Prediction of chatter stability in high speed milling using the numerical differentiation method

A numerical differentiation method is presented to predict the high speed milling stability of a two degrees of freedom (DOF) system based on the finite difference method and extrapolation method. The milling dynamics taking the regenerative effect into account are represented as linear periodic delayed differential equations (DDE) in the state space form. Then, each component of the first derivative of the state function versus time at the discretized sampling grids is approximated as a weighted linear sums of the state function values at its neighboring grid points, where the weight coefficients are calculated based on the extrapolation method. As such, the DDE on the forced vibration duration is approximately discretized as a series of algebraic equations. Thereafter, the Floquet transition matrix can be constructed on one tooth passing period by combining the analytical solution of the free vibration and the algebraic equations of the forced vibration. Finally, the milling stability is determined according to Floquet theory. The stability diagrams and convergence of critical eigenvalues in comparison with the benchmark algorithms (the semi-discretization method and numerical integration method) via experimentally verified examples are utilized to demonstrate the effectiveness and efficiency of the proposed method.