Research Papers

Analytical Prediction of the Critical Depth of Cut and Worst Spindle Speeds for Chatter in End Milling

[+] Author and Article Information
J.-J. Junz Wang

e-mail: jjwang@mail.ncku.edu.tw

C. F. Sung

Department of Mechanical Engineering,
National Cheng Kung University,
Tainan 701, Taiwan

1Corresponding author.

Manuscript received February 12, 2012; final manuscript received September 12, 2013; published online November 5, 2013. Assoc. Editor: Tony Schmitz.

J. Manuf. Sci. Eng 136(1), 011003 (Nov 05, 2013) (10 pages) Paper No: MANU-12-1047; doi: 10.1115/1.4025452 History: Received February 12, 2012; Revised September 12, 2013

The frequency response function (FRF) method has been well used to determine the worst spindle speeds and their critical limiting chip width for turning operation by finding the maximum negative real part of the FRF. In this study, a modified FRF concept is adapted for a 2 DOF milling system of planar isotropic dynamics to determine the worst spindle speeds and the critical limiting axial depth of cut in explicit, analytic formulas. Analogous to the formulation of worst spindle speeds, similar expression for the best spindle speeds is also obtained. The modified FRF is obtained by multiplying the original FRF of the structure with a complex scaling factor, corresponding to a scaling and a rotation of its original Nyquist plot. The scaling factor is determined analytically from the system characteristic equation with the radial cutting constant and radial immersion angle as the major system parameters. Through the presented method, it is also shown that the worst spindle speeds for a milling operation can be found without the prior knowledge of modal dynamics and stability lobe diagram. The proposed analytical expressions are confirmed by the existing stability models and experimentally verified.

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Fig. 1

Schematic of milling system showing tool dynamics and chip thickness variation in milling

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Fig. 2

System block diagram for milling with regenerative feedback

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Fig. 3

Variation of critical chatter frequency ratio with phase angle at various damping ratios

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Fig. 4

Real and imaginary parts of the measured and fitted FRF

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Fig. 5

Nyquist plots of FRF from the tool tip measurements and by simulation for the identified 2-mode and 1-mode dynamics

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Fig. 6

Stability lobe diagrams from ZOS and SDM with identified worst and best speeds

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Fig. 7

Critical depth of cut by the EDM, the average tooth angle method, and from simulated stability lobes

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Fig. 8

Worst speeds predicted by the presented EDM, the ATA method, and identified from the ZOS lobes for n = 0

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Fig. 9

Best speeds predicted from the presented EDM, the ATA method, and from the ZOS lobes for n = 0

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Fig. 10

Experimental results and the predicted stability lobe diagrams

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Fig. 11

Measured acceleration for (a) stable milling at ap = 0.5 mm, and (b) unstable milling at ap = 1.0 mm at 12,000 rpm




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