Differential Geometrical Conditions of Hypoid Gears with Conjugate Tooth Surfaces

Abstract

Hypoid gears are one of the most general form of gearing, and a theoretical solution for them has been studied by many researchers. Many hypotheses and theorems about these gears have been proposed—some of them correct and many of them wrong. The tooth surfaces are parts of general curved surfaces and they must have principal directions and principal curvatures on every contact point. However, there has been no detailed research on the fundamental elements of the surface. This study develops necessary conditions for determining these curvatures and principal directions for conjugate gearing with a contact line by introducing the concept of geodesic torsions. [S1050-0472(00)00503-1]

The Mesh Characteristics of Arch Cylindrical Gear and Its Digitization Analysis

“…The following conclusions can be made from Fig.7: (1)The meshing tooth-surfaces are coincident, i.e. 3D graphs described by equation (9) and equation (10) are coincident.…”

“…According to principles of gears engagement, if the meshing line expressed by equation (9) coincides with the meshing line expressed by equation (10), a conclusion can be get that a pair of gears whose tooth profiles meet the demands of equation (4) and (5) can engage well and the meshing gear pair is line contact. Because of the existence of trigonometric function variables, it is complicated to solve equation (9) and (10) directly.…”

Multiphysics Investigations on the Dynamics of Differential Hypoid Gears

“…Also, the lubricant inlet entrainment flow vector was assumed to be along the minor axis of the contact ellipse. However, experimental evidence [26] and numerical investigations [27][28][29] have suggested significant side-leakage flow from the contact footprint along the major axis of the contact ellipse. The repercussions of ignoring the side leakage flow is breach of continuity of flow condition, as well as errors introduced in the evaluation of contact temperatures due to the side leakage flow out of the contact.…”

Geometric and Meshing Properties of Conjugate Curves for Gear Transmission

“…In [10], Wu and Luo studied a geometric theory of conjugate tooth surfaces and derived curvatures equations in terms of the limit functions of the first kind and considering the mating surfaces subjected to relative screw motion with constant translational and rotational velocities. Ito and Takahashi [11] analyzed curvatures in hypoid gears starting from a classical differential geometry point of view, but then introducing kinematic relationships. By employing the theory of screws, Dooner [12] provided the third law of gearing and formulated the limiting relationship between the radii of curvature of conjugate surfaces, which is valid only for the reference pitch surfaces.…”