This paper investigates two different temporal finite element techniques, a multiple element (h-version) and single element (p-version) method, to analyze the stability of a system with a time-periodic coefficient and a time delay. The representative problem, known as the delayed damped Mathieu equation, is chosen to illustrate the combined effect of a time delay and parametric excitation on stability. A discrete linear map is obtained by approximating the exact solution with a series expansion of orthogonal polynomials constrained at intermittent nodes. Characteristic multipliers of the map are used to determine the unstable parameter domains. Additionally, the described analysis provides a new approach to extract the Floquet transition matrix of time periodic systems without a delay.

1.
Insperger
,
T.
,
Mann
,
B. P.
,
Stépán
,
G.
, and
Bayly
,
P. V.
, 2003, “
Stability of Up-Milling and Down-Milling, Part 1. Alternative Analytal Methods
,”
Int. J. Mach. Tools Manuf.
0890-6955,
43
, pp.
25
34
.
2.
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.
0890-6955,
43
, pp.
35
40
.
3.
Altintas
,
Y.
, and
Budak
,
E.
, 1995, “
Analytical Prediction of Stability Lobes in Milling
,”
CIRP Ann.
0007-8506,
44
(
1
), pp.
357
362
.
4.
Mann
,
B. P.
,
Garg
,
N. K.
,
Young
,
K. A.
, and
Helvey
,
A. M.
, 2005, “
Milling Bifurcations From Structural Asymmetry and Nonlinear Regeneration
,”
Nonlinear Dyn.
0924-090X,
42
(
4
), Dec, pp.
319
337
.
5.
Balachandran
,
B.
, 2001, “
Nonlinear Dynamics of Milling Process
,”
Proc. R. Soc. London, Ser. A
1364-5021,
359
, pp.
793
819
.
6.
Yang
,
B.
, and
Wu
,
X.
, 1998, “
Modal Expansion of Structural Systems With Time Delays
,”
AIAA J.
0001-1452,
36
(
12
), pp.
2218
2224
.
7.
Yakubovitch
,
V. A.
, and
Starzhinskii
,
V. M.
, 1975,
Linear Differential Equations With Periodic Coefficients
,
Wiley
, New York.
8.
Lindh
,
K. G.
, and
Likins
,
P. W.
, 1970, “
Infinite Determinant Methods for Stability Analysis of Periodic-Coefficient Differential Equations
,”
AIAA J.
0001-1452,
8
, pp.
680
686
.
9.
Brockett
,
R. W.
, 1970,
Finite Dimensional Linear Systems
,
John Wiley
, New York.
10.
Peters
,
D. A.
, and
Hohenemser
,
K. H.
, 1971, “
Application of Floquet Transition Matrix to Problems of Lifting Rotor Stability
,”
J. Am. Helicopter Soc.
0002-8711,
16
, pp.
25
33
.
11.
Hsu
,
C. S.
, and
Cheng
,
W. H.
, 1973, “
Application of the Theory of Impulsive Parametric Excitation and New Treatment of General Parametric Excitation Problems
,”
ASME J. Appl. Mech.
0021-8936,
40
, pp.
78
86
.
12.
Hsu
,
C. S.
, 1974, “
On Approximating a General Linear Periodic System
,”
J. Math. Anal. Appl.
0022-247X,
45
, pp.
234
251
.
13.
Sinha
,
S. C.
,
Chou
,
C. C.
, and
Denman
,
H. H.
, 1979, “
Stability Analysis of Systems With Periodic Coefficients: An Approximate Approach
,”
J. Sound Vib.
0022-460X,
64
, pp.
515
527
.
14.
Friedmann
,
P.
,
Hammond
,
C. C.
, and
Woo
,
T. H.
, 1977, “
Efficient Numerical Treatment of Periodic Systems With Applications to Stability Problems
,”
Int. J. Numer. Methods Eng.
0029-5981,
11
, pp.
1117
1136
.
15.
Gockel
,
M. A.
, 1972, “
Practical Solution of Linear Equations With Periodic Coefficients
,”
J. Am. Helicopter Soc.
0002-8711,
17
, pp.
2
10
.
16.
Gaonkar
,
G. H.
,
Prasad
,
D. S. S.
, and
Sastry
,
D.
, 1981, “
On Computing Floquet Transition Matrices of Rotocraft
,”
J. Am. Helicopter Soc.
0002-8711,
26
, pp.
56
61
.
17.
Nayfeh
,
A. H.
, 1973,
Perturbation Methods
,
Wiley
, New York.
18.
Jordan
,
D. W.
, and
Smith
,
P.
, 1977,
Nonlinear Ordinary Differential Equations
,
Clarendon Press
, Oxford.
19.
Sinha
,
S. C.
, and
Wu
,
D. H.
, 1991, “
An Efficient Computational Scheme for Analysis of Periodic Systems
,”
J. Sound Vib.
0022-460X,
151
(
1
), pp.
91
117
.
20.
Sinha
,
S. C.
, 1997, “
On the Analysis of Time-Periodic Nonlinear Dynamical Systems
,”
Sadhana: Proc., Indian Acad. Sci.
0256-2499,
22
(
3
), pp.
411
434
.
21.
Sinha
,
S. C.
,
Pandiyan
,
R.
, and
Bibb
,
J. S.
, 1996, “
Liapunov-Floquet Transformation: Computation and Applications to Periodic Systems
,”
ASME J. Vibr. Acoust.
0739-3717,
118
, pp.
209
219
.
22.
Sinha
,
S. C.
,
Wu
,
D. H.
,
Juneja
,
V.
, and
Joseph
,
P.
, 1991, “
An Approximate Analytical Solution for Systems With Periodic Coefficients via Symbolic Computation
,” AIAA/ASME/ASCE/AHS/ASC 32nd Structures, Structural Dynamics and Materials Conference, April, pp.
790
797
.
23.
Joseph
,
P.
,
Pandiyan
,
R.
, and
Sinha
,
S. C.
, 1993, “
Optimal Control of Mechanical Systems Subjected to Periodic Loading via Chebyshev Polynomials
,”
Opt. Control Appl. Methods
0143-2087,
14
, pp.
75
90
.
24.
Butcher
,
E. L.
,
Ma
,
H.
,
Bueler
,
E.
,
Averina
,
V.
, and
Szabo
,
Z.
, 2004, “
Stability of Linear Time-Periodic Delay-Differential Equations via Chebyshev Polynomials
,”
Int. J. Numer. Methods Eng.
0029-5981,
59
(
7
), pp.
895
922
.
25.
Ma
,
H.
,
Deshmukh
,
V.
,
Butcher
,
E. A.
, and
Averina
,
V.
, 2005, “
Delayed State Feedback and Chaos Control for Time-Periodic Systems via a Symbolic Approach
,”
Commun. Nonlinear Sci. Numer. Simul.
1007-5704,
10
, pp.
467
580
.
26.
Ma
,
H.
,
Butcher
,
E. A.
, and
Bueler
,
E.
, 2003, “
Chebyshev Expansion of Linear and Peicewise Linear Dynamic Systems With Time Delay and Periodic Coefficients Under Control Excitations
,”
ASME J. Dyn. Syst., Meas., Control
0022-0434,
125
, pp.
236
243
.
27.
Bueler
,
E.
,
Averina
,
V.
, and
Butcher
,
E. A.
, 2004, “
Periodic Linear DDEs: Collocation Approximation to the Monodromy Operator
,” SIAM Annual Meeting, Portland, Oregan.
28.
Szabo
,
S.
, and
Butcher
,
E. A.
, 2002, “
Stability Analysis of Delayed 2nd Order Odes Based on the Method of Chebyshev Polynomials
,” European Conference on Numerical Methods of Computational Mechanics, July 15–19, Miskolc, Hungary.
29.
Horng
,
I. R.
, and
Chou
,
J. H.
, 1985, “
Analysis Parameter Estimation and Optimal Control of Time-Delay Systems via Chebyshev Series
,”
Int. J. Control
0020-7179,
41
, pp.
1221
1234
.
30.
Chung
,
H. Y.
, and
Sun
,
Y. Y.
, 1987, “
Analysis of Time-Delay Systems Using an Alternative Technique
,”
Int. J. Control
0020-7179,
46
, pp.
1621
1631
.
31.
Mathieu
,
E.
, 1868, “
Memoire sur le Mouvement Vibratorie d’une Membrane de Forme Elliptique
,”
J. Math
,
13
, pp.
137
203
.
32.
Bellman
,
R.
, and
Cooke
,
K.
, 1963,
Differential-Difference Equations
,
Academic Press
, New York.
33.
Bhatt
,
S. J.
, and
Hsu
,
C. S.
, 1966, “
Stability Criteria for Second-Order Dynamical Systems With Time Lag
,”
Appl. Math. (Germany)
0862-7940,
33
, pp.
113
118
.
34.
Niemark
,
J.
, 1949, “
D-Subdivision and Spaces of Quasi-Polynomials
,”
Prikl. Mat. Mekh.
0032-8235,
13
, pp.
349
380
(in Russian).
35.
Andreev
,
A. F.
, 1958, “
Twelve Papers on Function Theory, Probability and Differential Equations
,” American Mathematical Society, Series 2, Volume 8.
36.
Insperger
,
T.
, and
Stépán
,
G.
, 2002, “
Stability Chart of the Delayed Mathieu Equation
,”
Proc. R. Soc. London, Ser. A
1364-5021,
458
, pp.
1989
1998
.
37.
Insperger
,
T.
, and
Stépán
,
G.
, 2001, “
Semi-Discretization of Delayed Dynamical Systems
,”
Proc. of ASME 2001 Design Engineering Technical Conferences and Computers and Information in Engineering Conference
, Pittsburgh, ASME, New York.
38.
Insperger
,
T.
, and
Stépán
,
G.
, 2002, “
Semi-Discretization Method for Delayed Systems
,”
Int. J. Numer. Methods Eng.
0029-5981,
55
(
5
), pp.
503
518
.
39.
Chang
,
P. Y.
,
Yang
,
S. Y.
, and
Wang
,
M. L.
, 1986, “
Solution of Linear Dynamic Systems by Generalized Orthogonal Polynomials
,”
Int. J. Syst. Sci.
0020-7721,
17
, pp.
1727
1740
.
40.
Sinha
,
S. C.
, and
Chou
,
C. C.
, 1976, “
An Approximate Analysis of Transient Response of Time Dependent Linear Systems by Orthogonal Polynomials
,”
J. Sound Vib.
0022-460X,
49
, pp.
309
326
.
41.
Lindlbauer
,
M.
, 1998, “
On the Rate of Convergence of the Laws of Markov Chains Associated With Orthogonal Polynomials
,”
J. Comput. Appl. Math.
0377-0427,
99
, pp.
287
297
.
42.
Halley
,
J. E.
, 1999, “
Stability of Low Radial Immersion Milling
,” Master’s thesis, Washigton University, Saint Louis.
43.
Bayly
,
P. V.
,
Halley
,
J. E.
,
Mann
,
B. P.
, and
Davis
,
M. A.
, 2003, “
Stability of Interrupted Cutting by Temporal Finite Element Analysis
,”
ASME J. Manuf. Sci. Eng.
1087-1357,
125
, pp.
220
225
.
44.
Mann
,
B. P.
, 2003, “
Dynamic Models of Milling and Broaching
,” Ph.D. dissertation, Washington University, Saint Louis, May.
45.
Mann
,
B. P.
,
Bayly
,
P. V.
,
Davies
,
M. A.
, and
Halley
,
J. E.
, 2004, “
Limit Cycles, Bifurcations, and the Accuracy of the Milling Process
,”
J. Sound Vib.
0022-460X,
277
, pp.
31
48
.
46.
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
, pp.
446
453
.
47.
Nayfeh
,
A. H.
, and
Balachandran
,
B.
, 1995,
Applied Non-linear Dynamics Analytical, Computational and Experimental Methods
,
Wiley
, New York, Wiley Series on Nonlinear Science.
48.
Hale
,
J. K.
, and
Lunel
,
S. V.
, 1993,
Introduction to Functional Differential Equations
,
Springer-Verlag
, Berlin.
49.
Hassard
,
B. D.
, 1997, “
Counting Roots of the Characteristic Equation for Linear Delay-Differential Systems
,”
J. Differ. Equations
0022-0396,
136
, pp.
222
235
.
50.
Mann
,
B. P.
,
Garg
,
N. K.
,
Young
,
K. A.
, and
Helvey
,
A. M.
, 2005, “
Milling Bifurcations From Structural Asymmetry and Nonlinear Regeneration
,”
Nonlinear Dyn.
0924-090X,
42
(
4
), pp.
319
337
.
51.
Zienkiewicz
,
O. C.
, and
Taylor
,
R. L.
, 2000,
The Finite Element Method
,
5th ed.
,
Butterworth Heinemann
, Oxford, Vol.
1
.
52.
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
, pp.
117
141
.
53.
Burnett
,
D. S.
, 1988,
Finite Element Analysis (From Concepts to Applications)
,
5th ed.
,
Addison-Wesley
, Reading, MA.
54.
Zienkiewicz
,
O. C.
,
Zhu
,
J. Z.
, and
Gong
,
N. G.
, 1989, “
Effective and Practical h-p Version Adaptive Analysis Procedures for the Finite Element Method
,”
Int. J. Numer. Methods Eng.
0029-5981,
28
, pp.
879
891
.
You do not currently have access to this content.