Tangency of one-sheeted hyperboloids as axoids of the hypoid gearing

One-sheeted hyperboloids of revolution are basic surfaces (axoids) of the input and output details of hypoid gears occupying a prominent place in the large variety of gear trains. The main feature of the hypoid gear is the skew axes of rotation. Such arrangement of the axes allows one to provide the smoothness of movement, quietness of the operation, and higher load capacity of the mechanism. In this work, geometrical aspects of modeling the hypoid gear with perpendicular axes are considered, namely, (i) for parameters of the hyperboloids, the condition providing the existence of a common rectilinear generator passing through a common point of their striction lines has been found; (ii) using this condition, an original proof of the following well-known fact has been obtained: rolling of hyperboloids with tangency along a rectilinear generator when they rotate about their axis with the corresponding ratio of velocities is possible only if the imaginary axes of the rotating hyperbolas forming the hyperboloids are equal; (iii) it has been proved that, at given displacements of the axes and gearing ratio, parameters of the hyperboloids are determined uniquely.