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Research Papers

Investigation of Period-Doubling Islands in Milling With Simultaneously Engaged Helical Flutes

[+] Author and Article Information
Firas A. Khasawneh1

Department of Engineering Mathematics,  University of Bristol, Bristol, BS8 1TR, United Kingdomfiras.khasawneh@bristol.ac.uk

Oleg A. Bobrenkov, Eric A. Butcher

Department of Mechanical and Aerospace Engineering,  New Mexico State University, Las Cruces, NM 88003

Brian P. Mann

Department of Mechanical Engineering and Materials Science,  Duke University, Durham, NC 27708

1

Corresponding author.

J. Vib. Acoust 134(2), 021008 (Jan 18, 2012) (9 pages) doi:10.1115/1.4005022 History: Received July 16, 2010; Accepted July 14, 2011; Published January 18, 2012; Online January 18, 2012

This paper investigates the stability of a milling process with simultaneously engaged flutes using the state-space TFEA and Chebyshev collocation methods. In contrast to prior works, multiple flute engagement due to both the high depth of cut and high step-over distance are considered. A particular outcome of this study is the demonstration of a different stability behavior in comparison to prior works. To elaborate, period-doubling regions are shown to appear at relatively high radial immersions when multiple flutes with either a zero or nonzero helix angle are simultaneously cutting. We also demonstrate stability differences that arise due to the parity in the number of flutes, especially at full radial immersion. In addition, we study other features induced by helical tools such as the waviness of the Hopf lobes, the sensitivity of the period-doubling islands to the radial immersion, as along with the orientation of the islands with respect to the Hopf lobes.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

The different cases associated with a helical milling tool. The gray area represents the cutting zone and, in graph (a) only a single flute is cutting at any instant, while in graphs (b) and (c) multiple flutes are cutting due to a high depth of cut, and a high radial step-over distance, respectively. Graph (d) shows the case of a high depth of cut combined with a high radial step-over distance. The variables shown in the figure are defined in Sec. 2.

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Figure 2

Illustrations of (a) upmilling, and (b) downmilling

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Figure 3

Schematic of a helical end mill with multiple flutes and the differential cutting forces in the axial, radial, and tangential directions

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Figure 4

A plot of the integration limits described in Eqs. 6 as a function of the flute rotation angle

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Figure 5

The stability criteria dictates that all the eigenvalues μ of the monodromy operator U should lie within the unit circle in the complex plane. Moreover, the manner in which the eigenvalues depart the unit circle produces different bifurcation behavior. For example, an eigenvalue leaving the unit circle through −1 or 1 results in a period-doubling bifurcation or a fold bifurcation, respectively, whereas two complex conjugate eigenvalues departing the unit circle result in a secondary Hopf bifurcation.

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Figure 6

A parametrically induced period-doubling island appears when downmilling at 0.05 radial immersion with a zero-helix 3-flute cutter. The stability of several points is also noted using a notation consistent with the one used in Fig. 5; circles were used to denote a stable region, whereas triangles were used to denote unstable period-doubling regions.

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Figure 7

Downmilling stability diagrams of Eq. 8 for cutters with 4, 6, and 8 flutes. The light line represents the zero-helix case while the thick line represents a helical tool with β = 30°. The parameters used to generate the plots are shown in Table 1 and the radial immersions used are (a) 0.05, (b) 0.50, (c) 0.75, and (d) 1.0. “H” indicates Hopf bifurcations, while “P” indicates period-doubling bifurcations.

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Figure 8

Downmilling stability diagrams for cutters with β = 30° and various odd number of flutes at various radial immersion levels

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Figure 10

Upmilling stability charts for the 4 cutting flutes and for the radial immersion values of 0.10, 0.20, 0.75, and 0.80 (columns) and the helix angle values of 1, 13, 23, 25, 45, 60, and 63° (rows)

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Figure 11

Diagrams showing the combinations of radial immersion and helix angle where islands occur (shaded region). Results are for 4 cutting flutes and (a) upmilling, and (b) downmilling.

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Figure 9

Snapshots of two helical mills at full radial immersion with (a) 4 flutes, and (b) 5 flutes. The snapshots illustrate the top view for an axial slice of the cutter during the beginning, middle, and end of one cutting period.

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