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Research Papers

Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels

[+] Author and Article Information
Eric A. Butcher

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001eab@nmsu.edu

Oleg A. Bobrenkov

Department of Mechanical and Aerospace Engineering, New Mexico State University, Las Cruces, NM 88001chaalis@nmsu.edu

Ed Bueler

Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK 99775ffelb@uaf.edu

Praveen Nindujarla

Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775ftpkn1@uaf.edu

J. Comput. Nonlinear Dynam 4(3), 031003 (May 19, 2009) (12 pages) doi:10.1115/1.3124088 History: Received January 30, 2008; Revised September 26, 2008; Published May 19, 2009

In this paper the dynamic stability of the milling process is investigated through a single degree-of-freedom model by determining the regions where chatter (unstable) vibrations occur in the two-parameter space of spindle speed and depth of cut. Dynamic systems such as milling are modeled by delay-differential equations with time-periodic coefficients. A new approximation technique for studying the stability properties of such systems is presented. The approach is based on the properties of Chebyshev polynomials and a collocation expansion of the solution. The collocation points are the extreme points of a Chebyshev polynomial of high degree. Specific cutting force profiles and stability charts are presented for the up- and down-milling cases of one or two cutting teeth and various immersion levels with linear and nonlinear regenerative cutting forces. The unstable regions due to both secondary Hopf and flip (period-doubling) bifurcations are found, and an in-depth investigation of the optimal stable immersion levels for down-milling in the vicinity of where the average cutting force changes sign is presented.

Copyright © 2009 by American Society of Mechanical Engineers
Topics: Force , Stability , Cutting , Milling
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Figures

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Figure 6

DDE23 simulations of Eq. 13 for the four parameter points A(13000,3.5), B(16800,3.5), C(18000,3.5), and D(23000,3.5) for down-milling, one tooth, and q=1 (linear cutting force)

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Figure 7

Optimal stable immersion levels for down-milling, one cutting tooth, tan γ=0.3, for q=1 (linear cutting force)

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Figure 8

Specific cutting forces and stability charts for up-milling (rows 1 and 2) and down-milling (rows 3 and 4) for two cutting teeth and immersion ratios of a/D=0.25, 0.5, 0.75, and 1; q=0.75 (nonlinear cutting force)

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Figure 9

DDE23 simulations of Eq. 13 for the four parameter points A(5000,3.5), B(7250,3.5), C(9750,3.5), and D(12000,3.5) for down-milling, two cutting teeth, and q=0.75 (nonlinear cutting force)

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Figure 10

Optimal stable immersion levels for down-milling, tan γ=0.3, two cutting teeth, and q=0.75 (nonlinear cutting force)

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Figure 13

Immersion ratio where ⟨h(t)⟩=0 (solid) and ⟨h(t)⟩=2mωn2ζ(ζ±1)/b for down-milling versus tan γ (dashed for b=5 mm; dotted for b=2 mm) for a single tooth with (a) q=1 (linear cutting force) and (b) q=0.75 (nonlinear cutting force)

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Figure 14

Average cutting force for (top curve) up- and (bottom curve) down-milling for tan γ=0.3, a single tooth, the linear case q=1 (dashed), and the nonlinear cases q=0.75 (solid) and q=0.5 (dotted)

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Figure 15

Stability charts for down-milling for tan γ=0.3, a single tooth, 76% immersion ratio, and (a) q=1, (b) q=0.8, (c) q=0.6, and (d) q=0.4

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Figure 5

Specific cutting forces and stability charts for up-milling (rows 1 and 2) and down-milling (rows 3 and 4) for one cutting tooth and immersion ratios of a/D=0.25, 0.5, 0.75, and 1; q=1 (linear cutting force)

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Figure 4

Chebyshev collocation expansion of the specific cutting force h(t) and collocation vectors on successive intervals of the solution x(t) for the case of a period of free vibration

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Figure 3

Illustrations of (a) up-milling and (b) down-milling

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Figure 2

Geometry of (a) cutting forces and (b) feed per tooth

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Figure 1

Diagrams of (a) Chebyshev collocation points as defined by projections from the unit circle and (b) collocation vectors on successive intervals

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Figure 12

(a) Specific cutting force profile for a/D=1.00 with two ways of averaging: throughout the entire period (long-dashed line) and inside the positive and negative parts separately (short-dashed line). b) The corresponding stability chart illustrating a negative depth of cut. The dashed lines are estimates of the minima/maxima of the Hopf lobes obtained from averaging over the period while the dotted curves are corresponding estimates for the period-doubling lobes obtained using the second averaging method.

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Figure 11

Average cutting forces for (top curve) up- and (bottom curve) down-milling for a single tooth with tan γ=0.22 (dashed), 0.3 (solid), and 0.38 (dotted)

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