High-order full-discretization methods for milling stability prediction by interpolating the delay term of time-delayed differential equations

The full-discretization method (FDM) has been proven effective for prediction on the regenerative chatter in many papers. However, the previous studies towards FDM just focused on high-order Lagrange interpolation for state term of time-delayed differential equations (DDEs), which formulates the dynamics model in milling process. It is well known that the discretization error caused by the delay term of DDEs would transmit to the state term inevitably; higher-order Lagrange interpolation for delay term is thus vital. In this paper, second-order, third-order, and fourth-order full-discretization methods using Lagrange interpolation for the delay term of DDEs (DFDMs) were firstly proposed. Then, influence on the accuracy, computational efficiency, and convergence rate of the proposed DFDMs was discussed in detail as the change of interpolation order. It was found that rise in accuracy and convergence rate of the proposed DFDM nearly stopped when the interpolation order for delay term was up to fourth order. Next, some researches on 2-degree-of-freedom (2-DOF) of dynamic system was studied and the results show that the proposed method using fourth-order Lagrange interpolation for the delay term of DDEs (4th DFDM) was effective. Finally, this paper verified the 4th DFDM by experiment and analyzed the prediction error of 4th DFDM, which may be caused by the modeling process of cutting force. The proposed DFDMs are developed to find a better method, which can update the existing FDM and make regenerative chatter's prediction more efficient and precise.