On optimizing the basic geometry of hypoid gears

• Dual vectors allow more reliability in formulation and interpretation.
• Calculation of hypoid gear dimensions and attributes without iteration
• Optimization of the friction loss coefficient, the theoretical efficiency etc
• Optimization of weighted axial and radial forces at the pinion

By the help of dual numbers and vectors, an intrinsic parametric representation of the Plücker conoid is derived that is the ruled surface consisting of all composite screw axes generated by two given skew screws. In the field of gearing an application results in the design of pitch surfaces of a double screw gearing and especially of hypoid gears. For basic geometric dimensions parametric representations are derived concerning pitch cones, contact normal, spiral angles, limit pressure angles, sliding velocity deviation, friction loss, and efficiency. These magnitudes simply depend on the introduced reference angle and length as variables. For hypoid gears, the paper shows that the free optimization problems for the absolute value of the composite relative velocity, the friction loss coefficient, and the theoretical efficiency are uniquely solvable. As an example for a constraint optimization problem the degree of mesh overlap is considered. Finally, the free optimization of weighted axial and radial forces at the pinion of a hypoid gear is solved.