Modeling of teeth surfaces of contacting details of a hypoid gear

Hypoid gears are intended for transmitting the rotation between skew shafts and are characterized by higher loading capacity, ease of movement, and operation quietness. Base surfaces (axoids) of such a gear are hyperboloids of revolution of one sheet. The surface of the tooth of the input component $S$ is obtained by helical motion of a circumference around the detail axis of rotation with a simultaneous decrease in the radius of this circumference; at the same time, centers of circumferences of the family must lie on the axoid of the input component, i.e., form a helical line on this hyperboloid. In this work, exact analytical equations of the surface $S$ are obtained and the input component tooth surface is found as an envelope of the family of surfaces $S$. This family is formed by rotations of the surface S around the axis of rotation of the input detail with a simultaneous rotation around the axis of the output detail (after a shift to the distance between the axes). The first and second rotations are performed at angles $\tau$ and $-\tau/i$, respectively, where $i$ is the gearing ratio. Parametric equations of the tooth contact line as a regular curve along which the envelope is tangential to the surface of the family (the characteristic) are obtained.