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Research Papers

Numerical Integration Method for Stability Analysis of Milling With Variable Spindle Speeds

[+] Author and Article Information
Ye Ding

Gas Turbine Research Institute;State Key Laboratory of Mechanical
System and Vibration,
School of Mechanical Engineering,
Shanghai Jiao Tong University,
Shanghai 200240, China

Jinbo Niu

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

LiMin Zhu

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

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 July 23, 2015; final manuscript received September 16, 2015; published online October 26, 2015. Assoc. Editor: Philippe Velex.

J. Vib. Acoust 138(1), 011010 (Oct 26, 2015) (11 pages) Paper No: VIB-15-1281; doi: 10.1115/1.4031617 History: Received July 23, 2015; Revised September 16, 2015

A semi-analytical method is presented in this paper for stability analysis of milling with a variable spindle speed (VSS), periodically modulated around a nominal spindle speed. Taking the regenerative effect into account, the dynamics of the VSS milling is governed by a delay-differential equation (DDE) with time-periodic coefficients and a time-varying delay. By reformulating the original DDE in an integral-equation form, one time period is divided into a series of subintervals. With the aid of numerical integrations, the transition matrix over one time period is then obtained to determine the milling stability by using Floquet theory. On this basis, the stability lobes consisting of critical machining parameters can be calculated. Unlike the constant spindle speed (CSS) milling, the time delay for the VSS is determined by an integral transcendental equation which is accurately calculated with an ordinary differential equation (ODE) based method instead of the formerly adopted approximation expressions. The proposed numerical integration method is verified with high computational efficiency and accuracy by comparing with other methods via a two-degree-of-freedom milling example. With the proposed method, this paper details the influence of modulation parameters on stability diagrams for the VSS milling.

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Figures

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Fig. 1

Schematic of end milling system

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Fig. 2

Sinusoidal spindle speed modulation

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Fig. 3

Comparisons of time delays obtained with different methods. Root-finding method is based on Eq. (13), ODE-based method is based on Eq. (20), linear approximation is based on Eq. (14), and quadratic approximation is based on Eq. (16). tRF and tODE represent the delay computational time using the root-finding method and the ODE-based method, respectively.

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Fig. 4

Comparisons of the proposed numerical integration method and the SDM

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Fig. 5

Stability diagrams obtained using different time delay calculation method: the ODE-based method, the quadratic approximation method, and the linear approximation method

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Fig. 6

Stability diagram verification using the time domain simulation. The square mark “□” and circle mark “○” represent the unstable and stable machining cases, respectively.

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Fig. 7

Eigenvalue loci for different time delay approximation methods. The maximum-modulus characteristic exponents are obtained with the proposed numerical integration method and drawn along with the unit circles in the right column subfigures. “meth.” and “appr.” are the abbreviations of “method” and “approximation,” respectively.

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Fig. 8

Maximum stable ADC with different modulation parameters

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