For the long epicycloid bevel gear, Litvin et al. Studied the pitch cone design of the hypoid gear machined by the end face hobbing method, and deduced the relationship and calculation formula between the geometric parameters on the pitch plane. After that, they established the mathematical model of machining equal height bevel gear by end face hobbing method, and simulated the gear tooth surface.
Zhang and Wu studied the tooth surface geometry of bevel gear machined by end face hobbing method, deduced the tooth root transition fillet equation, and accurately modeled the tooth surfaces of large and small wheels. Fan established the mathematical model of end face hobbing method, studied the contact analysis of bevel gear tooth surface, improved the traditional contact analysis algorithm of bevel gear tooth surface, and simplified the solution method of contact ellipse. Shih et al. Also established the numerical control machining mathematical model for the six axis numerical control gear milling machine, analyzed the influence of the movement change of each numerical control axis on the tooth surface shape of bevel gear, and established the unified mathematical model of end face hobbing method and the tooth surface contact analysis method.
Simon analyzed the influence of bevel gear tooth surface modification on the contact area and transmission error of Klinger Cycloid Bevel gear, and put forward the modification method of double circular arc of tool profile. Vimercati has systematically studied the mathematical model of face hobbing, tooth surface modeling, tooth root transition surface, tooth surface contact analysis and load contact analysis. Nishino establishes the mathematical model of machining bevel gear by end face hobbing method, deduces the equation of large gear surface, and deduces the equation of small gear surface by conjugate method. On this basis, the solid modeling of bevel gear pair is carried out, the curvature relationship between matching bevel gear tooth surfaces is analyzed, and the bearing contact analysis is studied.