A two degree of freedom model of the milling process is investigated. The governing equation of motion is decomposed into two parts: an ordinary differential equation describing the periodic chatter-free motion of the tool and a delay-differential equation describing chatter. The stability chart is derived by using the semi-discretization method for the delay-differential equation corresponding to the chatter motion. The periodic chatter-free motion of the tool and the associated surface location error (SLE) are obtained by a conventional solution technique of ordinary differential equations. It is shown that the SLE is large at the spindle speeds where the ratio of the dominant frequency of the tool and the tooth passing frequency is an integer. This phenomenon is explained by the large amplitude of the periodic chatter-free motion of the tool at these resonant spindle speeds. It is shown that large stable depths of cut with a small SLE can still be attained close to the resonant spindle speeds by using the SLE diagrams associated with stability charts. The results are confirmed experimentally on a high-speed milling center.
Issue Section:
Technical Papers
1.
Tlustý
, J.
, Poláček
, A.
, Danêk
, C.
, Špaček
, J.
, 1962, Selbsterregte Schwingungen an Werkzeugmaschinen
, VEB Verlag Technik
, Berlin.2.
Tobias
, S. A.
, 1965, Machine Tool Vibration
, Blackie
, London.3.
Stépán
, G.
, 1989, Retarded Dynamical Systems
, Longman
, Harlow.4.
Smith
, S.
, and Tlustý
, J.
, 1991, “An Overview of Modeling and Simulation of the Milling Process
,” ASME J. Eng. Ind.
0022-0817, 113
, pp. 169
–175
.5.
Li
, H.
, and Li
, X.
, 2000, “Modelling and Simulation of Chatter in Milling Using a Predictive Force Model
,” Int. J. Mach. Tools Manuf.
0890-6955, 40
, pp. 2047
–2071
.6.
Zhao
, M. X.
, and Balachandran
, B.
, 2001, “Dynamics and Stability of Milling Process
,” Int. J. Solids Struct.
0020-7683, 38
(10–13
), pp. 2233
–2248
.7.
Campomanes
, M. L.
, and Altintas
, Y.
, 2003, “An Improved Time Domain Simulation for Dynamic Milling at Small Radial Immersions
,” ASME J. Manuf. Sci. Eng.
1087-1357, 125
(3
), pp. 416
–422
.8.
Xu
, A.-P.
, Qu
, Y.-X.
, Zhang
, D.-W.
, and Huang
, T.
, 2003, “Simulation and Experimental Investigation of the End Milling Process Considering the Cutter Flexibility
,” Int. J. Mach. Tools Manuf.
0890-6955, 43
, pp. 283
–292
.9.
Paris
, H.
, Peigné
, G.
, and Mayer
, R.
, 2004, “Surface Shape Prediction in High Speed Milling
,” Int. J. Mach. Tools Manuf.
0890-6955, 44
(15
), pp. 1591
–1597
.10.
Altintas
, Y.
, and Budak
, E.
, 1995, “Analytical Prediction of Stability Lobes in Milling
,” CIRP Ann.
0007-8506, 44
(1
), pp. 357
–362
.11.
Tian
, J.
, and Hutton
, S. G.
, 2001, “Chatter Instability in Milling Systems With Flexible Rotating Spindles—A New Theoretical Approach
,” ASME J. Manuf. Sci. Eng.
1087-1357, 123
(1
), pp. 1
–9
.12.
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
(2
), pp. 217
–225
.13.
Bayly
, P. V.
, Halley
, J. E.
, Mann
, B. P.
, and Davies
, M. A.
, 2003, “Stability of Interrupted Cutting by Temporal Finite Element Analysis
,” ASME J. Manuf. Sci. Eng.
1087-1357, 125
(2
), pp. 220
–225
.14.
Faassen
, R. P. H.
, van de Wouw
, N.
, Oosterling
, J. A. J.
, and Nijmeijer
, H.
, 2003, “Prediction of Regenerative Chatter by Modeling and Analysis of High-Speed Milling
,” Int. J. Mach. Tools Manuf.
0890-6955, 43
(14
), pp. 1437
–1446
.15.
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
(1
), pp. 25
–34
.16.
Szalai
, R.
, and Stépán
, G.
, 2003, “Stability Boundaries of High-Speed Milling Corresponding to Period Doubling Are Essentially Closed Curves
,” Proceedings of ASME International Mechanical Engineering Conference and Exposition
, Washington, DC, Paper No. IMECE2003-42122 (CD-ROM).17.
Wang
, J.-J. J.
, Zheng
, C. M.
, and Huang
, C. Y.
, 2003, The Effect of Harmonic Force Components on Regenerative Stability in End Milling
,” Proceedings of the 2003 ASME International Mechanical Engineering Congress and Exposition, Washington, DC
, Paper No. IMECE2003-42367 (CD-ROM).18.
Merdol
, S. D.
, and Altintas
, Y.
, 2004, “Multi Frequency Solution of Chatter Stability for Low Immersion Milling
, ASME J. Manuf. Sci. Eng.
1087-1357, 126
(3
), pp. 459
–466
.19.
Corpus
, W. T.
, and Endres
, W. J.
, 2004, “Added Stability Lobes in Machining Processes That Exhibit Periodic Time Variation—Part 1: An Analytical Solution
,” ASME J. Manuf. Sci. Eng.
1087-1357, 126
(3
), pp. 467
–474
.20.
Insperger
, T.
, Stépán
, G.
, Bayly
, P. V.
, and Mann
, B. P.
, 2003, “Multiple Chatter Frequencies in Milling Processes
,” J. Sound Vib.
0022-460X, 262
(2
), pp. 333
–345
.21.
Gradišek
, J.
, Friedrich
, R.
, Govekar
, E.
, and Grabec
, I.
, 2002, “Analysis of Data From Periodically Forced Stochastic Processes
,” Phys. Lett. A
0375-9601, 294
(3–4
), pp. 234
–238
.22.
Bravo
, U.
, Altuzarra
, O.
, López de Lacalle
, L. N.
, Sánchez
, J. A.
, and Campa
, F. J.
, 2005, “Stability Limits of Milling Considering the Flexibility of the Workpiece and the Machine
,” Int. J. Mach. Tools Manuf.
0890-6955, 45
, pp. 1669
–1680
.23.
Kivanc
, E. B.
, and Budak
, E.
, 2004, “Structural Modeling of End Mills for Form Error and Stability Analysis
,” Int. J. Mach. Tools Manuf.
0890-6955, 44
(11
), pp. 1151
–1161
.24.
Schmitz
, T. L.
, Davies
, M. A.
, and Kennedy
, M. D.
, 2001, “Tool Point Frequency Response Prediction for High-Speed Machining by RCSA
,” ASME J. Manuf. Sci. Eng.
1087-1357, 123
(4
), pp. 700
–707
.25.
Young
, K. A.
, and Helvey
, A. M.
, 2003, “Requirements for Consistent and Productive Performance in High Speed Milling
,” Proceedings of the 2003 ASME International Mechanical Engineering Congress and Exposition
, Washington, DC, Paper No. IMECE2003-41694 (CD-ROM).26.
Kline
, W. A.
, Devor
, R. E.
, and Shareef
, I.
, 1982, “Prediction of Surface Accuracy in End Milling
, ASME J. Eng. Ind.
0022-0817, 104
, pp. 272
–278
.27.
Sutherland
, J. W.
, and DeVor
, R. E.
, 1986, “An Improved Method for Cutting Force and Surface Error Prediction in Flexible End Milling Systems
, ASME J. Eng. Ind.
0022-0817, 108
, pp. 269
–279
.28.
Montgomery
, D.
, and Altintas
, Y.
, 1991, “Mechanism of Cutting Force and Surface Generation in Dynamic Milling
,” ASME J. Eng. Ind.
0022-0817, 113
, pp. 160
–168
.29.
Shirase
, K.
, and Altintas
, Y.
, 1996, “Cutting Force and Dimensional Surface Error Generation in Peripheral Milling With Variable Pitch Helical End Mills
,” Int. J. Mach. Tools Manuf.
0890-6955, 36
(5
), pp. 567
–584
.30.
Schmitz
, T.
, and Ziegert
, J.
, 1999, “Examination of Surface Location Error Due to Phasing of Cutter Vibrations
,” Precis. Eng.
0141-6359, 23
, pp. 51
–62
.31.
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
(1–2
), pp. 31
–48
.32.
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
.33.
Insperger
, T.
, and Stépán
, G.
, 2002, “Semi-Discretization Method for Delayed Systems
,” Int. J. Numer. Methods Eng.
0029-5981, 55
(5
), pp. 503
–518
.34.
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
(1
), pp. 117
–141
.35.
Stépán
, G.
, Szalai
, R.
, Mann
, B. P.
, Bayly
, P. V.
, Insperger
, T.
, Gradišek
, J.
, and Govekar
, E.
, 2005, “Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics and Experiments
,” ASME J. Vibr. Acoust.
0739-3717, 127
(2
), pp. 197
–203
.36.
Lipski
, J.
, Litak
, G.
, Rusinek
, R.
, Szabelski
, K.
, Teter
, A.
, Warmiński
, J.
, and Zaleski
, K.
, 2002, “Surface Quality of a Work Material’s Influence on the Vibrations of the Cutting Process
,” J. Sound Vib.
0022-460X, 252
(4
), pp. 729
–737
.37.
Peigne
, G.
, Paris
, H.
, Brissaud
, D.
, and Gouskov
, A.
, 2004, “Impact of the Cutting Dynamics of Small Radial Immersion Milling Operations on Machined Surface Roughness
, Int. J. Mach. Tools Manuf.
0890-6955, 44
(11
), pp. 1133
–1142
.38.
Lazoglu
, I.
, and Liang
, S. Y.
, 1997, “Analytical Modeling of Ball-End Milling Forces
,” Mach. Sci. Technol.
1091-0344, 1
(2
), pp. 219
–234
.39.
Budak
, E.
, and Altintas
, Y.
, 1998, “Analytical Prediction of Chatter Stability in Milling—Part I: General Formulation
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 120
, pp. 22
–30
.40.
Budak
, E.
, and Altintas
, Y.
, 1998, “Analytical Prediction of Chatter Stability in Milling—Part II: Application of the General Formulation to Common Milling Systems
,” ASME J. Dyn. Syst., Meas., Control
0022-0434, 120
, pp. 31
–36
.41.
Gradišek
, J.
, Kalveram
, M.
, and Weinert
, K.
, 2004, “Mechanistic Identification of Specific Force Coefficients for a General End Mill
,” Int. J. Mach. Tools Manuf.
0890-6955, 44
(4
), pp. 401
–414
.42.
Gradišek
, J.
, Kalveram
, M.
, Insperger
, T.
, Weinert
, K.
, Stépán
, G.
, Govekar
, E.
, and Grabec
, I.
, 2005, “On Stability Prediction for Milling
,” Int. J. Mach. Tools Manuf.
0890-6955, 45
(7–8
), pp. 741
–991
.43.
Li
, H. Z.
, Liu
, K.
, and Li
, X. P.
, 2001, “A New Method for Determining the Undeformed Chip Thickness in Milling
,” J. Mater. Process. Technol.
0924-0136, 113
, pp. 378
–384
.44.
Faassen
, R.
, van de Wouw
, N.
, Oosterling
, H.
, and Nijmeijer
, H.
, 2005, “Updated Tool Path Modelling With Periodic Delay for Chatter Prediction in Milling
,” Proceedings of the Fifth EUROMECH Nonlinear Dynamics Conference
, Eindhoven
, The Netherlands, pp. 1080
–1089
.Copyright © 2006
by American Society of Mechanical Engineers
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