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Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients

Publication ,  Journal Article
Bobrenkov, OA; Butcher, EA; Mann, BP
Published in: JVC/Journal of Vibration and Control
March 1, 2013

In this paper, the Liapunov-Floquet transformation (LFT) is applied to a time-periodic delay differential equation (DDE) discretized by the Chebyshev spectral continuous time approximation. The proposed combined approach allows for the stability and time-response analysis of a constant non-delayed analog of the original periodic DDE by applying the LFT to an equivalent large-order system of time-periodic ordinary differential equations. The implementation issues are analyzed for the time-delayed Mathieu's equation which is used as an example. It is shown that an order reduction procedure in which only the dominant modes of the infinite-dimensional DDE are retained in the LFT is necessary. The application of the proposed technique is studied in the presence of delay and parametric resonances for the delayed Mathieu's equation, as well as for a double inverted pendulum subjected to a time-periodic retarded follower force. © 2012 The Author(s).

Duke Scholars

Published In

JVC/Journal of Vibration and Control

DOI

EISSN

1741-2986

ISSN

1077-5463

Publication Date

March 1, 2013

Volume

19

Issue

4

Start / End Page

521 / 537

Related Subject Headings

  • Acoustics
  • 4017 Mechanical engineering
  • 4005 Civil engineering
  • 0913 Mechanical Engineering
  • 0906 Electrical and Electronic Engineering
 

Citation

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Bobrenkov, O. A., Butcher, E. A., & Mann, B. P. (2013). Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients. JVC/Journal of Vibration and Control, 19(4), 521–537. https://doi.org/10.1177/1077546311433914
Bobrenkov, O. A., E. A. Butcher, and B. P. Mann. “Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients.” JVC/Journal of Vibration and Control 19, no. 4 (March 1, 2013): 521–37. https://doi.org/10.1177/1077546311433914.
Bobrenkov OA, Butcher EA, Mann BP. Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients. JVC/Journal of Vibration and Control. 2013 Mar 1;19(4):521–37.
Bobrenkov, O. A., et al. “Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients.” JVC/Journal of Vibration and Control, vol. 19, no. 4, Mar. 2013, pp. 521–37. Scopus, doi:10.1177/1077546311433914.
Bobrenkov OA, Butcher EA, Mann BP. Application of the Liapunov-Floquet transformation to differential equations with time delay and periodic coefficients. JVC/Journal of Vibration and Control. 2013 Mar 1;19(4):521–537.
Journal cover image

Published In

JVC/Journal of Vibration and Control

DOI

EISSN

1741-2986

ISSN

1077-5463

Publication Date

March 1, 2013

Volume

19

Issue

4

Start / End Page

521 / 537

Related Subject Headings

  • Acoustics
  • 4017 Mechanical engineering
  • 4005 Civil engineering
  • 0913 Mechanical Engineering
  • 0906 Electrical and Electronic Engineering