线性Birkhoff方程的随机与区间不确定性保辛算法及其对比研究  

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

作　　者：邱志平[1] 祝博 QIU ZhiPing;ZHU Bo(School of Aeronautic Science and Engineering,Beihang University,Beijing 100191,China)

机构地区：[1]北京航空航天大学航空科学与工程学院,北京100191

出　　处：《中国科学：物理学、力学、天文学》2020年第8期104-120,共17页Scientia Sinica Physica,Mechanica & Astronomica

基　　金：国家自然科学基金(编号:11432002);国防基础科研计划(编号:JCKY2018601B001)资助项目。

摘　　要：随着工程结构复杂度的提高,数值计算在结构分析中的地位日渐凸显.Birkhoff系统的保辛算法在精度与稳定性方面优于传统算法,并且一般的力学系统理论上都可转化为Birkhoff系统.不确定性的影响在实际工程中不可忽略,然而考虑不确定性的Birkhoff系统保辛算法的研究却尚显薄弱.本文考虑参数不确定性,研究了线性Birkhoff方程的随机与区间不确定性保辛算法以及两种方法计算结果之间的相容性.基于参数摄动理论,建立了线性Birkhoff方程的不确定性摄动方程,推导了线性Birkhoff系统不确定性响应数字特征的计算式,结合Birkhoff保辛算法,提出了线性Birkhoff系统不确定性响应数值计算的随机与区间保辛摄动方法.基于Chebyshev不等式定义了不确定性响应的概率边界,进而证明了所提随机与区间方法数值结果之间的相容性.最后通过算例验证了所提方法在不确定性动力学响应计算方面的有效性以及相比于传统数值方法的优越性.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 b

关 键 词：线性Birkhoff系统 动力学响应 辛算法 不确定性 参数摄动方法 

分 类 号：O411.1[理学—理论物理]