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Technical Briefs

Stability Analysis of Milling Via the Differential Quadrature Method

[+] Author and Article Information
Ye Ding

e-mail: y.ding@sjtu.edu.cn

LiMin Zhu

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

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 Manufacturing Engineering Division of ASME for publication in the Journal of Manufacturing Science and Engineering. Manuscript received December 25, 2012; final manuscript received April 18, 2013; published online July 17, 2013. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 135(4), 044502 (Jul 17, 2013) (7 pages) Paper No: MANU-12-1368; doi: 10.1115/1.4024539 History: Received December 25, 2012; Revised April 18, 2013; Accepted April 23, 2013

This paper presents a time-domain semi-analytical method for stability analysis of milling in the framework of the differential quadrature method. The governing equation of milling processes taking into account the regenerative effect is formulated as a linear periodic delayed differential equation (DDE) in state space form. The tooth passing period is first separated as the free vibration duration and the forced vibration duration. As for the free vibration duration, the analytical solution is available. As for the forced vibration duration, this time interval is discretized by sampling grid points. Then, the differential quadrature method is employed to approximate the time derivative of the state function at a sampling grid point within the forced vibration duration by a weighted linear sum of the function values over the whole sampling grid points. The Lagrange polynomial based algorithm (LPBA) and trigonometric functions based algorithm (TFBA) are employed to obtain the weight coefficients. Thereafter, the DDE on the forced vibration duration is discretized as a series of algebraic equations. By combining the analytical solution of the free vibration duration and the algebraic equations of the forced vibration duration, Floquet transition matrix can be constructed to determine the milling stability according to Floquet theory. Simulation results and experimentally validated examples are utilized to demonstrate the effectiveness and accuracy of the proposed approach.

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Figures

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Fig. 1

Two-degrees-of-freedom milling system

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Fig. 2

Stability diagrams determined by the first-order (1st) SDM, LPBA, and TFBA. The stability lobes presented by grey color for reference are determined by first-order SDM with n = 200.

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Fig. 3

Convergence of the critical eigenvalues for the proposed LPBA, TFBA, the CCM, and the IEBSM with different computational parameters

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Fig. 4

Computational time of the proposed LPBA, TFBA with different computational parameters for ap = 1 mm, the spindle speed fixed as 5000 rpm and the radial immersion ratio fixed as ae/D = 1

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Fig. 5

Comparison of stability lobes for two-degrees-of-freedom milling system. The stability lobes presented by grey color for reference are determined by the TFEA method with n = 4.

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