1. Geartrains
Materials taken from several sources including: Building Robots with LEGO Mindstroms by Ferrari, Ferrari, and Hempel

2. DC motor output characteristics: Speed
Speed: Usually specified as the speed in rotations per minute (RPM) of the motor when it is unloaded, or running freely, at its specified operating voltage.
The speed of the motor is proportional to the applied voltage. At max speed, the motor draws relatively little current.
Typical DC motors run at high speeds up to twenty thousand RPM or more.

3. DC motor output characteristics: Torque
Torque: The rotary force produced on the output shaft. When a motor is stalled it is producing the maximum amount of torque that it can produce. Hence the torque rating is usually taken when the motor has stalled and is called the stall torque. The motor torque is measured in ounce-inches (in the English system). One ounce-inch means that the motor is exerting a tangential force of one ounce at a radius of one inch from the center of its shaft.
The torque of the motor is proportional to the current flowing through it. When stalled, the motor is drawing near max current.

4. DC motor output characteristics: Power
Power: The product of motor speed and torque. The power output is greatest somewhere between the unloaded speed (maximum speed, no torque) and the stalled state (maximum torque, no speed).
PM = T x w
PMmax ~= ¼ x Tmax x wmax

5. Torque, Speed and Power

6. Efficiency
A DC motor is a transducer which changes voltage and current into speed and torque.
To calculate a motor's efficiency, you measure its mechanical output power and divide it by the electrical input power.
Effeciency = PM / Pe ~= 50%-60%
Pe = V x I
PM = T x w
Implications:
Efficiency is highest in the middle of the torque range so you need to oversize your motor to run with high efficiency.
Most robots require relatively low speed and high torque so motor output is usually geared down.

7. Characteristics of the LEGO motor
Lego 9V Technic Motors

8. Gears
A typical DC motor operates at speeds that are far too high to be useful, and at torques that are far too low. Gear reduction is the standard method by which a motor is made useful.
LEGO gears: Gears are classified by the number of teeth they have; the description of which is then shortened to form their name. For instance, a gear with 24 teeth becomes "a 24t gear."

9. Fundamental Properties
The gears are transferring motion from one axle to the other.
If their teeth match well, there is only a small amount of friction. LEGO parts are designed to match properly at standard distances.
The two axles turn in opposite directions: one clockwise and the other counterclockwise.
The two axles revolve at different speeds. When you turn the 8t, the 24t turns more slowly, while turning the 24t makes the 8t turn faster.

10. Gearing Up and Down
Gearing is able to convert torque to velocity. The more velocity gained, the more torque sacrifice. The ratio is exactly the same: if you get three times your original angular velocity, you reduce the resulting torque to one third.
This conversion is symmetric: we can also convert velocity to torque at the same ratio.
The price of the conversion is power loss due to friction.

11. Gear Ratio Gear Ratio = # teeth input gear / # teeth output gear
You can think of gear ratio as a multiplier on speed and a divider on torque.
You can calculate the gear ratio by using the number of teeth of the “driving gear" (a.k.a. the input gear) divided by the number of teeth of the “driven gear” (a.k.a. the output gear).
We gear up when we increase velocity and decrease torque.
Ratio: 3:1
We gear down when we increase torque and reduce velocity.
Ratio: 1:3
driving gear
driven gear
Gear Ratio = # teeth input gear / # teeth output gear
= torque in / torque out = speed out / speed in

12. Building Geartrains
The largest LEGO gear is the 40t, while the smallest is the 8t, thus, the highest ratio we can obtain is 8:40, or 1:5.
If you need an even higher ratio you can use a multistage reduction system, called a geartrain.

13. Idler Gear
What's the ratio of the geartrain in the adjoining figure? Starting from the 8t, the first stage performs an 8:24 reduction, while the second is a 24:40. Multiplying the two fractions, you get 8:40, or 1:5, the same result you'd get meshing the 8t directly to the 40t. The intermediate 24t is an idler gear, which doesn't affect the gear ratio.
Idler gears are quite common in machines, usually to help connect distant axles. Idler gears have one very important effect: They change the direction of rotation.

14. Blacklash
Backlash is the amount of oscillation a gear can endure without affecting its meshing gear.
Backlash is amplified when gearing up, and reduced when gearing down. It generally has a bad effect on a system, reducing the precision with which you can control the output axle.

15. Special Gears: The Worm Gear
The worm gear leads to an assymetric system; that is, you can use it to turn other gears, but it can't be turned by other gears.
For every turn of the worm gear, the mating gear rotates by exactly one tooth. The worm gear is a 1t gear. If mated with a 24t, you get a 1:24 ratio with a single stage.

16. Special Gears: Clutch Gear
A thick 24t white gear, which has strange markings on its face
The clutch gear limits the strength you can get from a geartrain; this helps to preserve your motors, and other parts.
The mysterious "2.5-5 Ncm" indicates that this gear can transmit a maximum torque of about 2.5 to 5 Ncm.
When exceeding this limit its internal clutch mechanism starts to slip.
The maximum torque produced by a system with a clutch gear results from the maximum torque of the clutch gear multiplied by the ratio of the gear stages that follow it.

17. Special Gears: Bevel and Crown Gears
There's a class of gears designed to transfer motion from one axle to another axle perpendicular to it, called bevel gears.
The 24t gear also exists in the form of a crown gear, a special gear with front teeth that can be used like an ordinary 24t, but can also combine with another straight gear to transmit motion in an orthogonal direction.

18. Pulleys and Belts
An advantage of pulleys over gear wheels is that their distance is not as critical. They can be used to transfer motion to a distant axle.
Pulleys are not very suitable to transmitting high torque, because the belts tend to slip. The amount of slippage is not easy to estimate; it depends on many factors, including the torque and speed, the tension of the belt, the friction between the belt and the pulley, and the elasticity of the belt.
The rule with pulleys is that the reduction ratio is determined by the ratio between their diameters.

19. Special Gears: Differential Gear
The differential gear is used to help cars turn corners. The differential gear (placed midway between the two wheels) allows one wheel to turn at a greater speed than the other. Even though the wheels may be turning at different speeds, the action of the differential means that the torque generated by the motor is distributed equally between the half-axles upon which the wheels are mounted.