 Gear ratio

A gear ratio states the relation between the rotational motions of connected gears. Because conventions differ, specify the input, output and direction. Here the positive reduction ratio isso $i>1$ means that the output has lower angular speed. For a single external pair on fixed axes, tooth-passage counting givesThe minus sign assumes the same positive rotation convention on both parallel axes. KHK uses the driven-to-driver tooth-count ratio for its speed-reduction examples.^[ref]

Ratios multiply through successive stages of a gear train. Consider a 12-tooth input driving a 36-tooth gear, with a 15-tooth gear rigidly attached to that intermediate rotating body and driving a 45-tooth output. Each stage reduces angular speed by three, so the combined reduction ratio is nine. An input of $1800$ revolutions per minute yields an output of $200$ revolutions per minute. Two external meshes reverse direction twice, so input and output rotate in the same direction.

An intermediate gear used only to pass rotation onwards gives a different result. If a 12-tooth input drives a 36-tooth intermediate gear that itself drives a 48-tooth output, the ratio is $(36/12)(48/36)=4$: the intermediate tooth count cancels. The previous ninefold reduction required two distinct tooth counts rigidly sharing the intermediate rotation.

For steady ideal transmission, conservation of energy gives $|{\tau }_{\mathrm{out}}|=i|{\tau }_{\mathrm{in}}|$. If gear efficiency is $\eta =P_{\mathrm{out}}/P_{\mathrm{in}}$, the corresponding relation is $|{\tau }_{\mathrm{out}}|=\eta i|{\tau }_{\mathrm{in}}|$. With $i=9$, $\eta =0.90$ and input torque $2\mathrm{N}\mathrm{m}$, the output torque is $16.2\mathrm{N}\mathrm{m}$. This power balance assumes steady operation: accelerating rotating components can temporarily store kinetic energy. A measured gear ratio establishes kinematics, while torque capacity and losses require additional mechanical information.