Experimental Modeling of the Bifurcation Set Equation of the Chip-Splitting Catastrophe in Symmetrical Straight Double-Edged Cutting

Abstract

The chip-splitting catastrophe (CSC) generated by symmetrical cutting with a straight double-edged tool will lead to a significant reduction in cutting force. This has enormous potential for energy-saving machining and for the design of energy efficient cutting tools. The premise of the utilization is to establish a mathematical model that can predict the critical conditions of CSC. However, no related literature has studied the prediction model of CSC. Therefore, this paper proposes an experimental method based on catastrophe theory to establish a model of CSC bifurcation set equations that can predict critical conditions. A total of 355 groups of experiments are conducted to observe the critical conditions of CSC in symmetrical straight double-edged cutting, and 22 groups of experimental data of the critical conditions were acquired. The modeling process is converted into the optimal solution of the function coefficient value when the mapping function from a set of actual control parameters to theoretical control parameters (u, v, w) is a linear function. The bifurcation set equation of CSC is established, which can predict CSC in the symmetrical cutting of a straight double-edged turning tool with any combination of edge angle and rake angle. With verification, it is found that the occurrence of CSC has obvious regularity, and the occurrence of CSC will lead to a maximum reduction of 64.68% in the specific cutting force. The predicted values of the critical cutting thickness for the CSC of the established equation are in good agreement with the experimental results (the average absolute error is 5.34%). This study lays the foundation for the energy-saving optimization of tool geometry and process parameters through the reasonable utilization of CSC.