Abstract
When shaft angle Σ, offset E and gear ratio i0 are given, it is clarified how the lengthwise contact ratio common to all kinds of gear is defined and how the pitch cones of hypoid gear are chosen to obtain the equal lengthwise contact ratios of drive and coast sides as follows. (1) Σ, E and i0 determine the field of relative velocity in the static space, points Pw having the same relative velocity Vrsw draw a line element Lpw of cylinder around the instantaneous axis and the Lpw is also the intersection of the surfaces of action. In the case of cylindrical and bevel gears, the Lpw coincides with the instantaneous axis because of Σ → 0 or π or E → 0. (2) By rotating the Lpw around the axes of pinion and gear, the pitch hyperboloids are drawn respectively, which are the pitch surfaces common to all kinds of gear because the Lpw is the intersection of the surfaces of action. (3) The tooth trace is defind as the curve which the path of contact put on the Lpw draws in the coordinate system which rotates with each gear respectively. (4) The lengthwise contact ratio is defined as the one calculated along the Lpw. (5) In the case of hypoid gears, equal lengthwise contact ratios of drive and coast sides can be realized by choosing the pitch cones contacting at Pw so that the intersection between the pitch cone and the surface of action coincides with the Lpw and the resultant pitch angle of cone is different from Wildhaber's one and almost equal to the inclination angle of the instantaneous axis, which is shown in the design examples.
