A symplectic algorithm for the stability analysis of nonlinear parametric excited systems

Z. G. Ying, Yiqing Ni, Z. H. Chen

Research output: Journal article publicationJournal articleAcademic researchpeer-review

3 Citations (Scopus)

Abstract

A symplectic numerical approach to the stability analysis of nonlinear parametric excited systems with multi-degree-of-freedom is developed and the stability of a nonlinear vibration system depends on the dynamic behavior of its perturbation. The nonlinear parameter-varying differential equations for the perturbation motion are expressed in the form of Hamiltonian equations with time-varying Hamiltonian, and are converted further into the classical Hamiltonian equations with extended time-invariant Hamiltonian by augmenting the state variables. The solution to the augmented Hamiltonian equations has the symplectic structure in terms of the symplectic transformation. Then the difference equations of the symplectic Runge-Kutta algorithm with a sufficient condition are constructed, which is proved to preserve the intrinsic symplectic structure of the original solution. In particular, the symplectic Gauss-Runge-Kutta algorithm with stage 2 and order 4 is proposed and applied to the stability analysis of a nonlinear system. Unstable regions based on the nonlinear periodic-parameter perturbation equation are obtained by using the symplectic Gauss-Runge-Kutta algorithm, analytical solution method and non-symplectic conventional Runge-Kutta algorithm to verify the higher accuracy of the proposed algorithm. Unstable regions based on the nonlinear perturbation are given to illustrate the improvement over those based on the linear perturbation. The developed symplectic approach to the stability analysis can preserve the symplectic structure of the original system and is applicable to nonlinear parametric excited systems with multi-degree-of-freedom.
Original languageEnglish
Pages (from-to)561-584
Number of pages24
JournalInternational Journal of Structural Stability and Dynamics
Volume9
Issue number3
DOIs
Publication statusPublished - 1 Sep 2009

Keywords

  • Augmented Hamiltonian equations
  • Gauss-Runge-Kutta algorithm
  • Nonlinear parametric excited system
  • Stability
  • Symplectic solution

ASJC Scopus subject areas

  • Civil and Structural Engineering
  • Building and Construction
  • Aerospace Engineering
  • Ocean Engineering
  • Mechanical Engineering
  • Applied Mathematics

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