The Third Law of Gearing: A Surface Defining Maximum Relative Curvature of Conjugate Gear Flanks

Abstract Gear flanks are a special case of two surfaces in direct contact; the restriction being that the surface normal between the two flanks maintains a desired gear speed ratio. This restriction is commonly referred to as reciprocity and the surfaces are considered conjugate. At any instant, there exists a maximum and minimum relative curvature for conjugated gear flanks. These extreme curvature values are affected by tooth profile modifications (viz., profile relief and lead crown). The third law of gearing establishes limiting values on the extreme relative curvatures for unmodified gear flanks. The lower limit is zero implying line contact between the gear flanks. The upper limit on the relative curvature between gear flank depends on the instantaneous pressure angle, spiral angle, and gear architecture (viz., speed ratio, shaft angle, and offset). An overview of these relations is presented. Introduced is a curvature surface that illustrates all candidate maximum relative curvatures by considering all possible pressure angles. This surface is independent of the tooth profile geometry (viz., involute, cycloid, or other). An automotive hypoid gear set is presented to illustrate the curvature surface.