Stability of linear time-periodic delay-differential equations via Chebyshev polynomials

International Journal for Numerical Methods in Engineering

Volumn 59, Issue 7, 2004, Pages 895-922

  Butcher, Eric A ^a   Ma, Haitao ^a   Bueler, Ed ^b   Averina, Victoria ^b   Szabo, Zsolt ^c  

^a University of Alaska Fairbanks   (United States)

^b UNIVERSITY OF ALASKA FAIRBANKS   (United States)

^c BUDAPEST UNIVERSITY OF TECHNOLOGY AND ECONOMICS   (Hungary)

Author keywords

Chebyshev polynomials; Delay differential equations (DDEs); Stability properties

Indexed keywords

CHEBYSHEV APPROXIMATION; CONVOLUTION; DIFFERENCE EQUATIONS; DIFFERENTIAL EQUATIONS; POLYNOMIALS; TIME VARYING SYSTEMS;

PARAMETRIC EXCITATION; PERIODIC DELAY;

SYSTEM STABILITY;

MATHEMATICAL METHOD;

EID: 1442313921     PISSN: 00295981     EISSN: None     Source Type: Journal    
DOI: 10.1002/nme.894     Document Type: Article

References (31)

1. Niculescu SI. Delay Effects on Stability-A Robust Control Approach. Springer: London, 2001.
2. Stepan G. Retarded Dynamical Systems. Longman: Harlow, UK, 1989.
3. Bayly PV, Halley JE, Mann BP, Davies MA. Stability of interrupted cutting by temporal finite element analysis. Proceedings of ASME DETC 2001, 9-13 September 2001, Pittsburgh, PA.
4. Budak E, Altintas Y. Analytical prediction of chatter stability in milling-Part I: General formulation. Journal of Dynamic Systems, Measurement, and Control 1998; 120:22-30.
5. Insperger T, Stepan G. Stability of the milling process. Periodica Polytechnica Series of Mechanical Engineering 2000; 44(1):47-57.
6. Insperger T, Stepan G. Stability of high-speed milling. Proceedings of 2000 ASME IMECE, Orlando, FL, 5-10 November, 2000.
7. Minis I. Yanushevsky R. A new theoretical approach for the prediction of machine tool chatter in milling. Journal of Engineering for Industry 1993; 115:1-8.
8. Opitz H, Bernardi F. Investigation and calculation of the chatter behavior of lathes and milling machines. Annals of the CIRP 1970; 18:335-343.
9. Segalman DJ, Butcher EA. Suppression of regenerative chatter via impedance modulation. Journal of Vibration and Control 2000; 6:243-256.
10. Zhao M-X, Balachandran B. Dynamics and stability of milling process. International Journal of Solids and Structures 2001; 38:2233-2248.
11. Insperger T, Stepan G. Remote control of periodic robot motion. Proceedings of the 13th CISM-IFTOMM Symposium, Zakopane, Poland, 2000.
12. Hagan MT, Demuth HB, Beale M. Neural Network Design. PWS: Boston, 1996.
13. Deshmukh V, Ma H-T, Butcher EA. Optimal control of parametrically excited linear delay differential systems via Chebyshev polynomials. Proceedings of the 2003 American Control Conference 4-6 June 2003, Denver, CO.
14. Schmitz TL, Medicus K, Dutterer B. Exploring once-per-revolution audio signal variance as a chatter indicator. Machining Science and Technology 2002; 6(2):207-225.
15. Insperger T, Stépán G. Stability chart for the delayed Mathieu equation. Proceedings of The Royal Society, Mathematical, Physical and Engineering Sciences 2002; 458(2024):1989-1998.
16. Insperger T, Stépán G. Semi-discretization method for general delayed systems. International Journal for Numerical Methods in Engineering 2002; 55:503-518.
17. Sanders JA, Verhulst F. Averaging Methods in Nonlinear Dynamical Systems. Springer: New York, 1985.
18. Hahn W. On difference differential equations with periodic coefficients. Journal of Mathematical Analysis and Application 1961; 3:70-101.
19. Hale J, Verduyn Lunel SM. Introduction to Functional Differential Equations. Springer: Berlin, 1993.
20. Fox L, Parker IB. Chebyshev Polynomials in Numerical Analysis. Oxford University Press: London, 1968.
21. Averina V. Symbolic stability of delay different al equations. M.S. Thesis, Department of Mathematical Sciences. University of Alaska Fairbanks, August 2002.
22. Sinha SC, Butcher EA. Solution and stability of a set of pth order linear differential equations with periodic coefficients via Chebyshev polynomials. Mathematical Problems in Engineering 1996; 2:165-190.
23. Bueler E, Butcher E. Stability of periodic linear delay-differential equations and the Chebyshev approximation of fundamental solutions. Technical Report, No. 02-03, Department of Mathematical Sciences, University of Alaska Fairbanks, 2002.
24. Yakubovich VA, Starzhinski VM, Linear Differential Equations with Periodic Coefficients, Part I. Wiley: New York, 1975.
25. Sinha SC, Wu DH. An efficient computational scheme for the analysis of periodic systems. Journal of Sound and Vibration 1991; 151(1):91-117.
26. Sinha SC, Butcher EA. Symbolic computation of fundamental solution matrices for linear time-periodic dynamical systems. Journal of Sound and Vibration 1997; 206(1):61-85.
27. Szabo S, Butcher EA. Stability analysis of delayed 2nd order ODEs based on the method of Chebyshev polynomials. Euro Conference on Numerical Methods and Computational Mechanics, Miskolc, Hungary, 15-19 July 2002.
28. Burden RL, Faires D. Numerical Analysis. Brooks/Cole: Pacific Grove, CA, 1997.
29. Mohler RH. Nonlinear Systems, Volume I: Dynamics and Control. Prentice-Hall: New York, 1991.
30. Hsu CS, Bhatt SJ. Stability charts for second-order dynamical systems with time lag. Journal of Applied Mechanics 1966; 33E:119-124.
31. Van de Pol F, Strutt MJO. On the stability of the solutions of Mathieu's equation. Philosophical Magazine, and Journal of Science 1928; 5:18-38.