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.
Issue Section:
Research Papers
1.
Tobias
, S. A.
, 1965, Machine Tool Vibration
, Wiley
, New York
.2.
Tlusty
, J.
, 1986, “Dynamics of High-Speed Milling
,” ASME J. Eng. Ind.
0022-0817, 108
, pp. 59
–67
.3.
4.
Halley
, J. E.
, Helvey
, A. M.
, and Smith
, K. S.
, 1999, “The Impact of High-Speed Machining on the Design and Fabrication of Aircraft Components
,” 17th ASME Biennial Conference on Mechanical Vibration and Noise
, Las Vegas, NV, Sept. 12–15.5.
Hahn
, W.
, 1961, “On Difference Differential Equations With Periodic Coefficients
,” J. Math. Anal. Appl.
0022-247X, 8
, pp. 70
–101
.6.
Minis
, I.
, and Yanushevsky
, R.
, 1993, “A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling
,” ASME J. Eng. Ind.
0022-0817, 115
, pp. 1
–8
.7.
Altintas
, Y.
, and Budak
, E.
, 1995, “Analytical Prediction of Stability Lobes in Milling
,” CIRP Ann.
0007-8506, 44
, pp. 357
–362
.8.
Davies
, M. A.
, Pratt
, J. R.
, Dutterer
, B.
, and Burns
, T. J.
, 2002, “Stability Prediction for Low Radial Immersion Milling
,” ASME J. Manuf. Sci. Eng.
1087-1357, 124
, pp. 217
–225
.9.
Zhao
, M. X.
, and Balachandran
, B.
, 2001, “Dynamics and Stability of Milling Process
,” Int. J. Solids Struct.
0020-7683, 38
, pp. 2233
–2248
.10.
Davies
, M. A.
, Dutterer
, B.
, Pratt
, J. R.
, and Schaut
, A.
, 1998, “On the Dynamics of High Speed Milling With Long, Slender Endmills
,” CIRP Ann.
0007-8506, 47
, pp. 55
–60
.11.
Davies
, M. A.
, Dutterer
, B.
, Pratt
, J. R.
, and Burns
, T. J.
, 2000, “The Stability of Low Radial Immersion Milling
” CIRP Ann.
0007-8506, 49
(1
), pp. 37
–40
.12.
Insperger
, T.
, and Stépán
, G.
, 2000, “Stability of the Milling Process
,” Period. Polytech., Mech. Eng.-Masinostr.
0324-6051, 44
(1
), pp. 47
–57
.13.
Insperger
, T.
, and Stépán
, G.
, 2004, “Vibration Frequencies in High-Speed Milling Processes or a Positive Answer to Davies, Pratt, Dutterer and Burns
,” ASME J. Manuf. Sci. Eng.
1087-1357, 126
, pp. 481
–487
.14.
Insperger
, T.
, and Stépán
, G.
, 2004, “Updated Semi-Discretization Method for Periodic Delay-Differential Equations With Discrete Delay
,” Int. J. Numer. Methods Eng.
0029-5981, 61
, pp. 117
–141
.15.
Mann
, B. P.
, Bayly
, P. V.
, Davies
, M. A.
, and Halley
, J. E.
, 2004, “Limit Cycles, Bifurcations, and Accuracy of the Milling Process
,” J. Sound Vib.
0022-460X, 277
, pp. 31
–48
.16.
Mann
, B. P.
, Young
, K. A.
, Schmitz
, T. L.
, and Dilley
, D. N.
, 2005, “Simultaneous Stability and Surface Location Error Predictions in Milling
,” ASME J. Manuf. Sci. Eng.
1087-1357, 127
(3
), pp. 446
–453
.17.
Mann
, B. P.
, Garg
, N. K.
, Young
, K. A.
, and Helvey
, A. M.
, 2005, “Milling Bifurcations From Structural Asymmetry and Nonlinear Regeneration
,” Nonlinear Dyn.
0924-090X, 42
(4
), pp. 319
–337
.18.
Garg
, N. K.
, Mann
, B. P.
, Kim
, N. H.
, and Kurdi
, M. H.
, 2007, “Stability of a Time-Delayed System With Parametric Excitation
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 129
(2
), pp. 125
–135
.19.
Insperger
, T.
, Mann
, B. P.
, Stépán
, G.
, and Bayly
, P. V.
, 2003, “Stability of Up-Milling and Down-Milling, Part 1: Alternative Analytical Methods
,” Int. J. Mach. Tools Manuf.
0890-6955, 43
, pp. 25
–34
.20.
Mann
, B. P.
, Insperger
, T.
, Bayly
, P. V.
, and Stépán
, G.
, 2003, “Stability of Up-Milling and Down-Milling, Part 2: Experimental Verification
,” Int. J. Mach. Tools Manuf.
0890-6955, 43
, pp. 35
–40
.21.
Insperger
, T.
, Stépán
, G.
, Bayly
, P. V.
, and Mann
, B. P.
, 2003, “Multiple Chatter Frequencies in Milling Processes
,” J. Sound Vib.
0022-460X, 262
, pp. 333
–345
.22.
Szalai
, R.
, and Stépán
, G.
, 2006, “Lobes and Lenses in the Stability Chart of Interrupted Turning
,” ASME J. Comput. Nonlinear Dyn.
1555-1423, 1
(3
), pp. 205
–211
.23.
Zatarain
, M.
, Munoa
, J.
, Peigne
, G.
, and Insperger
, T.
, 2006, “Analysis of the Influence of Mill Helix Angle on Chatter Stability
,” CIRP Ann.
0007-8506, 55
(1
), pp. 365
–368
.24.
Patel
, B. R.
, Mann
, B. P.
, and Young
, K. A.
, 2008, “Uncharted Islands of Chatter Instability in Milling
,” Int. J. Mach. Tools Manuf.
0890-6955, 48
, pp. 124
–134
.25.
Long
, X. -H.
, Balachandran
, B.
, and Mann
, B. P.
, 2007, “Dynamics of Milling Processes With Variable Time Delays
,” Nonlinear Dyn.
0924-090X, 47
(1–3
), pp. 49
–63
.26.
Olgac
, N.
, and Sipahi
, R.
, 2007, “Dynamics and Stability of Variable-Pitch Milling
,” J. Vib. Control
1077-5463, 13
(7
), pp. 1031
–1043
.27.
Insperger
, T.
, Stépán
, G.
, Hartung
, F.
, and Turi
, J.
, 2005, “State-Dependent Regenerative Delay in Milling Processes
,” Proceedings of the ASME IDETC
, Long Beach, CA, Sept. 24–28.28.
Sinha
, S. C.
, and Wu
, D. -H.
, 1991, “An Efficient Computational Scheme for the Analysis of Periodic Systems
,” J. Sound Vib.
0022-460X, 151
, pp. 91
–117
.29.
Butcher
, E. A.
, Ma
, H.
, Bueler
, E.
, Averina
, V.
, and Szabó
, Z.
, 2004, “Stability of Linear Time-Periodic Delay-Differential Equations via Chebyshev Polynomials
,” Int. J. Numer. Methods Eng.
0029-5981, 59
, pp. 895
–922
.30.
Bellen
, A.
, 1984, “One-Step Collocation for Delayed Differential Equations
,” J. Comput. Appl. Math.
0377-0427, 10
, pp. 275
–283
.31.
Ito
, K.
, Tran
, H. T.
, and Manitus
, A.
, 1991, “A Fully Discrete Spectral Method for Delay Differential Equations
,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429, 28
, pp. 1121
–1140
.32.
Engelborghs
, K.
, Luzyanina
, T.
, In‘T Hout
, K. J.
, and Roose
, D.
, 2000, “Collocation Methods for the Computation of Periodic Solutions of Delay Differential Equations
,” SIAM J. Sci. Comput. (USA)
1064-8275, 22
, 1593
–1609
.33.
Luzyanina
, T.
, and Engelborghs
, K.
, 2002, “Computing Floquet Multipliers for Functional Differential Equations
,” Int. J. Bifurcation Chaos Appl. Sci. Eng.
0218-1274, 12
, pp. 2977
–2989
.34.
Engelborghs
, K.
, and Roose
, D.
, 2002, “On Stability of LMS Methods and Characteristic Roots for Delay Differential Equations
,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429, 40
(2
), pp. 629
–650
.35.
Hale
, J. K.
, and Lunel
, M. V.
, 1993, Introduction to Functional Differential Equations
, Springer
, New York
.36.
Trefethen
, L. N.
, 2000, Spectral Methods in MATLAB
(Software-Environment-Tools Series
), SIAM
, Philadelphia, PA
.37.
Bueler
, E.
, 2007, “Error Bounds for Approximate Eigenvalues of Periodic-Coefficient Linear Delay Differential Equations
,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
0036-1429, 45
(6
), pp. 2510
–2536
.38.
Gilsinn
, D. E.
, and Potra
, F. A.
, 2006, “Integral Operators and Delay Differential Equations
,” J. Integral Equ. Appl.
0897-3962, 18
(3
), pp. 297
–336
.39.
Butcher
, E. A.
, Nindujarla
, P.
, and Bueler
, E.
, 2005, “Stability of Up- and Down-Milling Using Chebyshev Collocation Method
,” Proceedings of the Fifth International Conference on Multibody Systems, Nonlinear Dynamics, and Control, ASME DETC’05
, Long Beach, CA, Sept. 24–28.40.
Deshmukh
, V.
, Butcher
, E. A.
, and Bueler
, E.
, 2008, “Dimensional Reduction of Nonlinear Delay-Differential Equations With Periodic Coefficients Using Chebyshev Polynomials
,” Nonlinear Dyn.
0924-090X, 52
, pp. 137
–149
.41.
Deshmukh
, V.
, 2008, “Spectral Collocation-Based Optimization in Parameter Estimation for Nonlinear Time-Varying Dynamical Systems
,” ASME J. Comput. Nonlinear Dyn.
1555-1423, 3
, p. 011010
.42.
Bueler
, E.
, 2005, “Guide to DDEC: Stability of Linear, Periodic DDEs Using the DDEC Suite of MATLAB Codes
,” http://www.dms.uaf.edu/~bueler/DDEcharts.htmhttp://www.dms.uaf.edu/~bueler/DDEcharts.htm.43.
Shampine
, L. F.
, and Thompson
, S.
, 2001, “Solving Delay Differential Equations With DDE23
,” Appl. Numer. Math.
0168-9274, 37
(4
), pp. 441
–458
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