This paper provides time domain simulation and experimental results for surface location error (SLE) and surface roughness when machining under both stable (forced vibration) and unstable (period-2 bifurcation) conditions. It is shown that the surface location error follows similar trends observed for forced vibration, so zero or low error conditions may be selected even for period-2 bifurcation behavior. The surface roughness for the period-2 instability is larger than for stable conditions because the surface is defined by every other tooth passage and the apparent feed per tooth is increased. Good agreement is observed between simulation and experiment for stability, surface location error, and surface roughness results.

References

1.
Arnold
,
R. N.
,
1946
, “
The Mechanism of Tool Vibration in the Cutting of Steel
,”
Proc. Inst. Mech. Eng.
,
154
(1), pp. 261–284.
2.
Doi
,
S.
, and
Kato
,
S.
,
1956
, “
Chatter Vibration of Lathe Tools
,”
Trans. ASME
,
78
, pp.
1127
1134
.
3.
Tobias
,
S. A.
, and
Fishwick
,
W.
,
1958
, “
The Chatter of Lathe Tools Under Orthogonal Cutting Conditions
,”
Trans. ASME
,
80
, pp.
1079
1088
.
4.
Tlusty
,
J.
, and
Polacek
,
M.
,
1963
, “
The Stability of Machine Tools Against Self-Excited Vibrations in Machining
,”
ASME International Research in Production Engineering Conference
, Pittsburgh, PA, pp.
465
474
.
5.
Tobias
,
S. A.
,
1965
,
Machine Tool Vibration
,
Wiley
,
New York
.
6.
Merritt
,
H. E.
,
1965
, “
Theory of Self-Excited Machine-Tool Chatter
,”
ASME J. Eng. Ind.
,
87
(
4
), pp.
447
454
.
7.
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
.
8.
Hohn
,
R. E.
,
Shridar
,
R.
, and
Long
,
G. W.
,
1968
, “
A Stability Algorithm for a Special Case of the Milling Process: Contribution to Machine Tool Chatter Research—6
,”
ASME J. Eng. Ind.
,
90
(2), pp.
326
329
.
9.
Shridar
,
R.
,
Hohn
,
R. E.
, and
Long
,
G. W.
,
1968
, “
A Stability Algorithm for the General Milling Process: Contribution to Machine Tool Chatter Research—7
,”
ASME J. Eng. Ind.
,
90
(
2
), pp.
330
334
.
10.
Hanna
,
N. H.
, and
Tobias
,
S. A.
,
1974
, “
A Theory of Nonlinear Regenerative Chatter
,”
ASME J. Eng. Ind.
,
96
(
1
), pp.
247
255
.
11.
Tlusty
,
J.
, and
Ismail
,
F.
,
1981
, “
Basic Non-Linearity in Machining Chatter
,”
Ann. CIRP
,
30
(
1
), pp.
299
304
.
12.
Tlusty
,
J.
, and
Ismail
,
F.
,
1983
, “
Special Aspects of Chatter in Milling
,”
ASME J. Vib., Stress Reliab. Des.
,
105
(
1
), pp.
24
32
.
13.
Tlusty
,
J.
,
1985
, “
Machining Dynamics
,”
Handbook of High-Speed Machining Technology
,
R. I.
King
, ed.,
Chapman and Hall
,
New York
, pp.
48
153
.
14.
Tlusty
,
J.
,
1986
, “
Dynamics of High-Speed Milling
,”
ASME J. Eng. Ind.
,
108
(
2
), pp.
59
67
.
15.
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
.
16.
Altintas
,
Y.
, and
Budak
,
E.
,
1995
, “
Analytical Prediction of Stability Lobes in Milling
,”
Ann. CIRP
,
44
(
1
), pp.
357
362
.
17.
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
.
18.
Moon
,
F. C.
, and
Kalmár-Nagy
,
T.
,
2001
, “
Nonlinear Models for Complex Dynamics in Cutting Materials
,”
Philos. Trans. R. Soc.
, A,
359
(
1781
), pp.
695
711
.
19.
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
.
20.
Moon
,
F. C.
,
1994
, “
Chaotic Dynamics and Fractals in Material Removal Processes
,”
Nonlinearity and Chaos in Engineering Dynamics
,
J.
Thompson
, and
S.
Bishop
, ed.,
Wiley
, New York, pp.
25
37
.
21.
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
.
22.
Stépán
,
G.
, and
Kalmár-Nagy
,
T.
,
1997
, “
Nonlinear Regenerative Machine Tool Vibrations
,”
Proceedings of the ASME Design Engineering Technical Conference on Vibration and Noise
, Sacramento, CA, Sept. 14–17.
23.
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
, New York, pp.
193
213
.
24.
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
, New York, pp.
125
163
.
25.
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
, New York, pp.
3
32
.
26.
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
.
27.
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
.
28.
Zhao
,
M. X.
, and
Balachandran
,
B.
,
2001
, “
Dynamics and Stability of Milling Process
,”
Int. J. Solids Struct.
,
38
(10–13), pp.
2233
2248
.
29.
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
.
30.
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
.
31.
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
.
32.
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
.
33.
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
.
34.
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
.
35.
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
.
36.
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
.
37.
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
.
38.
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
.
39.
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
.
40.
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
.
41.
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, June 12–13, Vancouver, BC, Canada, pp. 12–13.
42.
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
.
43.
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
.
44.
Honeycutt
,
A.
, and
Schmitz
,
T.
,
2015
, “
The Extended Milling Bifurcation Diagram
,”
Proc. Manuf.
,
1
, pp.
466
476
.
45.
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
.
46.
Kline
,
W.
,
DeVor
,
R.
, and
Shareef
,
I.
,
1982
, “
The Prediction of Surface Accuracy in End Milling
,”
ASME J. Eng. Ind.
,
104
(
3
), pp.
272
278
.
47.
Kline
,
W.
,
DeVor
,
R.
, and
Lindberg
,
J.
,
1982
, “
The Prediction of Cutting Forces in End Milling With Application to Cornering Cuts
,”
Int. J. Mach. Tool Des. Res.
,
22
(
1
), pp.
7
22
.
48.
Tlusty
,
J.
,
1985
, “
Effect of End Milling Deflections on Accuracy
,”
Handbook of High Speed Machining Technology
,
R. I.
King
, ed.,
Chapman and Hall
,
New York
, pp.
140
153
.
49.
Sutherland
,
J.
, and
DeVor
,
R.
,
1986
, “
An Improved Method for Cutting Force and Surface Error Prediction in Flexible End Milling Systems
,”
ASME J. Eng. Ind.
,
108
(
4
), pp.
269
279
.
50.
Montgomery
,
D.
, and
Altintas
,
Y.
,
1991
, “
Mechanism of Cutting Force and Surface Generation in Dynamic Milling
,”
ASME J. Eng. Ind.
,
113
(
2
), pp.
160
168
.
51.
Altintas
,
Y.
,
Montgomery
,
D.
, and
Budak
,
E.
,
1992
, “
Dynamic Peripheral Milling of Flexible Structures
,”
ASME J. Eng. Ind.
,
114
(
2
), pp.
137
145
.
52.
Tarng
,
Y.
,
Liao
,
C.
, and
Li
,
H.
,
1994
, “
A Mechanistic Model for Prediction of the Dynamics of Cutting Forces in Helical End Milling
,”
Int. J. Model. Simul.
,
14
(
2
), pp.
92
97
.
53.
Schmitz
,
T.
, and
Ziegert
,
J.
,
1999
, “
Examination of Surface Location Error Due to Phasing of Cutter Vibrations
,”
Prec. Eng.
,
23
(
1
), pp.
51
62
.
54.
Altintas
,
Y.
,
2000
,
Manufacturing Automation
,
Cambridge University Press
,
Cambridge, UK.
55.
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
.
56.
Schmitz
,
T.
,
Couey
,
J.
,
Marsh
,
E.
,
Mauntler
,
N.
, and
Hughes
,
D.
,
2007
, “
Runout Effects in Milling: Surface Finish, Surface Location Error, and Stability
,”
Int. J. Mach. Tools Manuf.
,
47
(
5
), pp.
841
851
.
57.
Yun
,
W.-S.
,
Ko
,
J.
,
Cho
,
D.-W.
, and
Ehmann
,
K.
,
2002
, “
Development of a Virtual Machining System—Part 2: Prediction and Analysis of a Machined Surface Error
,”
Int. J. Mach. Tools Manuf.
,
42
(
15
), pp.
1607
1615
.
58.
Schmitz
,
T.
, and
Mann
,
B.
,
2006
, “
Closed-Form Solutions for Surface Location Error in Milling
,”
Int. J. Mach. Tools Manuf.
,
46
(
12–13
), pp.
1369
1377
.
59.
Schmitz
,
T.
, and
Smith
,
K. S.
,
2009
,
Machining Dynamics: Frequency Response to Improved Productivity
,
Springer
,
New York
.
60.
Dombovari
,
Z.
, and
Stépán
,
G.
,
2015
, “
On the Bistable Zone of Milling Processes
,”
Philos. Trans. R. Soc., A
,
373
(
2051
), p.
20140409
.
61.
Bachrathy
,
D.
,
Munoa
,
J.
, and
Stépán
,
G.
,
2016
, “
Experimental Validation of Appropriate Axial Immersions for Helical Mills
,”
Int. J. Adv. Manuf. Technol.
,
84
(5), pp.
1295
1302
.
62.
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
.
63.
Ransom
,
T.
,
Honeycutt
,
A.
, and
Schmitz
,
T.
,
2016
, “
A New Tunable Dynamics Platform for Milling Experiments
,”
Prec. Eng.
,
44
, pp.
252
256
.
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