A new geometric meshing theory for a closed-form vector representation of the face-milled generated gear tooth surface and its curvature analysis

• A new geometric meshing theory is proposed to obtain the mathematical model of the face-milled generated gear.
• The tooth surface is obtained as a closed-form vector representation.
• Curvature analysis is directly conducted with differential geometry approaches.
• The proposed method is more efficient and straightforward than previous methods.

The tooth surface and its curvature are fundamental inputs to evaluate the contact and transmission of spiral bevel gears or hypoid gears. Currently, the tooth surface of the face-milled generated gear is represented as an implicit form of three parameters, and one of which can be eliminated according to the well-known equation of meshing. Unfortunately, it is not easy to solve the equation of meshing due to the complex process of gear generation. Moreover, it is complicated to calculate the derivatives of the equation of meshing, and this makes it inefficient to use the results of well-established differential geometry to conduct curvature analysis. To address this problem, a new geometric meshing theory is proposed to obtain a closed-form (explicit) vector representation of the tooth surface, and curvature analysis is directly implemented with differential geometry equations. As a consequence, the calculation is straightforward and efficient. The example of a face-milled spiral bevel gear is presented as the application of the proposed method.