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.
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June 2003
Technical Papers
Chebyshev Expansion of Linear and Piecewise Linear Dynamic Systems With Time Delay and Periodic Coefficients Under Control Excitations
Haitao Ma,
e-mail: fthm@uaf.edu
Haitao Ma
Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775
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Eric A. Butcher, Assoc. Mem. ASME,
e-mail: ffeab@uaf.edu
Eric A. Butcher, Assoc. Mem. ASME
Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775
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Ed Bueler
Ed Bueler
Department of Mathematical Sciences, University of Alaska Fairbanks, Fairbanks, AK 99775
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Haitao Ma
Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775
e-mail: fthm@uaf.edu
Eric A. Butcher, Assoc. Mem. ASME
Department of Mechanical Engineering, University of Alaska Fairbanks, Fairbanks, AK 99775
e-mail: ffeab@uaf.edu
Ed Bueler
Department of Mathematical Sciences, University of Alaska Fairbanks, Fairbanks, AK 99775
Contributed by the Dynamic Systems and Control Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received by the ASME Dynamic Systems and Control Division, June 2002; final revision, January 2003. Associate Editor: N. Olgac.
J. Dyn. Sys., Meas., Control. Jun 2003, 125(2): 236-243 (8 pages)
Published Online: June 4, 2003
Article history
Received:
June 1, 2002
Revised:
January 1, 2003
Online:
June 4, 2003
Citation
Ma, H., Butcher, E. A., and Bueler, E. (June 4, 2003). "Chebyshev Expansion of Linear and Piecewise Linear Dynamic Systems With Time Delay and Periodic Coefficients Under Control Excitations ." ASME. J. Dyn. Sys., Meas., Control. June 2003; 125(2): 236–243. https://doi.org/10.1115/1.1570449
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