线性随机结构在随机激励下动力响应分析

DYNAMIC RESPONSE OF LINEAR STOCHASTIC STRUCTURESUNDER RANDOM EXCITATION

利用虚拟激励法对随机结构正交展开理论进行扩展,并在Ritz向量子空间中对扩阶系统方程进行动力聚缩,提出了一类可以快速高效地进行线性随机结构复合随机振动分析的计算方法．算例分析表明,该法可以方便地分析随机结构在平稳或非平稳随机激励下的复合随机振动问题,且分析结果与 Monte Carlo模拟分析结果符合良好;与均值参数确定性结构传统随机振动分析计算结果相比,随机结构在相同随机激励下响应自谱密度曲线具有峰值降低、谱宽增大的特点．

In this paper a new method to analyze the dynamic response of structures withuncertain parameters under external random excitation is proposed. This method expands theorthogonal expansion method with the pseudo-excitation method. Considering the engineeringbackground of most practical structures, it is rational to assume that the temporal variability ofexternal excitation and the spatial variability of stochastic structural parameters are statisticallyindependent of each other, therefore the different source of stochasticity can be tackled with dif-ferent methods. Firstly the structural response vectors are decomposed in a probability sub-spacespanned by spatial random variables as the summation of orthogonal polynomial series. Using therecurrence relationship and the orthogonality properties of the orthogonal polynomial series, thedynamic equilibrium equation of the original stochastic structural system can be expanded iato anorder-expanded equation whose right-side order-expanded load vector is temporally stochastic, andthe order-expanded matrices are all deterministic. This order-expanded equation can be consid-ered as a classic random vibration problem and solved with any classic random vibration analysismethod. Herein the pseudo-excitation method is used and the solution of this temporally randomequation can be transferred into deterministic dynamic integration analysis. Corresponding to theorder-expanded load vector, a set of deterministic pseudo loading vectors that varies with frequencyis formed. The power spectral density (PSD) matrix of the order-expanded response vector canbe obtained with the pseudo-responses of the order-expanded equation under the deterministicpseudo-excitation. Then considering the recurrence relationship and the orthogonality propertiesof the orthogonal polynomial series, the PSD matrix of the original structures can be calculatedfrom the PSD matrix of the order-expanded response vector.In order to reduce the computation cost of the orthogonal expansion method to analyze thedynamic response of stochastic structures, the Arnoldi algorithm is used to generate a set of Ritzmodal vectors as the approximate orthonormal basis of the solution subspace. With the help ofthe generalized coordinate transformation matrix formed by the Ritz vectors, the computationalefforts will be reduced to an acceptable degree.A reinforced concrete shear wall subjected to random stationary or non-stationary excitationis used as numerical examples to show the applicability of the proposed method. It is found thatthe peak value of structural response auto-PSD of stochastic structures tend to be lower than thecorresponding deterministic nominal structures, while the shape of the auto-PSD is wider. Theanalysis results converge quickly with the number of Ritz modal vectors. And a few Ritz modalvectors can yield computation results which are in good agreement with those of Monte Carlosimulation method, so the computational efficiency is greatly enhanced.