A novel method with Newton polynomial-Chebyshev nodes for milling stability prediction

As the approximation orders enlarge, the numerical algorithms of milling stability prediction are becoming increasingly complex with more and more calculation time being consuming. Above all, the effects of convergence and accuracy do not continuously improve for the presence of Runge phenomenon. This paper suggests a novel predictive scheme immune from this undesirable effect based on the Newton polynomial-Chebyshev nodes. Firstly, the processing dynamics embracing the self-excited vibration is described as the state-space equation with a separate time delay. Then n-order Newton polynomial with linear sampling nodes is chosen to unfold the monolithic non-homogeneous part to estimate the continuous response into linear discrete map, which designates the state transition of the system within a single cutting period. Taking advantage of this closed-form map, two coefficients that have considerable coupled elements with each other are established to generate the transition matrix, and the stability limits is searched out depending on Floquet theory. A set of comparisons are conducted using the experimentally validated examples to determine the optimal approximated order with exhibiting the features of the proposed methods. It is also disclosed that there is an obvious Runge effect existing in the presented high-order methods, which seriously lowers the estimated accuracy. In the interests of eliminating this unwanted effect, a nonlinear sampling technique based on Chebyshev nodes is adopted to substitute for the origin uniform time joints to establish a novel transition matrix with fewer calculated loads for the milling stability prediction, and the verifications illuminate the reconstituted methods are of satisfactory performance. The proposed methods are beneficial for the path planers to avoid some unsuited parameters which cause the machining chatter.