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When a pair of paths of contact intersecting at the design point P0 are given on the normal plane Sn which is perpendicular to the relative velocity at Po, the conditions for the pair of paths of contact to realize equal strengths of tooth and bearing on both drive and coast sides under the same transmitted torque are discussed. Those conditions are expressed by 5 equations with 7 independent variables, which draw 5 separate reference paths of contact on the plane Sn geometrically. In designing gears, 3 variables (shaft angle, offset and gear ratio) are given, so that the equations have no solutions generally. However, cylindrical, crossed helical (including worm) and straight bevel gears can satisfy those conditions because their 5 reference paths of contact are superimposed and become at most one or two lines rectangular each other. Spiral bevel and hypoid gears with a rectangular shaft angle can not satisfy those conditions and their reference paths of contact vary according to the variables (offset, gear ratio and spiral angle), so that they have been designed as similar gears to ones whose performances are already assured by fixing the variables in some degree.

nihon kikai gakkai ronbunshū. c

Theory on Designing Gears for Power Transmission (5th Report, A Pair of Paths of Contact realizing Equal Strengths of Tooth and Bearing on Both Drive and Coast Sides)