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Research Papers

# Numerical Integration Method for Prediction of Milling Stability

[+] Author and Article Information
Ye Ding, LiMin Zhu

State Key Laboratory of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China

XiaoJian Zhang

State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan 430074, China

Han Ding1

State Key Laboratory of Mechanical System and Vibration School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, China e-mail: hding@sjtu.edu.cn

1

Corresponding author.

J. Manuf. Sci. Eng 133(3), 031005 (Jun 08, 2011) (9 pages) doi:10.1115/1.4004136 History: Received March 23, 2010; Revised April 13, 2011; Published June 08, 2011; Online June 08, 2011

## Abstract

This paper presents a numerical scheme to predict the milling stability based on the integral equation and numerical integration formulas. First, the milling dynamics taking the regenerative effect into account is represented in the form of integral equation. Then, the tooth passing period is precisely divided into the free vibration phase during which the analytical solution is available and the forced vibration phase during which an approximate solution is needed. To obtain the numerical solution of the integral equation during the forced vibration phase, the time interval of interest is equally discretized. Over each small time interval, Newton-Cotes integration formulas or Gauss integration formulas are employed to approximate the integral term in the integral equation. After establishing the state transition matrix of the system in one period, the milling stability is predicted by using Floquet theory. The benchmark examples are utilized to verify the proposed approach. The results demonstrate that it is highly efficient and accurate.

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## Figures

Figure 1

Convergence of the eigenvalues for different approximation parameters n for the proposed numerical integration method (NIM) via Eq. 18 and Eq. 30, the zeroth-order (0th) SDM [16], and the first-order (1st) SDM [17]

Figure 2

Comparison of the proposed NIM via Eq. 30 (with n = 10) and the first-order (1st) SDM (with m = 30) [17] for two DOF milling model

Figure 3

Comparison of the proposed NIM via Eq. 30 (with n = 10) and the TFEA method (with four elements in the cut) for two DOF milling model

Figure 4

Stability prediction for the variable pitch cutter

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