Volume 13, Issue 6
A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications

Adv. Appl. Math. Mech., 13 (2021), pp. 1535-1557.

Published online: 2021-08

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• Abstract

In this paper, a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations, which mainly originate from parameters dispersions and measurement errors. Taking the perturbations into account, the perturbed system is regarded as a modification of the nominal system. By combining the perturbation series expansion method with the deterministic linear Hamiltonian system, the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived. Eventually, the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes. The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system. The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm. Numerical examples illustrate the superiority of the proposed method in accuracy and stability, especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.

74H15, 37N15, 65P10, 65M06

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@Article{AAMM-13-1535, author = {Qiu , Zhiping and Jiang , Nan}, title = {A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2021}, volume = {13}, number = {6}, pages = {1535--1557}, abstract = {

In this paper, a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations, which mainly originate from parameters dispersions and measurement errors. Taking the perturbations into account, the perturbed system is regarded as a modification of the nominal system. By combining the perturbation series expansion method with the deterministic linear Hamiltonian system, the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived. Eventually, the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes. The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system. The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm. Numerical examples illustrate the superiority of the proposed method in accuracy and stability, especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2020-0282}, url = {http://global-sci.org/intro/article_detail/aamm/19434.html} }
TY - JOUR T1 - A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications AU - Qiu , Zhiping AU - Jiang , Nan JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1535 EP - 1557 PY - 2021 DA - 2021/08 SN - 13 DO - http://doi.org/10.4208/aamm.OA-2020-0282 UR - https://global-sci.org/intro/article_detail/aamm/19434.html KW - Linear Hamiltonian system, perturbation series expansion method, symplectic structure, symplectic algorithm, structural dynamic response. AB -

In this paper, a novel symplectic conservative perturbation series expansion method is proposed to investigate the dynamic response of linear Hamiltonian systems accounting for perturbations, which mainly originate from parameters dispersions and measurement errors. Taking the perturbations into account, the perturbed system is regarded as a modification of the nominal system. By combining the perturbation series expansion method with the deterministic linear Hamiltonian system, the solution to the perturbed system is expressed in the form of asymptotic series by introducing a small parameter and a series of Hamiltonian canonical equations to predict the dynamic response are derived. Eventually, the response of the perturbed system can be obtained successfully by solving these Hamiltonian canonical equations using the symplectic difference schemes. The symplectic conservation of the proposed method is demonstrated mathematically indicating that the proposed method can preserve the characteristic property of the system. The performance of the proposed method is evaluated by three examples compared with the Runge-Kutta algorithm. Numerical examples illustrate the superiority of the proposed method in accuracy and stability, especially symplectic conservation for solving linear Hamiltonian systems with perturbations and the applicability in structural dynamic response estimation.

Zhiping Qiu & Nan Jiang. (1970). A Symplectic Conservative Perturbation Series Expansion Method for Linear Hamiltonian Systems with Perturbations and Its Applications. Advances in Applied Mathematics and Mechanics. 13 (6). 1535-1557. doi:10.4208/aamm.OA-2020-0282
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