The stability charts of high-speed milling are constructed. New unstable regions and vibration frequencies are identified. These are related to flip bifurcation, i.e. period doubling vibrations occur apart of the conventional self-excited vibrations well-known for turning or low-speed milling with multiple active teeth. The Semi-Discretization method is applied for the delayed parametric excitation model of milling providing the connection of the two existing and experimentally verified results of machine tool chatter research. The two extreme models in question, that is, the traditional autonomous delayed model of time-independent turning, and the recently introduced discrete map model of time-dependent highly interrupted machining, are both involved as special cases in the universal approach presented in this study.

1.
Taylor
,
F. W.
,
1907
, “
On the Art of Cutting Metals
,”
Trans. ASME
,
28
, pp.
31
350
.
2.
Tlusty, J., Polacek, A., Danek, C., and Spacek, J., 1962, Selbsterregte Schwingungen an Werkzeugmaschinen, VEB Verlag Technik, Berlin.
3.
Tobias, S. A., 1965, Machine Tool Vibration, Blackie, London.
4.
Kudinov, V. A., 1967, Dynamics of Tool-Lathe (in Russian), Mashinostroenie, Moscow.
5.
Moon, F. C., 1998, Dynamics and Chaos in Manufacturing Processes, Wiley, New York.
6.
Seagalman
,
D. J.
, and
Butcher
,
E. A.
,
2000
, “
Suppression of Regenerative Chatter via Impendance Modulation
,”
Pramana, Suppl.
,
6
, pp.
243
256
.
7.
Ste´pa´n
,
G.
,
2001
, “
Modelling Nonlinear Regenerative Effects in Metal Cutting
,”
Philos. Trans. R. Soc. London
,
359
, pp.
739
757
.
8.
Gouskov
,
A. M.
,
Voronov
,
S. A.
,
Paris
,
H.
, and
Batzer
,
S. A.
,
2002
, “
Nonlinear Dynamics of a Machining System With Two Interdependent Delays,”
Commun. Nonlinear Sci. Numer. Simul.
,
7
(
4
), pp.
207
221
.
9.
Ste´pa´n, G., 1989, Retarded Dynamical Systems, Longman, Harlow.
10.
Ste´pa´n, G., 1998, Delay-differential Equation Models for Machine Tool Chatter, in Dynamics and Chaos in Manufacturing Processes, Moon, F. C., ed., Wiley, New York, pp. 165–192.
11.
Shi
,
H. M.
, and
Tobias
,
S. A.
,
1984
, “
Theory of Finite Amplitude Machine Tool Instability
,”
Int. J. Mach. Tool Des. Res.
,
24
, pp.
45
69
.
12.
Ste´pa´n, G., and Kalma´r-Nagy, T., 1997, “Nonlinear Regenerative Machine Tool Vibration,” Proceedings of the 1997 ASME Design Engineering Technical Conferences, Sacramento, California, paper no. DETC97/VIB-4021 (CD-ROM).
13.
Ste´pa´n, G., Szalai, R., and Insperger, T., 2004, Nonlinear Dynamics of High-Speed Milling Subjected to Regenerative Effect, in Nonlinear Dynamics of Production Systems, Radons, G., ed., Wiley VCH, Weinheim, pp. 111–128.
14.
Kalma´r-Nagy, T., Pratt, J. R., Davies, M. A., and Kennedy, M., 1999, “Experimental and Analytical Investigation of the Subcritical Instability in Metal Cutting,” Proceedings of the ASME 1999 Design Engineering Technical Conferences, Sacramento, California, paper no. DETC99/VIB-8060, (CD-ROM).
15.
Balachandran
,
B.
,
2001
, “
Nonlinear Dynamics of Milling Process
,”
Philos. Trans. R. Soc. London
,
359
, pp.
793
820
.
16.
Metallidis
,
P.
, and
Natsiavas
,
S.
,
2000
, “
Vibration of a Continuous System With Clearance and Motion Constraints
,”
Int. J. Non-Linear Mech.
,
35
(
4
), pp.
675
690
.
17.
Batzer, S. A., Gouskov, A. M., and Voronov, S. A., 1999, “Modelling the Vibratory Drilling Process,” Proceedings of the 1999 ASME Design Engineering Technical Conferences, Las Vegas, Nevada, paper no. DETC99/VIB-8024, (CD-ROM).
18.
Davies
,
M. A.
, and
Balachandran
,
B.
,
2000
, “
Impact Dynamics in the Milling of Thin-Walled Structures
,”
Nonlinear Dyn.
,
22
(
4
), pp.
375
392
.
19.
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.
,
124
(
2
), pp.
217
225
.
20.
Hale, J. K., and Lunel, S. M. V., 1993, Introduction to Functional Differential Equations, Springer-Verlag, New York.
21.
Farkas, M., 1994, Periodic Motions, Springer-Verlag, New York.
22.
Minis
,
I.
, and
Yanushevsky
,
R.
,
1993
, “
A New Theoretical Approach for the Prediction of Machine Tool Chatter in Milling
,”
ASME J. Ind.
,
115
, pp.
1
8
.
23.
Altintas
,
Y.
, and
Budak
,
E.
,
1995
, “
Analytical Prediction of Stability Lobes in Milling
,”
CIRP Ann.
,
44
(
1
), pp.
357
362
.
24.
Budak
,
E.
, and
Altintas
,
Y.
,
1998
, “
Analytical Prediction of Chatter Stability in Milling—Part I: General Formulation
,”
ASME J. Dyn. Syst., Meas., Control
,
120
, pp.
22
30
.
25.
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
,
120
, pp.
31
36
.
26.
Corpus, W. T., and Endres, W. J., 2000, “A High-Order Solution for the Added Stability Lobes in Intermittent Machining,” Proceedings of the Symposium on Machining Processes, Orlando, Florida, MED-11, pp. 871–878.
27.
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.
,
123
(
1
), pp.
1
9
.
28.
Insperger
,
T.
, and
Ste´pa´n
,
G.
,
2000
, “
Stability of the Milling Process
,”
Period. Polytech., Mech. Eng.-Masinostr.
,
44
(
1
), pp.
47
57
.
29.
Insperger
,
T.
, and
Ste´pa´n
,
G.
,
2002
, “
Semi-Discretization Method for Delayed Systems
,”
Int. J. Numer. Methods Eng.
,
55
(
5
), pp.
503
518
.
30.
Bayly, P. V., Halley, J. E., Mann, B. P., and Davies, M. A., 2001, “Stability of Interrupted Cutting by Temporal Finite Element Analysis,” Proceedings of the ASME 2001 Design Engineering Technical Conferences, Pittsburgh, Pennsylvania, paper no. DETC2001/VIB-21581 (CD-ROM).
31.
Smith
,
S.
, and
Tlusty
,
J.
,
1991
, “
An Overview of Modeling and Simulation of the Milling Process
,”
ASME J. Ind.
,
113
, pp.
169
175
.
32.
Gradisˇek
,
J.
,
Govekar
,
E.
, and
Grabec
,
I.
,
1998
, “
Time Series Analysis in Metal Cutting: Chatter Versus Chatter-Free Cutting
,”
Mech. Syst. Signal Process.
,
12
(
6
), pp.
839
854
.
33.
Schmitz
,
T. L.
,
Davies
,
M. A.
,
Medicus
,
K.
, and
Snyder
,
J.
,
2001
, “
Improving High-Speed Machining Material Removal Rates by Rapid Dynamic Analysis
,”
CIRP Ann.
,
50
(
1
), pp.
263
268
.
34.
Gradisˇek
,
J.
,
Friedrich
,
R.
,
Govekar
,
E.
, and
Grabec
,
I.
,
2002
, “
Analysis of Data From Periodically Forced Stochastic Processes
,”
Phys. Lett. A
,
294
(
3–4
), pp.
234
238
.
35.
Insperger
,
T.
,
Mann
,
B. P.
,
Ste´pa´n
,
G.
, and
Bayly
,
P. V.
,
2003
, “
Stability of Up-Milling and Down-Milling, Part 1: Alternative Analytical Methods
,”
Int. J. Mach. Tools Manuf.
,
43
(
1
), pp.
25
34
.
36.
Zhao
,
M. X.
, and
Balachandran
,
B.
,
2001
, “
Dynamics and Stability of Milling Process
,”
Int. J. Solids Struct.
,
38
(
10–13
), pp.
2233
2248
.
37.
Insperger
,
T.
,
Ste´pa´n
,
G.
,
Bayly
,
P. V.
, and
Mann
,
B. P.
,
2003
, “
Multiple Chatter Frequencies in Milling Processes
,”
J. Sound Vib.
,
626
(
2
), pp.
333
345
.
38.
Mann
,
B. P.
,
Insperger
,
T.
,
Bayly
,
P. V.
, and
Ste´pa´n
,
G.
,
2002
, “
Stability of Up-Milling and Down-Milling, Part 2: Experimental Verification
,”
Int. J. Mach. Tools Manuf.
,
43
(
1
), pp.
35
40
.
39.
Laczik, B., 1986, “Vibration Monitoring of Cutting Processes,” (in Hungarian), PhD Thesis, Technical University of Budapest, Hungary.
40.
Kondo
,
E.
,
Ota
,
H.
, and
Kawai
,
T.
,
1992
, “
Regenerative Chatter Vibrations of Turning Workpiece
,”
Trans. Jpn. Soc. Mech. Eng., Ser. A
,
58
, pp.
1251
1265
.
41.
Halley, J. E., Helvey, A. M., Smith, K. S., and Winfough, W. R., 1999, “The Impact of High-Speed Machining on the Design and Fabrication of Aircraft Components,” Proceedings of 1999 ASME Design and Technical Conferences, Las Vegas, Nevada, paper no. DETC99/VIB-8057 (CD-ROM).
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