Geometrical theory of cutting stock with torus end mills in five-axis CNC machining and its applications in machining simulation

We have established the geometric theory of cutting stock with flat end mills in five-axis CNC machining and an accurate and efficient approach to 3D geometric modeling of un-deformed chips in this machining (Chang et al. Comput Aided Des 88:42–59, 2017). The work has laid a theoretical foundation for geometrical and physical simulation for five-axis milling. However, fillet and ball end mills are more popular than flat end mills. The equations of the instantaneous cutting edges of a fillet end mill are more complicated in the geometric theory, and the boundary of the area covered by the cutting edges is more difficult to determine. Therefore, this article is to formulate the instantaneous cutting edges and their critical points in five-axis milling and to illustrate how to use the boundary construction diagram to determine the boundary with two examples of the geometric and the physical simulations. This work is verified with the valid results of the examples, and the results indicate that this model can take into account the errors of cutter and machine tool. Therefore, the geometric theory is viable, and its application on the five-axis milling simulation is feasible for high-performance machining.