Regularization of mathematical model for chip flow angle catastrophe

Abstract

The chip flow angle (CFA) catastrophe in double-edged cutting results in a significant reduction in the cutting force, which can benefit the applications. However, established potential functions (i.e., cutting power calculation functions) of mathematical models for the CFA catastrophe are presented in the form of transcendental functions with two control parameters and one state parameter, which are extremely complex. A method is proposed herein to realize the regularization of the potential functions and establish mathematical models in a standard form and with complete content for the CFA catastrophe. Using this method, the potential function of the CFA catastrophe is expanded into a k-order Taylor polynomial at each midpoint of $N$ end-to-end equally partitioned intervals of the state parameter using the Taylor function provided in MATLAB. The potential function after piecewise Taylor expansion is transformed into the same form as the potential function of the standard cusp catastrophe model by truncating the first five terms of the Taylor polynomial and eliminating the third-order term of the state parameter with elementary transformation. Hence, the regularization of potential function is realized. Subsequently, the regularization of equilibrium surface and bifurcation set can be realized based on the conclusions of the catastrophe theory. Regularization errors of the potential function, equilibrium surface, and bifurcation set are defined to evaluate the effectiveness of this regularization method. The problem of calculating regularization errors is regarded as an optimization problem. The “simulannealbnd” function provided in MATLAB is used to solve the problem. Applying the proposed method, the regularization of a mathematical model for the CFA catastrophe established by the predecessor is completed; a mathematical model (i.e., standard cusp catastrophe model) in a standard form and with complete content for the CFA catastrophe is established; and the corresponding regularization errors are analyzed. The regularization errors of the potential function, equilibrium surface, and bifurcation set curves are 5.485 5 × 10⁻⁴%, 0.320 6%, and 4.653 9%, respectively. Based on the equilibrium surface and the bifurcation set curves constructed using the regularized mathematical model for the CFA catastrophe, the mechanism of the CFA catastrophe and the specific approach to render the cutting system operable in a low-energy consumption state by controlling the historical change path of the control point are analyzed. This study will promote the rational use of the CFA catastrophe.

Keywords

Chip flow angle (CFA) catastrophe / Mathematical model / Regularization / Potential function / Catastrophe theory / Cusp catastrophe model

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