= Idler gear
An <idler gear> is an intermediate <gear> that passes rotation between other gears without using a separately sized, rigidly coupled gear to form another reduction stage. In a <simple gear train>, its tooth count cancels from the ideal endpoint ratio, but it changes placement, intermediate speed and the number of direction reversals.
For example, a20-tooth input driving a30-tooth <idler gear> that drives a50-tooth output gives
$$
\frac{\omega_{\rm out}}{\omega_{\rm in}}
=\left(-\frac{20}{30}\right)\left(-\frac{30}{50}\right)=0.4.
$$
At750 input revolutions per minute, the idler turns at500 in the opposite direction and the output at300 in the original direction. Replacing the idler by60 teeth preserves output speed but changes idler speed to250 revolutions per minute.<KHK Gear Nomenclature>[Basic Guide B §1.1, pp.4–5, tooth-count speed relations and external-mesh reversals]{ref}
With standard module $2\,\mathrm{mm}$ and collinear centers, the30-tooth idler gives input/output axis spacing
$$
a_{\rm total}=\frac m2(N_{\rm in}+2N_{\rm idler}+N_{\rm out})=130\,\mathrm{mm}.
$$
The60-tooth replacement gives190mm, despite the unchanged output ratio.
An <idler support reaction> also remains necessary even when its net transmitted <torque> is zero. For the30-tooth arrangement, suppose input torque is $2\,\mathrm{N\,m}$, so both ideal mesh tangential loads are $100\,\mathrm N$. Their moments about the idler's center cancel, but in this collinear layout the two tangential forces point in the same direction on the idler. Equal, opposite radial mesh components cancel, leaving a mesh-induced shaft-support reaction of $200\,\mathrm N$ opposite the tangential forces.
\Image[/idler-gear-force-balance.png]
{title=Pitch-circle sketch of cancelling idler torques and a nonzero mesh-induced support reaction}
The drawing shows tangential mesh components and their balancing support reaction, rather than physical tooth profiles. Additional shaft loads are outside this ideal balance. The idler therefore affects geometry, forces and potential losses even though its count cancels from the ideal speed ratio.
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