Mathematical modeling and parametric solution of the flute geometry for solid end mills

Abstract

The flute of a solid end mill directly determines parameters such as the rake angle, core diameter, and flute opening angle, thereby affecting the cutting performance of the tool. To address the complexity of spatial analytical models and the possibility that direct radial projection may neglect helical phase compensation, the grinding-wheel–blank process system was investigated. Grinding-wheel and blank models were established in a local normal plane, the normal-plane section was obtained according to the envelope principle, and the radial-section curve was then obtained through inverse coordinate transformation. A model for solving the flute parameters was subsequently established, and UG secondary-development technology was used to develop flute-grinding software for solid end mills. To verify the effectiveness of the proposed method, grinding experiments were conducted under identical operating conditions using a 1A1 grinding wheel and a GU25UF carbide blank. A radial-projection baseline (RP), in which only the helical phase compensation was removed, was compared with the proposed normal-plane envelope–inverse helical transformation method (NP) through a controlled geometric ablation comparison based on the same source geometry. The results showed that the core diameter, rake angle, and flute opening angle obtained by the NP method were 4.038 mm, −11.24°, and 89.68°, respectively, with errors of 2.70%, 5.55%, and 3.05% relative to the physical measurements. Compared with the RP method, the NP method reduced the rake-angle and flute-opening-angle errors by 74.90% and 81.18%, respectively, and the geometric offset at the flute-opening endpoint was 1.27 mm. The helix-angle sensitivity analysis further showed that the RP projection error increased markedly with increasing helix angle. The proposed method reduces the geometric error caused by flute-section projection; the maximum relative error among the three parameters was 5.55%, demonstrating good engineering applicability.