Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/20456
Title: A symplectic algorithm for the stability analysis of nonlinear parametric excited systems
Authors: Ying, ZG
Ni, YQ 
Chen, ZH
Keywords: Augmented Hamiltonian equations
Gauss-Runge-Kutta algorithm
Nonlinear parametric excited system
Stability
Symplectic solution
Issue Date: 2009
Publisher: World Scientific
Source: International journal of structural stability and dynamics, 2009, v. 9, no. 3, p. 561-584 How to cite?
Journal: International journal of structural stability and dynamics 
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.
URI: http://hdl.handle.net/10397/20456
ISSN: 0219-4554
EISSN: 1793-6764
DOI: 10.1142/S0219455409003156
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