A symplectic homotopy perturbation method for random structural dynamics response analysis based on Birkhoffian systems

Abstract

With the enhancement of the engineering structure complexity, numerical computation plays a prominent role in structural dynamics analysis. In theory, general mechanical systems can be transformed into Birkhoffian forms, whose matching symplectic numerical algorithms perform well in accuracy and stability. Besides, the influence of uncertainty cannot be ignored in practical engineering, but the research on uncertainty symplectic algorithms of Birkhoffian systems is still weak. This paper proposes a symplectic homotopy perturbation method (SHPM) for general Birkhoffian equations with random uncertainties. Different from the traditional perturbation method, the proposed method does not depend on small parameters. Based on the homotopy perturbation method, embedded parameters are introduced to establish the matrix and parameter perturbation equations of random Birkhoffian equations, respectively. Then the formulas for calculating the mean value and standard deviation of the response are derived with probability theory. Combined with Birkhoffian symplectic preserving algorithms, a symplectic homotopy perturbation method for the numerical solution of random Birkhoffian equations is proposed. Finally, numerical examples are given to verify the applicability and effectiveness of the proposed method and its superiority compared with the traditional non-symplectic method.