结构随机振动时域响应统计特征分析的多项式维数分解法  

A polynomial dimensional decomposition method for analyzing statistical characteristics of structural random vibration responses in the time-domain

作　　者：刘凡 肖进 韩波 赵岩[2,4] LIU Fan;XIAO Jin;HAN Bo;ZHAO Yan(School of Civil Engineering,Central South University of Forestry and Technology,Changsha 410004,China;Department of Engineering Mechanics,Dalian University of Technology,Dalian 116024,China;Beijing Aerospace System Engineering Institute,Beijing 100076,China;Ningbo Research Institute of Dalian University of Technology,Ningbo 315016,China)

机构地区：[1]中南林业科技大学土木工程学院,长沙410004 [2]大连理工大学工程力学系,大连116024 [3]北京宇航系统工程研究所,北京100076 [4]大连理工大学宁波研究院,宁波315016

出　　处：《计算力学学报》2023年第5期672-677,共6页Chinese Journal of Computational Mechanics

基　　金：国家自然科学基金(11772084,U1906233);国家重点研发计划(2017YFC0307203);中央高校基础研究经费(DUT22ZD209)资助项目。

摘　　要：针对结构在随机激励作用下的动力响应统计特征分析问题,提出了结构平稳和非平稳随机振动时域响应分析的多项式维数分解PDD(polynomial dimensional decomposition)法,高效地实现了结构随机振动响应统计矩和概率密度的计算。首先,采用三角级数叠加法模拟随机激励,将其中的随机相位作为结构系统的随机输入,并将结构随机振动时域响应视为关于随机相位的函数。其次,将结构响应函数采用成员函数(Component function)进行维数分解,并对成员函数进行Fourier多项式展开,从而构造出结构响应预测的PDD展开模型。最后,为了解决构造PDD展开模型时面临的高维积分问题,采用降维积分方法降低积分维度,显著提高了计算效率。在数值算例中,进行了单自由度系统和20层框架结构的随机振动时域响应分析,并将本文方法的计算结果与Monte Carlo模拟结果进行对比,验证了建立方法的精确性和高效性。This paper aims at the statistical characteristics analysis of the dynamic response of structures under random excitation,proposes a polynomial dimensional decomposition(PDD)method for the time-domain response analysis of structural stationary and non-stationary random vibration and effectively calculates the statistical moments and probability density of the structural random vibration response.Firstly,the trigonometric series superposition method is used to simulate the random excitation,the random phases in the method are taken as the random input of the structural system,and the random vibration response in the time-domain of the structure is regarded as a function of the random phases.Secondly,the structural response function is decomposed by component functions,and the component functions are expanded by the Fourier polynomials,then the PDD expansion model for the prediction of structural responses is constructed.Finally,in order to solve the problem of high-dimensional integration when constructing the PDD expansion model,the dimension reduction integration method is introduced to reduce the integration dimension and significantly improves the computational efficiency.In the numerical examples,the random vibration response analyses in the time-domain of a single-degree-of-freedom system and twenty-story frame structure are carried out,and the results calculated by the proposed method are compared with the results of Monte Carlo simulation to verify the accuracy and efficiency of the proposed method.

关 键 词：随机振动 多项式维数分解 三角级数叠加 PDD展开模型 

分 类 号：O324[理学—一般力学与力学基础]