On the analysis of time-periodic nonlinear Hamiltonian dynamical systems

Eric A. Butcher, S. C. Sinha

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Abstract

In this paper, some analysis techniques for general time-periodic nonlinear Hamiltonian dynamical systems have been presented. Unlike the traditional perturbation or averaging methods, these techniques are applicable to systems whose Hamiltonians contain ‘strong’ parametric excitation terms. First, the well-known Liapunov-Floquet (L-F) transformation is utilized to convert the time-periodic dynamical system to a form in which the linear part is time invariant. At this stage two viable alternatives are suggested. In the first approach, the resulting dynamical system is transformed to a Hamiltonian normal form through an application of permutation matrices. It is demonstrated that this approach is simple and straightforward as opposed to the traditional methods where a complicated set of algebraic manipulations are required. Since these operations yield Hamiltonians whose quadratic parts are integrable and timeinvariant, furdier analysis can be carried out by the application of action-angle coordinate transformation and Hamiltonian perturbation theory. In the second approach, the resulting quasilinear time-periodic system (with a time-invariant linear part) is directly analyzed via time-dependent normal form theory. In many instances, the system can be analyzed via time-independent normal form theory or by the method of averaging. Examples of a nonlinear Mathieu’s equation and coupled nonlinear Mathieu’s equations are included and some preliminary results are presented.

Original languageEnglish (US)
Title of host publication15th Biennial Conference on Mechanical Vibration and Noise - Vibration of Nonlinear, Random, and Time-Varying Systems
PublisherAmerican Society of Mechanical Engineers (ASME)
Pages375-386
Number of pages12
ISBN (Electronic)9780791817186
DOIs
StatePublished - 1995
Externally publishedYes
EventASME 1995 Design Engineering Technical Conferences, DETC 1995, collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium - Boston, United States
Duration: Sep 17 1995Sep 20 1995

Publication series

NameProceedings of the ASME Design Engineering Technical Conference
Volume3A-1995

Conference

ConferenceASME 1995 Design Engineering Technical Conferences, DETC 1995, collocated with the ASME 1995 15th International Computers in Engineering Conference and the ASME 1995 9th Annual Engineering Database Symposium
CountryUnited States
CityBoston
Period9/17/959/20/95

ASJC Scopus subject areas

  • Mechanical Engineering
  • Computer Graphics and Computer-Aided Design
  • Computer Science Applications
  • Modeling and Simulation

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