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Technical Brief

Stability Analysis of Milling Processes With Periodic Spindle Speed Variation Via the Variable-Step Numerical Integration Method

[+] Author and Article Information
Jinbo Niu, LiMin Zhu, Han Ding

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

Ye Ding

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

1Corresponding author.

Manuscript received October 18, 2015; final manuscript received March 10, 2016; published online June 23, 2016. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 138(11), 114501 (Jun 23, 2016) (11 pages) Paper No: MANU-15-1521; doi: 10.1115/1.4033043 History: Received October 18, 2015; Revised March 10, 2016

This paper proposes a general method for the stability analysis and parameter optimization of milling processes with periodic spindle speed variation (SSV). With the aid of Fourier series, the time-variant spindle speeds of different periodic modulation schemes are unified into one framework. Then the time-varying delay is derived implicitly and calculated efficiently using an accurate ordinary differential equation (ODE) based algorithm. After incorporating the unified spindle speed and time delay into the dynamic model, a Floquet theory based variable-step numerical integration method (VNIM) is presented for the stability analysis of variable spindle speed milling processes. By comparison with other methods, such as the semi-discretization method and the constant-step numerical integration method, the proposed method has the advantages of high computational accuracy and efficiency. Finally, different spindle speed modulation schemes are compared and the modulation parameters are optimized with the aid of three-dimensional stability charts obtained using the proposed VNIM.

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Figures

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Fig. 2

Spindle speeds and spindle acceleration of the sinusoidal modulation (abbreviated as Sine), the original triangular modulation (abbreviated as OTri) and the approximate triangular modulation (abbreviated as ATri)

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Fig. 1

Schematic of the 2DOF end milling system

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Fig. 3

Variable-step discretization scheme for VSS milling: (a) the directional cutting force coefficients in one modulation period and (b) the variable-step discretization of hxx

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Fig. 4

SLDs obtained with the proposed VNIM: (a) corresponds to Fig. 1(a) in Ref. [39] and (b) corresponds to Fig. 5(a) in Ref. [26]

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Fig. 5

Computational efficiency comparisons of the proposed VNIM, the CNIM, and the SDM. me denotes the equivalent number of the discrete points of CNIM and SDM that makes the discretization step of these three methods same.

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Fig. 6

Prediction accuracy verification using time domain simulation. The square marker “□” indicates unstable parameter combinations and the circle marker “○” indicates stable parameter combinations.

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Fig. 7

Convergence diagrams of the VNIM, the CNIM, and the SDM

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Fig. 8

Stability diagrams of VSS milling system with sinusoidal and triangular modulations when Ω0=6000 rpm

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Fig. 9

Stability diagrams of VSS milling system with sinusoidal and triangular modulations when Ω0=3600 rpm

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