A method for obtaining analytic bounds for period doubling and cyclic fold instability regions in linear time-periodic systems with piecewise constant coefficients and time delay is suggested. The method is based on the use of transition matrices for Meissner’s equation corresponding to the desired type of instability. Analytic expressions for the disconnected regions of fold and flip instability for two- and three-segment coefficients including both complex and real eigenvalues in Meissner’s equation are obtained. The proposed method when applied to the example of two-segment interrupted turning with complex eigenvalues in each segment yields the same results as those obtained recently for the boundaries of the flip regions (Szalai and Stepan, 2006, “Lobes and Lenses in the Stability Chart of Interrupted Turning,” J Comput. Nonlinear Dyn., 1, pp. 205–211). Next, the period-doubling instability regions for a particular delay differential equation related to the damped Meissner’s equation and the fold instabilities for a model of delayed position feedback control are analytically obtained. Finally, we extend the method to a single degree-of-freedom milling model with a three-piecewise-constant-segment approximation to the true specific cutting force in which lower bounds for and horizontal locations of the regions of flip instability are obtained. The analytic results are verified through numerical stability charts obtained using the temporal finite element method. Conditions for the existence of islands of instability are also obtained.

References

References
1.
Tobias
,
S. A.
, 1961,
Schwingungen an Werkzeugmaschinen
,
Carl Hanswer Verlag
,
Munchen, Germany
.
2.
Tlusty
,
J.
, 2000,
Manufacturing Process and Equipment
,
Prentice-Hall
,
Englewood Cliffs, NJ
.
3.
Neimark
,
J. I.
, 1959, “
On Some Cases of Periodic Motions Depending on Parameters
,”
Dokl. Akad. Nauk USSR
,
129
, pp.
736
739
.
4.
Sacker
,
R.
, 1964, “
On Invariant Surfaces and Bifurcation of Periodic Solutions of Ordinary Differential Equations
,”
New York University
, Report No. IMM-NYU 333.
5.
Davies
,
M. A.
,
Pratt
,
J. R.
,
Dutterer
,
B.
, and
Burns
,
T. J.
, 2002, “
Stability Prediction for Low Radial Immersion Milling
,”
J. Manuf. Sci. Eng.
,
125
, pp.
217
225
.
6.
Insperger
,
T.
,
Mann
,
B. P.
,
Surmann
,
T.
, and
Stepan
,
G.
, 2008, “
On the Chatter Frequencies of Milling Processes With Runout
,”
Int. J. Mach. Tools Manuf.
,
48
, pp.
1081
1089
.
7.
Insperger
,
T.
, and
Stepan
,
G.
, 2004, “
Stability Analysis of Turning With Periodic Spindle Speed Modulation via Semidiscretization
,”
J. Vib. Control
,
10
, pp.
1835
1855.
8.
Sims
,
N. D.
,
Mann
,
B. P.
, and
Huyanan
,
S.
, 2008, “
Analytical Prediction of Chatter Stability for Variable Pitch and Variable Helix Milling Tools
,”
J. Sound Vib.
,
317
, pp.
664
686
.
9.
Dombovari
,
Z.
,
Altintas
,
Y.
, and
Stepan
,
G.
, 2010, “
The Effect of Serration on Mechanics and Stability of Milling Cutters
,”
Int. J. Mach. Tools Manuf.
,
50
, pp.
511
520
.
10.
Merdol
,
S. D.
, and
Altintas
,
Y.
, 2004, “
Mechanics and Dynamics of Serrated Cylindrical and Tapered Endmills
,”
J. Manuf. Sci. Eng.
,
126
, pp.
317
326
.
11.
Patel
,
B. R.
,
Mann
,
B. P.
, and
Young
,
K. A.
, 2008, “
Uncharted Islands of Chatter Instability in Milling
,”
Int. J. Mach. Tools Manuf.
,
48
, pp.
124
134
.
12.
Mann
,
B. P.
,
Edes
,
B. T.
,
Easley
,
S. J.
,
Young
,
K. A.
, and
Ma
,
K.
, 2008, “
Chatter Vibration and Surface Location Error Prediction for Helical End Mills
,”
Int. J. Mach. Tools Manuf.
,
48
, pp.
350
361
.
13.
Insperger
,
T.
,
Munoa
,
J.
,
Zatarain
,
M.
, and
Peigne
,
G.
, 2006, “
Unstable Islands in the Stability Chart of Milling Processes due to the Helix Angle
,”
CIRP 2nd International Conference on High Performance Cutting
, June 12-13, 2006,
Vancouver
.
14.
Khasawneh
,
F. A.
,
Mann
,
B. P.
,
Bobrenkov
,
O. A.
, and
Butcher
,
E. A.
, 2009, “
Self-Excited Vibrations in a Delay Oscillator: Application to Milling With Simultaneously Engaged Helical Flutes
,”
Proceedings of the ASME IDETC/CIE
,
San Diego
, Aug. 30–Sep. 2, 2009.
15.
Szalai
,
R.
, and
Stepan
,
G.
, 2006, “
Lobes and Lenses in the Stability Chart of Interrupted Turning
,”
J Comput. Nonlinear Dyn.
,
1
, pp.
205
211
.
16.
Insperger
,
T.
, and
Stepan
,
G.
, 2004, “
Updated Semi-Discretization Method for Periodic Delay-Differential Equations With Discrete Delay
,”
Int. J. Numer. Methods Eng.
,
61
, pp.
117
141
.
17.
Mann
,
B. P.
, and
Patel
,
B. R.
, 2010, “
Stability of Delay Equations Written as State Space Models
,”
J. Vib. Control
,
16
(
7–8
), pp.
1067
1085
.
18.
Butcher
,
E. A.
,
Bobrenkov
,
O.
,
Bueler
,
E.
, and
Nindujarla
,
P.
, 2009, “
Analysis of Milling Stability by the Chebyshev Collocation Method: Algorithm and Optimal Stable Immersion Levels
,”
J. Comput. Nonlinear Dyn.
,
4
, pp.
1
12
.
19.
Meissner
,
E.
, 1918, “
Uber Schuttel-schwingungen in Systemen mit Periodisch Veranderlicher Elastizitat
,”
Schweizer Bauzeitung
,
72
, pp.
95
98
.
20.
Butcher
,
E. A.
, and
Mann
,
B. P.
, 2009, “
Stability Analysis and Control of Linear Periodic Delayed Systems Using Chebyshev and Temporal Finite Element Methods
,” in
Delay Differential Equations: Recent Advances and New Directions
,
B.
Balachandran
,
D.
Gilsinn
, and
T.
Kalmar-Nagy
, eds.,
Springer
,
New York
.
21.
Hochstadt
,
H.
, 1963, “
A Special Hill’s Equation With Discontinuous Coefficients
,”
Am. Math. Monthly
,
70
(
1
), pp.
18
26
.
22.
Zatarain
,
M.
,
Munoa
,
J.
,
Peigne
,
G.
, and
Insperger
,
T.
, 2006, “
Analysis of the Influence of Mill Helix Angle on Chatter Instability
,”
CIRP Ann.
,
55
, pp.
365
368
.
You do not currently have access to this content.