The optimal design mathematical model of single-stage bevel gear nutation reducer is as follows

Where 1 ≤ J ≤ 12.

The model is a discrete optimization design problem with 4 design variables and 12 optimization constraints. The optimization results can be obtained by lingo software.

The tooth surface equation of double circular arc spiral bevel gear is established, and then the three-dimensional model of single-stage nutation double circular arc spiral bevel gear is established according to the optimized gear parameters, as shown in the figure.