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

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

Correction and accuracy improvement of non-parallel shear zone model

When applying the non-parallel shear zone model to predict the cutting process parameters of carbon steel workpiece, it is found that there is a big error between the prediction results and the experimental values. And also, the former approach to obtain the relevant cutting parameters of the non-parallel shear zone model by applying coordinate transformation to the parallel shear zone model has a theoretical error – it erroneously regards the determinant (|J|) of the Jacobian matrix (J) in the coordinate transformation as a constant. The shape of the shear zone obtained when |J| is not constant is drew and it is found that the two boundaries of the shear zone are two slightly curved surfaces rather than two inclined planes. Also, the error between predicted values and experimental values of cutting force and cutting thrust is slightly smaller than that of constant |J|. A corrected model where |J| is a variable is proposed. Since the specific values of inclination of the shear zone (α, β), the thickness coefficient of the shear zone (as) and the constants related to the material (f0, p) are not given in the former work, a method to obtain the above-mentioned five constants by solving multivariable constrained optimization problem based on experimental data was also proposed; based on the obtained experimental data of AISI 1045 steel workpiece cutting force, cutting thrust, chip thickness, the results of five above-mentioned model constants are obtained. It is found that, compared with prediction from uncorrected model, the cutting force and cutting thrust of AISI 1045 steel predicted by the corrected model with the obtained constants has a better agreement with the experimental values obtained by Ivester

Correction and accuracy improvement of non-parallel shear zone model

When applying the non-parallel shear zone model to predict the cutting process parameters of carbon steel workpiece, it is found that there is a big error between the prediction results and the experimental values. And also, the former approach to obtain the relevant cutting parameters of the non-parallel shear zone model by applying coordinate transformation to the parallel shear zone model has a theoretical error – it erroneously regards the determinant (|J|) of the Jacobian matrix (J) in the coordinate transformation as a constant. The shape of the shear zone obtained when |J| is not constant is drew and it is found that the two boundaries of the shear zone are two slightly curved surfaces rather than two inclined planes. Also, the error between predicted values and experimental values of cutting force and cutting thrust is slightly smaller than that of constant |J|. A corrected model where |J| is a variable is proposed. Since the specific values of inclination of the shear zone (α, β), the thickness coefficient of the shear zone (as) and the constants related to the material (f0, p) are not given in the former work, a method to obtain the above-mentioned five constants by solving multivariable constrained optimization problem based on experimental data was also proposed; based on the obtained experimental data of AISI 1045 steel workpiece cutting force, cutting thrust, chip thickness, the results of five above-mentioned model constants are obtained. It is found that, compared with prediction from uncorrected model, the cutting force and cutting thrust of AISI 1045 steel predicted by the corrected model with the obtained constants has a better agreement with the experimental values obtained by Ivester