutch but for a direct drive clutch (2-way). But you likely will find the names "pawl" and "dog" which serve as components of one-way clutches.
So how does a freewheel work on a derailleur bicycle? It's a lot simpler than the above. The input comes directly from the sprocket cluster on the rear wheel and the output is just the rear wheel.
Above we've described the pawl-dog-ring type "clutch" (aka one-way clutch), and mentioned that these are used for output from the planetary gear system. But what about the two clutches needed for the input to the planetary gears from your leg muscles via the pedals, chain and sprocket? This input is either to the planet carrier or the ring gear.
The two clutches needed are combined so that a parts list will only show one clutch since it is actually only one clutch which can engage either of two different objects. This makes it function like two clutches. A simple type of clutch is used where the clutch is analogous to turning a bolt with a mechanics socket (part of a socket wrench set). When the socket is pushed onto the bolt head, the socket and bolt are joined together and turn together (the "clutch" is engaged). When one slides the socket off the bolt head, the socket and bolt are disconnected from each other (the "clutch" is disengaged). Of course, to engage them, the bolt head and socket must be first aligned with each other. In a bicycle hub, the socket may be welded to a rotating gear or carrier and concentric with it. Or it may just rotate directly around the main axle. In all cases the bolt and socket rotate around the main axle (and of course both are hollow in the center so that the axle can pass thru them). The fact that they rotate around the same axle, automatically aligns them with each other.
As an illustration, suppose the rear sprocket is welded to a deep hex socket that rotates around the main hub axle as the rider pumps the bicycle. Inside this socket is a hollow hex-head bolt with a nut welded onto the end of the bolt, resulting in a 2-headed bolt. Then weld another socket to say the center of the planet carrier. Then when the deep socket (called the driver) is driven by the pedals, if this double headed bolt is slid so that one head of it engages the socket welded to the carrier, the sprocket drives the carrier and the carrier and sprocket turn in unison and are locked together by the 2 sockets and the bolt with 2 heads. But if this bolt is slid so that it engages nothing on one end, then the clutch is disengaged and the carrier is not driven.
The above describes just one clutch, but what about the second clutch? That's easy. Just put another socket around the bolt. This socket has a has had it's flat-end part cut off so that it is just like a short piece of pipe with hexagonal walls. It's placed concentrically with the bolt shaft. The exterior cylindrical surface of the socket is also welded to a rotating part such as the ring gear. So now the double-headed bolt has two possibilities for engagement. It can be slide so that it engages only the ring gear socket or only the carrier socket. With 2 possible things to drive it's like having 2 clutches. It's just one clutch that can engage (lock) itself to either one of two possible sockets.
In an actual 3-speed, the sliding double headed bolt is called the clutch and it doesn't look much like a bolt, since the bolt analogy was used to illustrate the principle of operation using an object that most people are familiar with. The sliding of the clutch is controlled by a pull rod which moves back and forth inside the hollow rear axle. This pull rod is moved by the gear shift cable which is moved by a shifter lever near the hand-grips. This is the shifter that allows you to change speed. Of course, a cable can only pull in one direction so a spring inside the hub must will move the clutch in the other direction. This keeps the shift cable taught.
Now, how can the sliding of a rod (aka pull rod) inside a stationary hollow axle cause the sliding of a rotating clutch inside the hub? Well, the hollow axle can have a lengthwise double-sided slot in it and this internal sliding rod (see above paragraph) can be attached to a slider (aka collar = thrust ring) which is like a ring placed over the axle and part of it extends into the slot (the pull rod screws into this extension). The slider slides back and forth along the hub's axle and pushes (slides) the clutch back and forth with it so that the clutch can engage either with the planet carrier or with the ring gear.
How is the mechanical connection made between the pull rod and the slider? It's by a pin called a "key". This pin can be thought of as a rectangular nut that the pull rod (with a threaded end) screws into (inside the axle at the slot in the axle). This rectangular nut is several times wider than it is high and the edges are sometimes rounded so that it looks almost like a pin (or circular rod, a little larger in diameter than the pull rod) with a threaded hole thru its center of mass (which makes the pin a "nut"). When screwed onto the pull rod, the result look like a tee made of 2 rods. There are also a pair of holes in the slider ring for this pin-nut. To assemble, one positions the sliding ring on the axle over the slot in the axle and then pushes the pin thru one of the holes in the sliding ring so that the pin also goes thru the slots in the axle and comes out of the other hole in the sliding ring (located 180 degrees from the entry hole).
Some notes: The slot in the hollow axle (of the hub) is long enough so that the pin-nut (aka key) can move freely within the slot so as to move the sliding ring (and the clutch) back and forth far enough to shift gears. The pin-nut (key) isn't exactly round nor are the holes for it. This keeps the key from rotating and assures that the nut-hole in it is aligned with the pull rod. The key is made as short as possible and doesn't stick out any beyond the slider ring. There is some sort of a "keeper" to prevent this key from slipping out of position when the pull rod is unscrewed. Sturmey-Archer shapes the clutch (that fits over the slider ring) to also serve as a "keeper".
Note that in operation, the clutch is rotating with respect to the axle but the slider, attached to the sliding rod inside the axle, doesn't rotate. The reulting friction is insignificant since the force to propel the bicycle is not transmitted thru it. There is significant thrust force here only when gears are being shifted.
Real clutches of course don't actually use hex-head "bolts" and "sockets" and such words (including "head") are never found in diagrams, part lists, etc. Instead of a hex socket (6 points) they could use what looks more like a 12 point socket which fits over a 12 point "head". The more points these sockets have, the less load on each point since the points share the load. Two of the sockets are formed as part of the bodies of the planet carrier and ring gear and are not really welded on as described previously for simplicity. The "hollow bolt with 2 heads" might be called a "clutch gear" and the driver socket that one end of it fits into is the "spline" part of the driver but the word "spline" is nowhere to be found in the parts list since it's just part of the driver. Thus the clutch may look like a couple of gears on a short hollow shaft which explains the name "clutch gear". The gear "teeth" are flat on top and thus the "gear" is sometimes called a "spline". Such shaped "teeth" can't be used to drive another gear so the name "gear" is really a misnomer. But it looks almost like a gear.
The Torpedo hub by Sachs (now SRAM) called the "key" (pin-nut) a "sliding block" which also functioned as the sliding ring (although there was no ring part). It looked more like a rectangular nut and not like a pin and the edges of the slots in the axle guided its sliding. Its ends directly contacted (and rubbed against) the rotating clutch and moved it back and forth guided by an internal groove inside the hollow part of the clutch. The return spring acts directly on the sliding block so that the spring force is not transmitted thru any rotating part (in contrast to Sturmey-Archer) thereby reducing friction.
The Sturmey Archer AW 3-speed hub didn't actually use any spline since the sprocket turned (drove) a hollow drive shaft (like a piece of pipe) which had 4 wide axial slots in it for the ends of a metal cross clutch (like a + sign) to project out. The cross, had a hole in its center for the main axle to pass thru. This cross-clutch could slide back and forth and the ends of 4 arms of the cross could engage either the planet carrier or the ring gear. To engage the planet carrier, the 4 arms of the cross engaged with the 4 projecting ends of the planet round pins. It was a simple but crude design, since there were high forces in the clutch-pin contact. It thus tended to deform the ends of the pins and fortunately thereby reduced the original pressure of a flat object pushing against a round one. But it's claimed that misalignment of the clutch-pin contact would result in constant sliding of the contact points as it rotates. The resulting wear would eventually cause the clutch to suddenly slip out of high gear if the rider was standing up on the pedals (pumping hard). This is reported to have resulted in serious accidents where the rider falls over the handlebars when it suddenly slips out of high gear. See http://faqs.cs.uu.nl/na-dir/bicycles-faq/part4.html.
To take the output from the planet carrier, we must disable the pawls on the ring gear as previously explained so that these pawls no longer drive the hub shell and wheels. This happens in low gear where the drive will be taken off the planet carrier pawls and the faster spinning ring gear will not be allowed to drive the hub via its pawls.
Sturmey archer TCW-3 and AW hubs just retracted the pawls. The cross-clutch directly contacted the pawls to push up in back of the pawl pivot point so that the pawl would pivot so as to retract (come down). A pawl is like a teeter-totter: You push one half of it up and the other half comes down since it can rotate a little around its pin. The pawl is shaped and positioned such that the half that is pushed up doesn't touch the dogs. Thus in low gear, the cross-clutch both drives the ring gear and retracts the pawls. To do these two things at the same time it uses 2 of its 4 arms to drive the ring gear and uses the other 2 arms to retract the pawls.
The early Shimano hubs used a method similar to Sturmey Archer but made the pawls do double-duty. When the round clutch with 4 short lugs on it, moves inside the cylindrical pawl carrier for the ring gear, it both pushes the pawls up a little and engages the pawls so that the clutch can drive the ring gear via the pawls. Thus it pushes up the pawls only enough to disengage them, but leaves enough of them exposed so that the clutch can drive them. Thus besides using the pawls as a clutch to drive the dog ring (output,in high gear), the same pawls serve as part of the input clutch to drive the ring gear (input in low gear).
The Shimano clutch had a small part of it which was round like a pipe and had no lugs on it. The leading edge was tapered so as to push up the bottom of the pawls (retracting the tops of the pawls) when the clutch slid along the axle. When this non-lug part of the clutch was under the pawls, it only lifted up the pawls but didn't drive the ring gear since the lugs on the clutch were still driving the carrier. Thus you obtained direct drive from the carrier pawls by retracting the ring gear pawls. In this position, about 1/3 of the clutch is driving the carrier, another 1/3 of it is retracting the ring gear pawls, and the remaining 1/3 is in empty space between the ring gear and the carrier.
Sachs (SRAM) slid the pawl carrier so that the pawls no longer contacted the dogs but just slid on a smooth surface. The pawl carrier was free to slide axially and still be connected to the ring gear via pins. So the clutch would simply push the pawl carrier so that the pawls no longer made contact with the dogs. The pawls then would just freely slide along the smooth interior surface of the shell hub.
By first understanding planetary gearing and clutches it's easy to understand how the system works. Let's start in high gear. This means we must drive the carrier and take the output from the ring gear. All we need to do is to slide the clutch on the driveshaft to a position where it engages the carrier. This locks the carrier to the rear sprocket and chain. Then we need to take the output from the ring gear. This output is automatically put on the hub via the pawls and dog ring as previously explained since the ring gear rotates faster than the carrier. So all we need to do for this case is to just drive the carrier.
For the middle gear, we just want direct drive. So slide the clutch so that it slips out of driving the carrier and slips into driving the ring gear. Then the pawls on the ring gear drive the hub. The turning of the ring gear also drives the carrier at a slower speed than the hub, but the pawls on the carrier just freewheel and don't drive anything.
For low gear we continue to drive the ring gear and make the pawls on the carrier drive the hub by disabling the pawls on the ring gear. The clutch moves from the middle gear to low gear position but it still drives the ring gear. This new position activates a method for disabling the pawls on the ring gear. See disable-pawls
An alternative method for middle gear (direct drive) is to continue driving the carrier but to retract the pawls on the ring gear so that they no longer drive the hub. Then the hub is driven by the pawls on the carrier and since the clutch is driving the carrier it's direct drive. The old Shimano 3-speeds used this method.
I once hope that both coaster brakes and speeds greater than three could be covered in future revisions. But I'll never have time to do this so I'm asking for volunteers. Also, this article could have been better written by someone who has a lot of experience with 3-speed overhaul. I've only worked on a few of them, mostly not recently. How do the various brands of 3-speeds compare? Sturmey-Archer needs to be given credit for inventing it (It was designed by Sturmey and Archer) and their basic design has been copied by others, sometimes with only minor improvements. Shimano (Japan) seems to have made significant improvements but doesn't seem to adequately supply parts for repair (in contrast to Sturmey Archer) for their older hubs. SRAM (Germany) could be the best of them all. The author realizes that diagrams would better help explain this article but has neither the software, time, nor skills to create them.