Random and interval uncertain symplectic methods for linear Birkhoff equations and their comparison study

Abstract

With the increasing complexity and design requirement of engineering structures, the numerical computation plays an increasingly important role in the structural analysis. The symplectic algorithm for Birkhoffian systems has advantages over the traditional algorithm in accuracy and stability, and general mechanical systems can be transformed into Birkhoffian systems in theory. Additionally, the influence of uncertainties in practical engineering cannot be ignored. However, the research foundation of symplectic algorithms for Birkhoffian systems considering uncertainties is still weak. In this paper, the random and interval uncertain symplectic algorithms for linear Birkhoffian systems are proposed, respectively, which can be applied to the numerical calculation of the structural dynamic response with uncertainties. Furthermore, the compatibility between the two methods is studied. Based on the small parameter perturbation theory, the uncertain perturbation equation of the uncertain linear Birkhoffian system is established. According to different types of uncertainties, based on the probability theory and interval mathematical method, expressions of the mean and variance of random Birkhoffian systems, as well as the median value and radius of interval Birkhoffian systems are derived. By combining aforementioned methods with Birkhoffian symplectic algorithms, the random and interval perturbation methods for calculating the uncertain response of linear Birkhoffian systems are proposed, respectively. Since the uncertainty exists objectively, the uncertain features of the response should not be incompatible when characterization and analysis methods for uncertainties are different. Based on the Chebyshev inequality, this paper deduces the probabilistic and interval boundaries from the result calculated by proposed methods and the comparison between the two kinds of boundaries. It is mathematically proved that the probabilistic upper bound is less than the interval upper bound and the probabilistic lower bound is larger than the interval lower bound. Finally, examples are given to demonstrate the effectiveness of the proposed method in calculating the response of uncertain dynamics and its superiority over the traditional numerical methods.