An improved numerical integration method for prediction of milling stability using the Lagrange-Simpson interpolation scheme

Abstract

The stability prediction is usually used to avoid the unstable machining in milling process. According to the Lagrange-Simpson hybrid interpolation scheme, this paper improves a numerical integration method (NIM) to perform the milling chatter prediction accurately and efficiently. Firstly, the higher-order numerical integral formulas (NIFs) are constructed based on the Lagrange polynomial. Thus, the third-order and fourth-order NIMs are built and investigated respectively. Then, to improve the calculated performance of the NIMs, the Simpson scheme is introduced to decrease the local discretization error. Finally, the comparisons among the built NIMs and the existing discretization methods are carried out by calculating the convergence rate and the stability boundaries. Compared to the third-order NIM and the third-order full-discretization method, the proposed second-order Lagrange-Simpson NIM shows the better computational performance.

Appendix. Matrix E4

$${\mathbf{E}}_{4} { = }\left[ {\begin{array}{*{20}c} 0 & {} & {} & {} & {} & {} & {} \\ {251e^{{{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{1} } & {646{\mathbf{A}}_{2} } & { - 264e^{{ - {\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{3} } & {106e^{{ - 2{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{4} } & { - 19e^{{ - 3{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{4} } & {} & {} \\ {} & \ddots & \ddots & \ddots & \ddots & {} & {} \\ {} & {} & {251e^{{{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{m - 3} } & {646{\mathbf{A}}_{m - 2} } & { - 264e^{{ - {\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{{\text{m - 1}}} } & {106e^{{ - 2{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{m} } & { - 19e^{{ - 3{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{m + 1} } \\ {} & {} & {} & {360e^{{{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{m - 2} } & {360{\mathbf{A}}_{m - 1} } & {} & {} \\ {} & {} & {} & {} & {360e^{{{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{m - 1} } & {360{\mathbf{A}}_{m} } & {} \\ {} & {} & {} & {} & {} & {360e^{{{\mathbf{A}}_{0} \tau }} {\mathbf{A}}_{m} } & {360{\mathbf{A}}_{m + 1} } \\ \end{array} } \right]$$