= Gear ratio
{title2=$i=|\omega_{\rm in}|/|\omega_{\rm out}|$}
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 is
$$
i=\frac{|\omega_{\mathrm{in}}|}{|\omega_{\mathrm{out}}|},
$$
so $i>1$ means that the output has lower <angular speed>. For a single external pair on fixed axes, tooth-passage counting gives
$$
\frac{\omega_{\mathrm{out}}}{\omega_{\mathrm{in}}}=-\frac{N_{\mathrm{in}}}{N_{\mathrm{out}}},
\qquad i=\frac{N_{\mathrm{out}}}{N_{\mathrm{in}}}.
$$
The 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.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §1.1, printed pp. 4–5, equations (1.1)–(1.2)]{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> $2\,\mathrm{N\,m}$, the output <torque> is $16.2\,\mathrm{N\,m}$. 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.
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