非线性结构地震随机响应方差的简化计算方法
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摘要
对于多自由度的非线性随机系统,通过随机振动的响应计算得到精确的解析解有许多困难。该文提出一种计算非线性结构地震随机响应方差的方法,将随机响应的协方差矩阵用K-L(Karhunen-Loéve)向量来表示,结合逐步积分法求出下一时间步长的K-L向量。同时根据K-L向量的模型截断原理,将对计算影响很小的K-L向量省略,并且根据系统受到低频激励时,高阶振型的反应可以近似用静态形式来表示的假设,进行自由度的降阶,减小了协方差矩阵的规模,使计算得到简化。以24自由度剪切型滞迟系统为例,分别采用了非降阶计算、K-L向量截断计算和自由度降阶计算,并将计算结果进行对比分析。结果表明,用这种基于K-L展开的确定性积分方法计算非线性结构在地震激励下的随机响应的方差是可行的。
Stochastic dynamic has been applied mainly to response of dynamic systems in stochastic excitation.But it is not easy to calculate the stochastic response especially for multi-freedom nonlinear systems.There is more difficulty in receiving exact analytical solution.A calculation method to forecast covariance of stochastic response to earthquake for large complicated non-liner structures is studied in this paper.The method which is a deterministic integration method with the Karhunen-Loéve(K-L) expansion can avoid the storage of the full covariance matrix.The covariance of stochastic response to earthquake is represented by the K-L vectors,and then the following step K-L vectors can be integrated by available and appropriate deterministic step by step integration scheme.At the same time,the smaller ones of the K-L vectors will be omitted because of application of mode truncation.The efficiency of calculation will be increased, and the calculation time will be saved.It is suggested that the high frequency modes react essentially in a static manner when exited by low frequencies.Then the number of iterative freedom in the calculation can be reduced.This will improve the calculation efficiency further.A 24-degree of freedom non-linear hysteretic system has been calculated as an example.The normal,K-L vectors' mode truncation and reducing the degree of freedom have been used.The results indicate that it is feasible to calculate the covariance of stochastic response to earthquake for large complicated nonliner structures using the deterministic integration method with the K-L expansion.
Stochastic dynamic has been applied mainly to response of dynamic systems in stochastic excitation.But it is not easy to calculate the stochastic response especially for multi-freedom nonlinear systems.There is more difficulty in receiving exact analytical solution.A calculation method to forecast covariance of stochastic response to earthquake for large complicated non-liner structures is studied in this paper.The method which is a deterministic integration method with the Karhunen-Loéve(K-L) expansion can avoid the storage of the full covariance matrix.The covariance of stochastic response to earthquake is represented by the K-L vectors,and then the following step K-L vectors can be integrated by available and appropriate deterministic step by step integration scheme.At the same time,the smaller ones of the K-L vectors will be omitted because of application of mode truncation.The efficiency of calculation will be increased, and the calculation time will be saved.It is suggested that the high frequency modes react essentially in a static manner when exited by low frequencies.Then the number of iterative freedom in the calculation can be reduced.This will improve the calculation efficiency further.A 24-degree of freedom non-linear hysteretic system has been calculated as an example.The normal,K-L vectors' mode truncation and reducing the degree of freedom have been used.The results indicate that it is feasible to calculate the covariance of stochastic response to earthquake for large complicated nonliner structures using the deterministic integration method with the K-L expansion.
引文
[1]朱位秋.非线性随机动力学与控制——Hamilton理论体系框架[M].北京:科学出版社,2003.
[2]Lutes L D,Sarkani S.Stochastic Analysis of Structural and Mechanical Vibrations[M].[S.l.]:Prentice-hall,1997.
[3]Pradlwarter H J.Deterministic Integration Algorithms for Stochastic Response Computations of FE-systems[J].Computersand Structures,2002(80):1489-1502.
[4]Schenk C A,Pradlwarter H J,Schueller G I.On the Dynamic Stochastic Response of FE Models[J].Probabilistic EngineeringMechanics,2004(19):161-170.
[5]Pradlwarter H J,Schueller G I,Schenk C A.A Computational Procedure to Estimate the Stochastic Dynamic Response of LargeNon-linear FE-models[J].Computer Methods in Applied Mechanics and Engineering,2003,192:777-801.
[6]陈学前,张培强.K-L法在结构模型降阶及辨识中的应用[J].力学与实践,2004,26(2):54-57.
[7]Pradlwarter H J,Schueller G I.A Consistent Concept for High and Low-frequency Dynamics Based on Stochastic Modal Analy-sis[J].Journal of Sound and Vibration,2005,288:653-667.
[2]Lutes L D,Sarkani S.Stochastic Analysis of Structural and Mechanical Vibrations[M].[S.l.]:Prentice-hall,1997.
[3]Pradlwarter H J.Deterministic Integration Algorithms for Stochastic Response Computations of FE-systems[J].Computersand Structures,2002(80):1489-1502.
[4]Schenk C A,Pradlwarter H J,Schueller G I.On the Dynamic Stochastic Response of FE Models[J].Probabilistic EngineeringMechanics,2004(19):161-170.
[5]Pradlwarter H J,Schueller G I,Schenk C A.A Computational Procedure to Estimate the Stochastic Dynamic Response of LargeNon-linear FE-models[J].Computer Methods in Applied Mechanics and Engineering,2003,192:777-801.
[6]陈学前,张培强.K-L法在结构模型降阶及辨识中的应用[J].力学与实践,2004,26(2):54-57.
[7]Pradlwarter H J,Schueller G I.A Consistent Concept for High and Low-frequency Dynamics Based on Stochastic Modal Analy-sis[J].Journal of Sound and Vibration,2005,288:653-667.
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