= Gear train
A <gear train> is an interconnected arrangement of <gears> that transmits rotation through one or more meshes. Its behavior depends on which <gears> mesh and which are rigidly attached to the same rotating body. A <simple gear train> passes motion through successive individual <gears>; a <compound gear train> uses at least one intermediate rotating body carrying different <gears> rigidly together.<KHK Gear Nomenclature>[The ABC's of Gears, Basic Guide B, §1.1, printed pp.4–5, single- and two-stage gear trains, Figures1.1 and1.3]{ref}
For a chain of external meshes on fixed parallel axes, each mesh satisfies $\omega_{j+1}/\omega_j=-N_j/N_{j+1}$. Multiplication gives the overall signed <gear ratio>; two rigidly joined <gears> instead share exactly the same <angular velocity>. These are different connection rules and must not be interchanged.
In a <simple gear train> with tooth counts 16, 40 and 64, the overall relation is
$$
\frac{\omega_{\rm out}}{\omega_{\rm in}}
=\left(-\frac{16}{40}\right)\left(-\frac{40}{64}\right)=\frac14.
$$
The intermediate count cancels, and the two reversals give equal input/output rotation directions. That intermediate <gear> can still change spacing and mounting geometry even though it does not change this reduction ratio.
For a <compound gear train>, let a 16-tooth input drive a 48-tooth <gear>, rigidly joined to a 12-tooth <gear> that drives a 60-tooth output. Then
$$
\frac{\omega_{\rm out}}{\omega_{\rm in}}
=\left(-\frac{16}{48}\right)\left(-\frac{12}{60}\right)=\frac1{15}.
$$
An input of $1500$ revolutions per minute produces $100$ revolutions per minute at the output. The two intermediate tooth counts do not cancel because they belong to different <gears> sharing rotation, rather than one <gear> engaging both neighbors.
In steady operation, stage <gear efficiencies> multiply when all intervening losses are included in their stated power boundaries. If the two stages each have efficiency $0.96$, the total is $0.9216$; at input <torque> $1\,\mathrm{N\,m}$, the output <torque> is $15(0.9216)=13.824\,\mathrm{N\,m}$. These fixed-axis relations do not automatically describe arrangements whose gear axes move. In all cases, measured speeds test kinematics, while measured <power> is needed to establish losses.
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