 Reverted gear train

A reverted gear train returns the output rotation axis to the input axis, usually through a two-stage compound gear train. The input and output gears can rotate independently on coaxial shafts, while the intermediate gears share another rotating body. Coaxial axes are a layout condition in addition to the desired gear ratio.

For standard unshifted spur gears, let gears1 and2 mesh in the first stage and gears3 and4 in the second, with2 and3 rigidly connected. The common intermediate-axis separation requiresIf both stages use the same gear module, this reduces to $N_1+N_2=N_3+N_4$. The reduction ratio independently satisfies $i=(N_2/N_1)(N_4/N_3)$. Both conditions must be satisfied; multiplying arbitrary stage ratios does not ensure a reverted layout.^[ref]

For an original design example, choose counts20,60,16 and64, with module $2\mathrm{mm}$ in both stages. Both sums are80, so both separations are $80\mathrm{mm}$. The reduction is $3\times 4=12$, taking an input of1440 revolutions per minute to an output of120. Two external meshes give the same input/output rotation direction.

The gears occupy different axial planes so that coaxial input and output do not require them to be one rigidly joined gear. A practical design still needs independent shaft support, compatible tooth geometry in each stage and adequate clearances. Operating with shifted tooth geometry or another mounting changes the applicable separation equations; the simple tooth-sum condition belongs to the stated standard-gear model.