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Gear ratios are one of the most common design decisions that a team must master in order to optimize mechanical advantage in their designs.
Sourced from RoboMatter
Gear Ratiois defined as the ratio of the rotational speeds of the first and final gears in a train of gears or of any two meshing gears. Essentially, the ratio of the input speed to the output speed of a geared set of shafts. These can be decided to choose a ratio of Speed (large gear driving small gear) or Torque (small gear driving large gear).
Gear ratios are often simplified fractions of the number of teeth on each gear. This is because (at least in Vex) gears have a proportional number of teeth to their diameter which defines the difference in output speed. From the above image, there is a 12 tooth gear, driving a 36 tooth gear which would constitute a 3:1 ratio.
The benefit of gear ratios is the ability to choose speed vs torque in a system. These have an inverse relationship however where a faster output will have lower torque. With the above example, the output shaft has 3x the torque, but also 1/3 the speed.