Technical Brief

A Legendre Polynomials Based Method for Stability Analysis of Milling Processes

[+] Author and Article Information
Ye Ding

Gas Turbine Research InstituteSchool of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China
e-mail: y.ding@sjtu.edu.cn

XiaoJian Zhang, Han Ding

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

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received May 26, 2014; final manuscript received December 19, 2014; published online February 12, 2015. Assoc. Editor: Philip Bayly.

J. Vib. Acoust 137(2), 024504 (Apr 01, 2015) (7 pages) Paper No: VIB-14-1194; doi: 10.1115/1.4029460 History: Received May 26, 2014; Revised December 19, 2014; Online February 12, 2015

This paper presents a time-domain approach for a semi-analytical prediction of stability in milling using the Legendre polynomials. The governing equation of motion of milling processes is expressed as a delay-differential equation (DDE) with time periodic coefficients. After the DDE being re-expressed in state-space form, the state vector is approximated by a series of Legendre polynomials. With the help of the Legendre–Gauss–Lobatto (LGL) quadrature, a discrete dynamic map is formulated to approximate the original DDE, and utilized to predict the milling stability based on Floquet theory. With numerical examples illustrating the efficiency and accuracy of the proposed approach, an experimental example validates the method.

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Grahic Jump Location
Fig. 1

Schematic of the milling system

Grahic Jump Location
Fig. 4

The predicted stability lobes and the experimental results. “○,” “□,” and “⊗” represent the stable, unstable, and unclear cases, respectively.

Grahic Jump Location
Fig. 5

Measured cutting force signals and spectrums from unstable and stable tests: (a) the time-domain signal of cutting forces at (7700 rpm, 2.5 mm); (b) the frequency spectrum of Fy at (7700 rpm, 2.5 mm); (c) the time-domain signal of cutting forces at (9300 rpm, 3 mm); and (d) the frequency spectrum of Fy at (9300 rpm, 3 mm)

Grahic Jump Location
Fig. 2

Stability diagrams calculated by the 1st SDM and the proposed LPBM. The stability lobes presented by gray color are determined by the 1st SDM with n = 200 for reference.

Grahic Jump Location
Fig. 3

Convergence of the critical eigenvalues for different computational parameters: (a) ap=0.2 mm, Ω=5000 rpm and (b) ap=1 mm, Ω=5000 rpm



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