Regularization of mathematical model for chip flow angle catastrophe

• 作者: Shao-Nan Zhang Dong-Dong Cheng Liang-Shan Xiong
• 来源期刊: 先进制造进展（英文版）
• 年，卷（期）: 2021,（4）
• 页码范围: 568-579
• 页数: 12页
• 语种: 英文

摘要

The chip flow angle (CFA) catastrophe in dou-ble-edged cutting results in a significant reduction in the cutting force,which can benefit the applications.However,established potential functions (i.e.,cutting power calcu-lation 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 poten-tial 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.Subse-quently,the regularization of equilibrium surface and bifurcation set can be realized based on the conclusions of the catastrophe theory.Regularization errors of the poten-tial 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 "simulan-nealbnd" function provided in MATLAB is used to solve the problem.Applying the proposed method,the regular-ization of a mathematical model for the CFA catastrophe established by the predecessor is completed;a mathemat-ical model (i.e.,standard cusp catastrophe model) in a standard form and with complete content for the CFA catastrophe is established;and the corresponding regular-ization errors are analyzed.The regularization errors of the potential function,equilibrium surface,and bifurcation set curves are 5.4855 x 10-4％,0.3206％,and 4.6539％,respectively.Based on the equilibrium surface and the bifurcation set curves constructed using the regularized mathematical model for the CFA catastrophe,the mecha-nism of the CFA catastrophe and the specific approach to render the cutting system operable in a low-energy con-sumption state by controlling the historical change path of the control point are analyzed.This study will promote the rational use of the CFA catastrophe.