In this paper, a new efficient method is proposed to obtain the transient response of linear or piecewise linear dynamic systems with time delay and periodic coefficients under arbitrary control excitations via Chebyshev polynomial expansion. Since the time domain can be divided into intervals with length equal to the delay period, at each such interval the fundamental solution matrix for the corresponding periodic ordinary differential equation (without delay) is constructed in terms of shifted Chebyshev polynomials by using a previous technique that reduces the problem to a set of linear algebraic equations. By employing a convolution integral formula, the solution for each interval can be directly obtained in terms of the fundamental solution matrix. In addition, by combining the properties of the periodic system and Floquet theory, the computational processes are simplified and become very efficient. An alternate version, which does not employ Floquet theory, is also presented. Several examples of time-periodic delay systems, when the excitation period is equal to or larger than the delay period and for linear and piecewise linear systems, are studied. The numerical results obtained via this method are compared with those obtained from Matlab DDE23 software (Shampine, L. F., and Thompson, S., 2001, “Solving DDEs in MATLAB,” Appl. Numer. Math., 37(4), pp. 441–458.) An error bound analysis is also included. It is found that this method efficiently provides accurate results that find general application in areas such as machine tool vibrations and parametric control of robotic systems.

1.
Watanabe
,
D. S.
, and
Roth
,
M. G.
,
1985
, “
The Stability of Difference Formulas for Delay Differential Equations
,”
SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.
,
22
(
1
), pp.
132
145
.
2.
Hale, J., and Verduyn Lunel, S. M., 1993, Introduction to Functional Differential Equations, Springer, New York.
3.
Stepan, G., 1989, Retarded Dynamical Systems, Longman, Harlow, UK.
4.
Averina, V., 2002, “Symbolic Stability of Delay Differential Equations,” M.S. thesis, Dept. of Mathematical Sciences, Univ. of Alaska Fairbanks.
5.
Insperger
,
T.
, and
Stepan
,
G.
,
2000
, “
Stability of the Milling Process
,”
Periodica Polytechnica Ser. Mech. Eng.
,
44
(
1
), pp.
47
57
.
6.
Insperger, T., and Stepan, G., 2001, “Remote Control of Periodic Robot Motion,” Proc. of 13th CISM-IFToMM Symp. on Theory and Practice of Robots and Manipulators (Zakopane, 2000), A. Morecki, G. Bianchi, and C. Rzymkowsky, eds., Springer, Vienna. pp. 197–204.
7.
Horng
,
I.-R.
, and
Chou
,
J.-H.
,
1985
, “
Analysis, Parameter Estimation and Optimal Control of Time-Delay Systems via Chebyshev Series
,”
Int. J. Control
,
41
(
5
), pp.
1221
1234
.
8.
Xu
,
X.
, and
Agrawal
,
S. K.
,
1999
, “
Linear Time-Varying Dynamic Systems Optimization via Higher-Order Method Using Shifted Chebyshev’s Polynomials
,”
J. Vibr. Acoust.
,
121
, pp.
258
261
.
9.
Chung
,
H.-Y.
, and
Sun
,
Y.-Y.
,
1987
, “
Analysis of Time-Delay Systems Using an Alternative Technique
,”
Int. J. Control
,
46
(
5
), pp.
1621
1631
.
10.
Lehman, B., and Weibel, S., 1998, “Moving Averages for Periodic Delay Differential and Difference Equations,” Stability and Control of Time-Delay Systems, L. Dugard and E. I. Verriest, eds., Springer, London.
11.
Opitz
,
H.
, and
Bernardi
,
F.
,
1970
, “
Investigation and Calculation of the Chatter Behavior of Lathes and Milling Machines
,”
Annals of CIRP
,
18
, pp.
335
343
.
12.
Sankin
,
Y. N.
, 1984, “The Stability of Milling Machines During Cutting,” Soviet Engineering Research, 4(4), pp. 40–43.
13.
Budak
,
E.
, and
Altintas
,
Y.
,
1998
, “
Analytical Prediction of Chatter Stability in Milling, Part I: General Formulation
,”
ASME J. Dyn. Syst., Meas., Control
,
120
(
1
), pp.
22
30
.
14.
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
(
1
), pp.
31
36
.
15.
Segalman
,
D. J.
, and
Butcher
,
E. A.
,
2000
, “
Suppression of Regenerative Chatter via Impedance Modulation
,”
J. Vib. Control
,
6
, pp.
243
256
.
16.
Sinha
,
S. C.
, and
Wu
,
D. H.
,
1991
, “
An Efficient Computational Scheme for the Analysis of Periodic Systems
,”
J. Sound Vib.
,
151
(
1
), pp.
91
117
.
17.
Shampine
,
L. F.
, and
Thompson
,
S.
, (
2001
), “
Solving DDEs in Matlab
,”
Appl. Numer. Math.
,
37
(
4
), pp.
441
458
.
18.
Yakubovich, V. A., and Starzhinski, V. M., 1975, Linear Differential Equations with Periodic Coefficients, Parts I and II, Wiley, N.Y.
19.
Brogan, W. L., 1991, Modern Control Theory, 3rd Edition, Prentice-Hall.
20.
Fox, L., and Parker, I. B., 1968, Chebyshev Polynomials in Numerical Analysis, Oxford Univ. Press, London.
21.
Sinha
,
S. C.
, and
Butcher
,
E. A.
,
1997
, “
Symbolic Computation of Fundamental Solution Matrices for Linear Time-Periodic Dynamical Systems
,”
J. Sound Vib.
,
206
(
1
), pp.
61
85
.
22.
Davies
,
M. A.
, and
Balachandran
,
B.
,
2000
, “
Impact Dynamics in Milling of Thin-Walled Structures
,”
Nonlinear Dyn.
,
22
, pp.
375
392
.
23.
Chou
,
J.-H.
, and
Horng
,
I.-R.
,
1985
, “
Shifted Legendre Series Analysis of Linear Optimal Control Systems Incorporating Observers
,”
Int. J. Syst. Sci.
,
16
, pp.
863
867
.
24.
Natsiavas
,
S.
,
,
S.
, and
Goudas
,
I.
,
2000
, “
Dynamic Analysis of Piecewise Linear Oscillators With Time Periodic Coefficients
,”
Int. J. Non-Linear Mech.
,
35
, pp.
53
68
.
You do not currently have access to this content.