 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.^[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 isThe 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. ThenAn 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 $1\mathrm{N}\mathrm{m}$, the output torque is $15(0.9216)=13.824\mathrm{N}\mathrm{m}$. 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.