Newton-Simpson-based predictor-corrector methods for milling chatter stability prediction

Milling chatter, a form of self-excited vibration, can cause significant damage in machining and manufacturing processes. By selecting appropriate milling parameters, milling chatter can be effectively mitigated without sacrificing milling efficiency. Within the framework of the semi-discretization scheme, this paper introduces the Newton-Simpson-based predictor-corrector methods to compute milling stability lobe diagrams. Firstly, the milling delay differential equation is transformed into the state space form, and then the time-delayed term and the periodic coefficient matrix of the state space equation are treated as an operator. Secondly, the tooth passing period is divided into the free vibration period and the forced vibration period. During the forced vibration period, the time-delayed term and the periodic coefficient matrix are approximated as a holistic operator over two different time intervals using the Newton interpolation polynomials and the Simpson formula, respectively. Finally, the state transition matrix is constructed based on the predictor-corrector scheme, and the stability lobe diagrams are obtained by applying Floquet theory. The convergence rate and calculation accuracy of the proposed methods are compared with those of the existing predictor-corrector methods, semi-discretization, and full-discretization methods. The results show that the proposed Newton-Simpson-based predictor-corrector methods have a faster convergence rate. For the local stability lobe diagrams, the arithmetic mean of relative error (AMRE), mean squared error (MSE), and the sum of absolute error (SAE) of the proposed methods are in the ranges of 0.003 to 0.004, 2.66 x 10-10 to 6.40 x 10-10, and 6.41 x 10-4 to 9.34 x 10-4, respectively, which are much lower than those of the existing methods, indicating that the proposed methods have higher calculation accuracy than the existing methods. The current work has a broad application prospect in the field of milling stability prediction for precision machining and the selection of chatter-free milling parameters.