This paper describes the use of subharmonic sampling to distinguish between different instability types in milling. It is demonstrated that sampling time-domain milling signals at integer multiples of the tooth period enables secondary Hopf and period-n bifurcations to be automatically differentiated. A numerical metric is applied, where the normalized sum of the absolute values of the differences between successively sampled points is used to distinguish between the potential bifurcation types. A new stability map that individually identifies stable and individual bifurcation zones is presented. The map is constructed using time-domain simulation and the new subharmonic sampling metric.

References

References
1.
Honeycutt
,
A.
, and
Schmitz
,
T.
,
2016
, “
A New Metric for Automated Stability Identification in Time Domain Milling Simulation
,”
ASME J. Manuf. Sci. Eng.
,
138
(
7
), p.
074501
.
2.
Arnold
,
R. N.
,
1946
, “
The Mechanism of Tool Vibration in the Cutting of Steel
,”
Proc. Inst. Mech. Eng.
,
154
(
1
), pp.
261
284
.
3.
Doi
,
S.
, and
Kato
,
S.
,
1956
, “
Chatter Vibration of Lathe Tools
,”
Trans. ASME
,
78
(
5
), pp.
1127
1134
.
4.
Tobias
,
S. A.
, and
Fishwick
,
W.
,
1958
, “
The Chatter of Lathe Tools Under Orthogonal Cutting Conditions
,”
Trans. ASME
,
80
(
2
), pp.
1079
1088
.
5.
Tlusty
,
J.
, and
Polacek
,
M.
,
1963
, “
The Stability of Machine Tools Against Self-Excited Vibrations in Machining
,”
Intl. Res. Prod. Eng.
,
1
(
1
), pp.
465
474
.
6.
Tobias
,
S. A.
,
1965
,
Machine Tool Vibration
,
Wiley
,
New York
.
7.
Merritt
,
H. E.
,
1965
, “
Theory of Self-Excited Machine-Tool Chatter
,”
ASME J. Eng. Ind.
,
87
(
4
), pp.
447
454
.
8.
Tlusty
,
J.
, and
Polacek
,
M.
,
1968
, “
Experience With Analysing Stability of Machine Tool Against Chatter
,”
9th MTDR Conference
, pp.
521
570
.
9.
Shridar
,
R.
,
Hohn
,
R. E.
, and
Long
,
G. W.
,
1968
, “
A General Formulation of the Milling Process Equation
,”
ASME J. Eng. Ind.
,
90
(
2
), pp.
317
324
.
10.
Hohn
,
R. E.
,
Shridar
,
R.
, and
Long
,
G. W.
,
1968
, “
A Stability Algorithm for a Special Case of the Milling Process
,”
ASME J. Eng. Ind.
,
90
(
2
), pp.
326
329
.
11.
Shridar
,
R.
,
Hohn
,
R. E.
, and
Long
,
G. W.
,
1968
, “
A Stability Algorithm for the General Milling Process
,”
ASME J. Eng. Ind.
,
90
(
2
), pp.
330
334
.
12.
Hanna
,
N. H.
, and
Tobias
,
S. A.
,
1974
, “
A Theory of Nonlinear Regenerative Chatter
,”
ASME J. Eng. Ind.
,
96
(
1
), pp.
247
255
.
13.
Tlusty
,
J.
, and
Ismail
,
F.
,
1981
, “
Basic Non-Linearity in Machining Chatter
,”
Ann. CIRP
,
30
(
1
), pp.
299
304
.
14.
Tlusty
,
J.
, and
Ismail
,
F.
,
1983
, “
Special Aspects of Chatter in Milling
,”
ASME J. Vib. Stress Reliab. Des.
,
105
(
1
), pp.
24
32
.
15.
Tlusty
,
J.
,
1985
, “
Machine Dynamics
,”
Handbook of High-Speed Machining Technology
,
R. I.
King
, ed.,
Chapman and Hall
,
New York
, pp.
48
153
.
16.
Tlusty
,
J.
,
1986
, “
Dynamics of High-Speed Milling
,”
ASME J. Eng. Ind.
,
108
(
2
), pp.
59
67
.
17.
Minis
,
I.
, and
Yanusevsky
,
R.
,
1993
, “
A New Theoretical Approach for Prediction of Chatter in Milling
,”
ASME J. Eng. Ind.
,
115
(
1
), pp.
1
8
.
18.
Altintas
,
Y.
, and
Budak
,
E.
,
1995
, “
Analytical Prediction of Stability Lobes in Milling
,”
Ann. CIRP
,
44
(
1
), pp.
357
362
.
19.
Davies
,
M. A.
,
Dutterer
,
B. S.
,
Pratt
,
J. R.
, and
Schaut
,
A. J.
,
1998
, “
On the Dynamics of High-Speed Milling With Long, Slender Endmills
,”
Ann. CIRP
,
47
(
1
), pp.
55
60
.
20.
Moon
,
F. C.
, and
Kalmár-Nagy
,
T.
,
2001
, “
Nonlinear Models for Complex Dynamics in Cutting Materials
,”
Philos. Trans. R. Soc. London A
,
359
(
1781
), pp.
695
711
.
21.
Davies
,
M. A.
,
Pratt
,
J. R.
,
Dutterer
,
B. S.
, and
Burns
,
T. J.
,
2000
, “
The Stability of Low Radial Immersion Milling
,”
Ann. CIRP
,
49
(
1
), pp.
37
40
.
22.
Moon
,
F. C.
,
1994
, “
Chaotic Dynamics and Fractals in Material Removal Processes
,”
Nonlinearity and Chaos in Engineering Dynamics
,
J.
Thompson
and
S.
Bishop
, eds.,
Wiley
,
Hoboken, NJ
, pp.
25
37
.
23.
Bukkapatnam
,
S.
,
Lakhtakia
,
A.
, and
Kumara
,
S.
,
1995
, “
Analysis of Senor Signals Shows Turning on a Lathe Exhibits Low-Dimensional Chaos
,”
Phys. Rev. E
,
52
(
3
), pp.
2375
2387
.
24.
Stépán
,
G.
, and
Kalmár-Nagy
,
T.
,
1997
, “
Nonlinear Regenerative Machine Tool Vibrations
,” ASME Paper No. DETC 97/VIB-4021.
25.
Nayfey
,
A.
,
Chin
,
C.
, and
Pratt
,
J.
,
1998
, “
Applications of Perturbation Methods to Tool Chatter Dynamics
,”
Dynamics and Chaos in Manufacturing Processes
,
F. C.
Moon
, ed.,
Wiley
,
Hoboken, NJ
, pp.
193
213
.
26.
Minis
,
I.
, and
Berger
,
B. S.
,
1998
, “
Modelling, Analysis, and Characterization of Machining Dynamics
,”
Dynamics and Chaos in Manufacturing Processes
,
F. C.
Moon
, ed.,
Wiley
,
Hoboken, NJ
, pp.
125
163
.
27.
Moon
,
F. C.
, and
Johnson
,
M.
,
1998
, “
Nonlinear Dynamics and Chaos in Manufacturing Processes
,”
Dynamics and Chaos in Manufacturing Processes
,
F. C.
Moon
, ed.,
Wiley
,
Hoboken, NJ
, pp.
3
32
.
28.
Smith
,
K. S.
, and
Tlusty
,
J.
,
1991
, “
An Overview of Modeling and Simulation of the Milling Process
,”
ASME J. Eng. Ind.
,
113
(
2
), pp.
169
175
.
29.
Campomanes
,
M. L.
, and
Altintas
,
Y.
,
2003
, “
An Improved Time Domain Simulation for Dynamic Milling at Small Radial Immersions
,”
ASME J. Manuf. Sci. Eng.
,
125
(
3
), pp.
416
422
.
30.
Zhao
,
M. X.
, and
Balachandran
,
B.
,
2001
, “
Dynamics and Stability of Milling Process
,”
Int. J. Solids Struct.
,
38
(
10–13
), pp.
2233
2248
.
31.
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
.
32.
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.
,
43
(
1
), pp.
35
40
.
33.
Mann
,
B. P.
,
Insperger
,
T.
,
Bayly
,
P. V.
, and
Stépán
,
G.
,
2003
, “
Stability of Up-Milling and Down-Milling—Part 1: Alternative Analytical Methods
,”
Int. J. Mach. Tools Manuf.
,
43
(
1
), pp.
25
34
.
34.
Insperger
,
T.
,
Stépán
,
G.
,
Bayly
,
P. V.
, and
Mann
,
B. P.
,
2003
, “
Multiple Chatter Frequencies in Milling Processes
,”
J. Sound Vib.
,
262
(
2
), pp.
333
345
.
35.
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.
,
126
(
3
), pp.
481
487
.
36.
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.
,
277
(
1–2
), pp.
31
48
.
37.
Merdol
,
S. D.
, and
Altintas
,
Y.
,
2004
, “
Multi Frequency Solution of Chatter Stability for Low Immersion Milling
,”
ASME J. Manuf. Sci. Eng.
,
126
(
3
), pp.
459
466
.
38.
Govekar
,
E.
,
Gradišek
,
J.
,
Kalveram
,
M.
,
Insperger
,
T.
,
Weinert
,
K.
,
Stepan
,
G.
, and
Grabec
,
I.
,
2005
, “
On Stability and Dynamics of Milling at Small Radial Immersion
,”
Ann. CIRP
,
54
(
1
), pp.
357
362
.
39.
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.
,
45
(
7–8
), pp.
769
781
.
40.
Mann
,
B. P.
,
Garg
,
N. K.
,
Young
,
K. A.
, and
Helvey
,
A. M.
,
2005
, “
Milling Bifurcations From Structural Asymmetry and Nonlinear Regeneration
,”
Nonlinear Dyn.
,
42
(
4
), pp.
319
337
.
41.
Stépán
,
G.
,
Szalai
,
R.
,
Mann
,
B. P.
,
Bayly
,
P. V.
,
Insperger
,
T.
,
Gradisek
,
J.
, and
Govekar
,
E.
,
2005
, “
Nonlinear Dynamics of High-Speed Milling—Analyses, Numerics, and Experiments
,”
ASME J. Vib. Acoust.
,
127
(
2
), pp.
197
203
.
42.
Zatarain
,
M.
,
Muñoa
,
J.
,
Peigné
,
G.
, and
Insperger
,
T.
,
2006
, “
Analysis of the Influence of Mill Helix Angle on Chatter Stability
,”
Ann. CIRP
,
55
(
1
), pp.
365
368
.
43.
Insperger
,
T.
,
Munoa
,
J.
,
Zatarain
,
M. A.
, and
Peigné
,
G.
,
2006
, “
Unstable Islands in the Stability Chart of Milling Processes Due to the Helix Angle
,”
CIRP
2nd International Conference on High Performance Cutting
, Vancouver, Canada, pp.
12
13
.
44.
Patel
,
B. R.
,
Mann
,
B. P.
, and
Young
,
K. A.
,
2008
, “
Uncharted Islands of Chatter Instability in Milling
,”
Int. J. Mach. Tools Manuf.
,
48
(
1
), pp.
124
134
.
45.
Moradi
,
H.
,
Vossoughi
,
G.
, and
Movahhedy
,
M.
,
2014
, “
Bifurcation Analysis of Nonlinear Milling Process With Tool Wear and Process Damping: Sub-Harmonic Resonance Under Regenerative Chatter
,”
Int. J. Mech. Sci.
,
85
, pp.
1
19
.
46.
Honeycutt
,
A.
, and
Schmitz
,
T.
,
2015
, “
The Extended Milling Bifurcation Diagram
,”
Procedia Manuf.
,
1
, pp.
466
476
.
47.
Honeycutt
,
A.
, and
Schmitz
,
T.
,
2016
, “
A Numerical and Experimental Investigation of Period-n Bifurcations in Milling
,”
ASME J. Manuf. Sci. Eng.
,
139
(
1
), p.
011003
.
48.
Schmitz
,
T.
, and
Smith
,
K. S.
,
2009
,
Machining Dynamics: Frequency Response to Improved Productivity
,
Springer
,
New York
.
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