Synthesis of the base curves for N-lobed elliptical gears

Motivated by the need to synthesize the tooth profiles of noncircular gears, we approach the synthesis of the tooth profile of circular spur gears using their pitch circle, rather than their base circle. We do this by means of envelope theory. The proposed formulation gives the involute tooth profile and its well-known base circle for any pitch radius and profile angle of the rack cutter, which coincides with the pressure angle for circular gears. Then, the foregoing approach applies to the synthesis of the base curves of noncircular gears with involute tooth profiles and of their rack. We do this by resorting to basic differential geometry using the Euler-Savary Theorem, rather than to envelope theory. In particular, the formulation of both base curves for the right and left involute tooth profiles is obtained, for the first time, for N-lobed elliptical gears and their rack through the formulation of the pitch curves and their evolutes. The proposed formulation is illustrated with numerical results.