An improved numerical integration method to predict the milling stability based on the Lagrange interpolation scheme

To predict the milling stability accurately and efficiently, an improved numerical integration method (INIM) is proposed based on the Lagrange interpolation scheme. First, the milling dynamic model considering the regenerative chatter can be described as a delay linear differential equation. The tooth passing period of the milling cutter is divided into the free and forced vibration stages. Then, the forced vibration stage is equally discretized, and the INIMs are built based on the Lagrange interpolation scheme within the discretized intervals to construct the state transition matrix. Finally, the convergence rates and the stability lobes for the benchmark milling systems are calculated and discussed by using the proposed INIMs and the existing methods, respectively. The comparison results reveal that the proposed second-order INIM shows higher computational efficiency and accuracy compared with the related discretization methods, and in the meantime, it is more accurate under just slightly loss of the time cost compared with the existing NIMs.