Stability Analysis in Drilling Processes: Nonlinear Modeling and Prediction of Limit Cycle Whirling Oscillations

The high demand for increased productivity pushes machining processes to their stability limits. Drilling with twist drills is one of the most common machining operations. In drilling, two primary oscillation phenomena are observed, called whirling and chatter. This work focuses on the mathematical description of whirling oscillations to optimize the tool geometry and enhance the process understanding. A nonlinear physical model with two degrees of freedom is developed to represent the drilling process. The model includes a mechanistic cutting force approach with experimentally determined cutting force coefficients, process damping at the main cutting edges, caused by the compressed workpiece material between the cutting edge and the newly cut workpiece surface, radial forces on the chisel edge in the center of rotation of the tool, and a model to describe the supporting effect of the margins against the borehole wall. The numerical treatment of the resulting delay differential equations is discussed, with consideration given to the history of the evolution of the tool motion. To derive the cutting force coefficients, thrust force and drilling torque measurements were conducted with a standard twist drill. Three-dimensional coefficients were derived from the two load signals by considering the limits of the frictional behavior in the cutting process. To validate the physical model, boreholes were drilled with different pre-drilled borehole radii using test tools with three different margin widths. Additionally, a method is proposed to reconstruct the tool vibrations from a measured borehole geometry, enabling the calculation of the whirling frequencies in a post-processing step. By including the nonlinear bounding margin contact, the model accurately predicts limit cycles of whirling oscillations. It is demonstrated that whirling frequencies are crucial to obtain accurate margin contact conditions. The whirling frequencies and amplitudes are predicted with high accuracy for a wide range of process conditions. The model reveals a strong tendency of the borehole geometry to adapt to the twisted shape of the tool body. It also predicts the axial contact length of the margins, which was validated by friction marks on the used drilling tools. In addition, it reveals that the process damping at the main cutting edges is secondary to the process damping at the margin contact, regarding the whirling oscillations, given by the significantly smaller contact regions. The validated model is then used to optimize the margin geometry. Measurements with prototypes featuring the optimized margin design showed a reduction in the diameter deviations of up to 60.3 % and an improvement of the borehole circularity by up to 47.8 % compared to the reference tool with standard geometry. However, a limitation of the physical model was identified for the largest margin widths, where abrasion is strongly increased due to the excessive friction. Incorporating this effect into the model could enable predictions of the borehole diameter deviations. Furthermore, drill misalignment and grinding errors were not considered in the current model. The symmetry of the model prevents an asymmetric evolution of the borehole geometry. To predict the borehole straightness along the depth, it is suggested to implement such imperfections in future refinements of the model.