Modeling of Milling Force for Ball End Milling Cutter Based on Oblique Cutting

In order address the complex three-dimensional force-field distribution and the significant dynamic variation of undeformed chip thickness during cutting with a ball end mill under varying helix-angle conditions, this paper proposed a milling force modeling method that integrated oblique cutting theory with dynamic kinematic simulation, enabling high-accuracy modeling and prediction of cutting forces in multi-axis ball end milling. An analytical framework of oblique cutting mechanics was established based on the equivalent planar method, where three-dimensional cutting was converted into two-dimensional planar cutting through spatial coordinate transformation. A composite mechanical model that simultaneously incorporated both shear and ploughing effects was derived. This model reveals the control mechanism of the tool inclination angle on the material flow direction in the cutting zone, the morphology of the shear deformation zone, and the stress distribution. Subsequently, the geometric characteristics of the cutting-edge profile of the ball end mill were constructed. Combined with the tool-workpiece kinematic coupling model, the differential equations of the tool tooth motion were solved, and dynamic machining surface topography was simulated using an improved Z-MAP algorithm, enabling the extraction of time-varying undeformed chip thickness distribution. Furthermore, a multi-scale mechanical mapping strategy was proposed, where the cutting edge was discretized into micro-cutting units along the curved direction. Based on the analytical oblique-cutting model, iterative integration of tangential, radial, and axial forces for each micro-unit was performed, ultimately superimposing them to obtain complete three-dimensional milling force time-domain signals. Finally, an experiment was carried out for verification, the results indicate that the maximum prediction errors of milling forces in the axial, feed, and width directions are 18.3%, 10.8%, and 22.4%, respectively, verifying the accuracy and applicability of the model in force analysis of complex tool geometries. This research method combined macroscopic kinematic simulation with microscopic mechanical analysis and provided theoretical support for process parameter optimization, tool structure design, and machining stability enhancement of ball end milling.