Normal forms and the structure of resonance sets in nonlinear time-periodic systems

Eric Butcher, S. C. Sinha

Research output: Contribution to journalArticle

11 Citations (Scopus)

Abstract

The structure of time-dependent resonances arising in the method of time-dependent normal forms (TDNF) for one and two-degrees-of-freedom nonlinear systems with time-periodic coefficients is investigated. For this purpose, the Liapunov-Floquet (L-F) transformation is employed to transform the periodic variational equations into an equivalent form in which the linear system matrix is time-invariant. Both quadratic and cubic nonlinearities are investigated and the associated normal forms are presented. Also, higher-order resonances for the single-degree-of-freedom case are discussed. It is demonstrated that resonances occur when the values of the Floquet multipliers result in MT-periodic (M = 1, 2, ...) solutions. The discussion is limited to the Hamiltonian case (which encompasses all possible resonances for one-degree-of-freedom). Furthermore, it is also shown how a recent symbolic algorithm for computing stability and bifurcation boundaries for time-periodic systems may also be employed to compute the time-dependent resonance sets of zero measure in the parameter space. Unlike classical asymptotic techniques, this method is free from any small parameter restriction on the time-periodic term in the computation of the resonance sets. Two illustrative examples (one and two-degrees-of-freedom) are included.

Original languageEnglish (US)
Pages (from-to)35-55
Number of pages21
JournalNonlinear Dynamics
Volume23
Issue number1
DOIs
StatePublished - Sep 2000
Externally publishedYes

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Time varying systems
Periodic Systems
Normal Form
Degree of freedom
Floquet multipliers
Hamiltonians
Variational Equation
Bifurcation (mathematics)
Periodic Coefficients
Small Parameter
Linear systems
Parameter Space
Nonlinear systems
Bifurcation
Nonlinear Systems
Linear Systems
Nonlinearity
Transform
Higher Order
Restriction

ASJC Scopus subject areas

  • Mechanical Engineering
  • Mechanics of Materials
  • Computational Mechanics

Cite this

Normal forms and the structure of resonance sets in nonlinear time-periodic systems. / Butcher, Eric; Sinha, S. C.

In: Nonlinear Dynamics, Vol. 23, No. 1, 09.2000, p. 35-55.

Research output: Contribution to journalArticle

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