*Article* **Predicting Milling Stability Based on Composite Cotes-Based and Simpson's 3/8-Based Methods**

**Xu Du <sup>1</sup> , Pengfei Ren <sup>1</sup> and Junqiang Zheng 2,\***


**\*** Correspondence: zhengjunqiang@hdu.edu.cn

**Abstract:** Avoiding chatter in milling processes is critical for obtaining machined parts with high surface quality. In this paper, we propose two methods for predicting the milling stability based on the composite Cotes and Simpson's 3/8 formulas. First, a time-delay differential equation is established, wherein the regenerative effects are considered. Subsequently, it is discretized into a series of integral equations. Based on these integral equations, a transition matrix is determined using the composite Cotes formula. Finally, the system stability is analyzed according to the Floquet theory to obtain the milling stability lobe diagrams. The simulation results demonstrate that for the single degree of freedom (single-DOF) model, the convergence speed of the composite Cotes-based method is higher than that of the semi-discrete method and the Simpson's equation method. In addition, the composite Cotes-based method demonstrates high computational efficiency. Moreover, to further improve the convergence speed, a second method based on the Simpson's 3/8 formula is proposed. The simulation results show that the Simpson's 3/8-based method has the fastest convergence speed when the radial immersion ratio is large; for the two degrees of freedom (two-DOF) model, it performs better in terms of calculation accuracy and efficiency.

**Keywords:** milling stability; composite Cotes-based method; Simpson's 3/8-based method; Floquet theory
