Transmission
The word transmission refers to a transfer of something. In this case, that is kinetic energy. In basically every automobile, a rotational force (whether produced from a reciprocating motion, such as in a typical gasoline engine with pistons, or from a rotary engine, electric motor, or turbine engine) is both transmitted from the engine to one or more driven wheels, but it is also geared down (and sometimes also up) so that it can do useful work.
In less vague terms, in a car the transmission's job is to take the power from the engine and get it to the wheels. Because the engine only makes power at certain speeds, the transmission has to be able to trade distance for force. This is the same job done by a simple machine like a lever, where if the lever is twice as long on one end as it is on the other, then you can lift twice the load but you have to move the end of the lever twice as far, but in an automotive transmission it is almost always done with gears.
Gear Ratios
While I'm not going to go into depth on the subject of gear ratios, if you match up a gear with another gear with twice its circumference (the distance around the outside) then you will have to turn your gear twice to turn the other gear once. However, that gear is capable of pushing twice as much weight around that circle - it just does it twice as slow. This isn't really a problem; in fact, it's an opportunity because it means that the engine can now move twice as much weight. But since the engine has a maximum speed, we need to be able to change these gear ratios. This is what separates a transmission from a torque converter.
Manual Transmissions
In a manual transmission, we typically have one or more movable gears that can be moved into position to mesh with a different set of gears. Each set of gears (they come in pairs) has a different ratio. The lowest (first) gear is set up so that the engine is able to push the car up a steep hill. The engine is disengaged during shifting of gears through the use of a clutch plate and disc - basically two metal plates that turn together when they are pressed tightly against one another, and otherwise are disconnected. The highest gear, usually the fifth, is usually a little past a 1:1 gear ratio. For example, if the first gear has a 4:1 ratio, the fifth gear may have a 1:1.1 ratio (numbers like 1.04 and 1.08 are more common.) This helps save on fuel when the car is "cruising" at a constant speed. There is also a reverse gear which makes the car go backwards.
The closest thing to an automatic manual based on this design was found in a Volkswagen fastback labeled "Automatic" on the back door. A hydraulic system fitted to the stick shift would operate the clutch for you, although in general it did not do so good a job as a human.
Manual transmissions are small and fairly light (given that they're full of metal and oil, they are going to weigh something) and in the most common types you can shift from any gear into any other gear. There is, however, an exception.
Sequential Manual Transmissions
The sequential manual transmission is another type of manual transmission with a simpler design. Because of the way they are built, it is possible to shift them without using the clutch.1 However, they only allow you to shift into the next gear. Some of them let you shift up and down gears; sometimes they allow you to shift up, but you can only return to neutral, then shift up again. This is most common in motorcycles, which are the place you most commonly see a sequential gearbox. These transmissions generally do not have a reverse gear.2 In general only the most expensive automobiles, such as the Ferrari 355 F1 or the Tesla Electric Roadster use a sequential transmission.
Using computer control, it is possible to make some types of manual transmission behave like an automatic, and the sequential manual is the easiest. Ferrari was one of the first to do this, with a paddle-operated sequential manual gearbox that featured a computer-controlled clutch. The car could be put into fully automatic mode, or in manual mode it would automatically shift down if the engine speed would otherwise drop below idle. Other automakers have since followed suit, but so far this feature has been primarily restricted to high-dollar vehicles. A notable exception is the Subaru Legacy B4, which utilizes a Porsche-designed "tiptronic" sequential gearbox which can be operated in manual or automatic mode.
Automatic Transmissions
"Basic" automatic transmissions have been around for a long time and are quite a marvel of engineering. They contain a pair of planetary gears wrapped in clutch bands so that they can be (dis)engaged, and a fluidic computer that operates hydraulic cylinders (built into the transmission's casing) in order to engage and disengage belts, and to shift the planetary gears between each of their two gear ratios. The "computer" contains various spring-loaded ball bearing check valves and consists otherwise of grooves cut into metal plates that channel fluid around through the unit. Pressure in these passages controls the transmission.
Originally, all automatic transmissions were based on the same design. This persisted until the 1960s when Honda invented the Hondamatic transmission, which was based on somewhat different principles. However, there are now numerous kinds of automatic transmissions, including electronically-controlled models.
Automatic transmissions are convenient because they are automatic, but they have numerous failings. Because they are fluidically controlled they are utterly dependent on the stability of the hydraulic fluid used - if its viscosity rises or falls too far the transmission will operate incorrectly, or not at all; shift points change, shifts never occur, and so on. For this reason automatic transmissions have for the most part been cooled somehow since about the 1960s or 1970s, but it wasn't until the invention of the electronically-controlled automatic that this particular problem was (more or less) wiped out. And because they use planetary gears they are heavier and tend to be larger than a manual transmission which can properly handle the same amount of power and torque.
Electronically-controlled Automatic Transmissions
While the use of purely fluid-controlled transmissions persisted into the 1980s, by the 1990s nearly all automatic transmissions had become electronically controlled. These systems use pressure sensors and solenoid valves to control some or all of the functionality that is ordinarily handled by the fluidic computer. This was the first "new" type of automatic transmission to come along, since it did not require scrapping any existing designs. This not only improves reliability but also allows the car to shift at programmed points which can be changed. Many vehicles incorporate a "tow/haul" switch which raises the shift points to increase RPMs (and thus engine braking, and power in a given gear.) Some vehicles sense when the pedal is being operated rapidly, and move shift points to give the illusion of added power.
Of course, there are some disadvantages; Muscle car enthusiasts have no love for these transmissions because they complicate the vehicle, necessitating the inclusion of a computer which controls them. Muscle cars have lots of torque, so the reduced number of gears in an automatic are not a problem. And since they are electronically controlled, a failure of the electrical system results in a failure of the transmission (although you're not going anywhere without it anyway.)
Continuously Variable Transmissions
One of the more annoying tendencies of the automatic transmission is the additional weight for adding more gears. In addition, adding gears to an automatic transmission sharply increases the complexity because unlike a manual transmission there is not a single clutch, but a clutch per gear set, and many other parts besides. Ideally, of course, you would have a transmission that let you infinitely vary the gear ratio while the vehicle was under way, so that you could always keep the engine in its power band.
The so-called continuously variable transmission or CVT gets you as close as possible to this goal while still utilizing basically a traditional powertrain. The first CVT used on any vehicle actually was continuously variable; a subway train with a belt drive had a belt that could be moved back and forth across a pair of cones which tapered in opposite directions. Moving the belt from one side to the other produced a smooth, even change in ratio between the driving and driven sides.
Honda's CVT design is similar in a certain respect, although the placement of the cones varies: the two cones are blunt and pointed at one another, and they can move closer or further apart. This forms a spool with no center hub, whose effective diameter can be changed. A sort of belt/chain hybrid rides on this pulley. This design allows a wide range of ratios, and changes very smoothly, but can only handle small amounts of power. As far as I know they have not used it with any engine in excess of 1.6 liters in displacement.
There are numerous other designs; for example Nissan has a toroidal system that is based on the interface between two curved spools. Based on where they intersect, the gear ratio is altered.
Non-mechanical power transmission
While the method has not been used in any production automobile of which I am aware, it is possible to replace the transmission system with an electrical system. The motor feeds into a generator, which is then wired to a motor which drives the wheels. This system is actually used in diesel locomotives today, and they are in fact properly called diesel-electric locomotives. The efficiency is actually much better than you might think, and as transmissions, drive shafts, and the like tend to be very heavy, it can actually save weight. However, what they are really saving in the case of trains is the gearbox; if trains had to do the full work of pushing and pulling through a transmission they would rapidly destroy it given today's loads.
This is worthy of mention when it comes to automobiles because this power system is soon to be used in a whole generation of series hybrid vehicles. It enables you to run the engine at its most efficient speed, and moreover to design the engine to run at one speed for maximum efficiency, while entirely or almost entirely eliminating the need for a mechanical transmission. It also lets you use a smaller engine and a larger electric motor - the electric motor being vastly more efficient. And of course, hybrids can perform regenerative braking since in cases where gasoline engines perform engine braking, electric motors (can) behave as generators, and the more load you put on a generator, the harder it is to turn. Diesel-electric locomotives, however, simply put that power into a carbon pile underneath the locomotive and turn it into heat, because there's too much to store in batteries effectively.
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