Spiral bevel gears usually are cut by the machine tools which are specifically made for manufacturing spiral bevel gear. For some practical applications, such as repair, prototype, and small batch, there are only several spiral bevel gears need to be machined. It seems not wise to buy a gear manufacturing machine tool for cutting those several gears. Consequently, some other manufacturing methods are introduced to cope with this situation. Recent advances in computer numerical control (CNC) milling machine tools make it possible to manufacture good quality spiral bevel gears. Moreover, CNC milling machine tools are usually cheaper than the specific gear manufacturing machine tools. Even more, they are not only cut gears but also other parts. Therefore, computer numerical control (CNC) milling is introduced in industry to cut spiral bevel gears.

End milling and flank milling are two modes of CNC milling. Compared to end milling, flank milling has been highlighted in quality enhancement, manufacturing time and cost reduction. Theoretically, the result of flank milling a developable ruled surface does not contain geometric deviations. Hence, flank milling has been widely used to machine the parts with ruled surface or the surface close to ruled surface, such as turbines and blades. Due to the fact that the tooth surfaces of spiral bevel gears are close to ruled surface (the tooth surface of Format-cut spiral bevel is part of a cone), flank milling can also be used to cut spiral bevel gears.

However, there is no literature about flank milling spiral bevel gears. Moreover, the current existing tooth surface models (except the tooth surface of Format-cut spiral bevel gear) are implicit, which makes further calculation inefficient and difficult. To address these two problems, a new ruled tooth surface design is proposed to spiral bevel gears design. Subsequently, the design model is used to be machined with five-axis flank milling by introducing a new tool path planning approach. To obtain the simulate machined tooth surface, a new geometric envelope approach is proposed to calculate the cutter envelope surface as a closed-form representation. Then the result of geometric deviations, which are obtained by comparing the design tooth surface and the simulate machined tooth surface, shows the flank milling is appropriate to cut spiral bevel gears. Furthermore, the conventional face-milled model is also used as the design model for flank milling, and another result of geometric deviations is obtained. Subsequently, both results are compared, and the comparison shows that the proposed design reduces the geometric deviations effectively. Moreover, the closed-form simulate machined tooth surface can only be obtained by using the proposed design.