随机结构非平稳响应的正交解法
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摘要
解随机结构扩阶系统动力方程一直是随机结构响应求解中的难点。借助Gegenbauer多项式逼近法,将随机参数系统动力响应计算问题转化为与其等价的确定扩阶系统的响应计算问题。用精细积分对过滤白噪声的非平稳随机激励进行K_L分解,利用K_L向量的能量集中的特点,用少量的K_L向量参与扩阶系统的响应计算即可获得较精确的响应值,显著提高了计算的效率。仿真验证了方法的正确性,并对随机参数概率密度函数的差异对响应均方值的影响等进行了研究,获得了一些有工程参考价值的结论。
Solving an expanded order-system dynamic-equation is the difficulty to obtain the dynamic response of a random structure.By using Gegenbauer polynomial approximation,the calculation problem of dynamic responses of a random parameter system is transformed to system's response calculation of an equivalent order certainty expansion.The Precise Integration Method is used to obtain the K_L decomposition vectors of the non-stationary filtered white noise random excitation,with characteristics of energy concentration,a small amount of K_L vector used to compute response of the extended order system can obtain a very accurate result,and it significantly improves the calculation efficiency.Simulation verifies the correctness of the method,and the effects of the probability density function of random parameters to the response mean square value are studied,and some engineering valuable conclusions are obtained.
Solving an expanded order-system dynamic-equation is the difficulty to obtain the dynamic response of a random structure.By using Gegenbauer polynomial approximation,the calculation problem of dynamic responses of a random parameter system is transformed to system's response calculation of an equivalent order certainty expansion.The Precise Integration Method is used to obtain the K_L decomposition vectors of the non-stationary filtered white noise random excitation,with characteristics of energy concentration,a small amount of K_L vector used to compute response of the extended order system can obtain a very accurate result,and it significantly improves the calculation efficiency.Simulation verifies the correctness of the method,and the effects of the probability density function of random parameters to the response mean square value are studied,and some engineering valuable conclusions are obtained.
引文
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[25]De La Fuente E.An efficient procedure to obtain exact solutions in random vibration analysis of linear structures[J].Engineering Structures,2008,30(11):2981―2990.
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[2]Soong T,Bogdanoff J.On the natural frequencies of a disordered linear chain of N degrees of freedom[J].International Journal of Mechanical Sciences,1963,5(3):237―265.
[3]陈塑寰.随机参数结构的振动理论[M].长春:吉林科学技术出版社,1992.Chen Suhuan.The vibration theory of random parameters[M].Changchun:Jilin Science and Technology Press,1992.(in Chinese)
[4]Shinozuka M,Jan C.Digital simulation of random processes and its applications[J].Journal of Sound and Vibration,1972,25(1):111―128.
[5]Shinozuka M,Wen Y.Monte Carlo solution of nonlinear vibrations[J].AIAA Journal,1972,10(1):37―40.
[6]Sun T.A finite element method for random differential equations with random coefficients[J].SIAM Journal on Numerical Analysis,1979,16(6):1019―1035.
[7]Ghanem R,Spanos P.Polynomial chaos in stochastic finite elements[J].Journal of Applied Mechanics,1990,57(1):197―202.
[8]Ghanem R,Spanos P.Spectral stochastic finite-element formulation for reliability analysis[J].Journal of Engineering Mechanics,1991,117(10):2351―2372.
[9]Jensen H,Iwan W.Response variability in structural dynamics[J].Earthquake Engineering&Structural Dynamics,2006,20(10):949―959.
[10]Jensen H.Response of systems with uncertain parameters to stochastic excitation[J].Journal of Engineering Mechanics,1992,118(5):1012―1025.
[11]李杰.随机结构分析的扩阶系统方法(I)扩阶系统方程[J].地震工程与工程振动,1995,15(3):111―118.Li Jie.Expanded order system method of stochastic structures analysis,Part I:equation of expanded order system[J].Earthquake Engineering and Engineering Vibration,1995,15(3):111―118.(in Chinese)
[12]李杰.随机结构分析的扩阶系统方法(II)结构动力分析[J].地震工程与工程振动,1995,15(4):27―35.Li Jie.Expanded order system method of stochastic structures analysis,Part II:dynamic analysis of structures[J].Earthquake Engineering and Engineering Vibration,1995,15(4):27―35.(in Chinese)
[13]李杰.复合随机振动分析的扩阶系统方法[J].力学学报,1996,28(1):66―75.Li Jie.The expanded order system method of combined random vibration analysis[J].Acta Mechanica Sinica,1996,28(1):66―75.(in Chinese)
[14]李杰.随机结构动力分析的扩阶系统方法[J].工程力学,1996,13(1):93―102.Li Jie.An expanded order system method for the dynamic analysis of stochastic structures[J].Engineering Mechanics,1996,13(1):93―102.(in Chinese)
[15]李杰.随机结构正交分解分析方法[J].振动工程学报,1999,12(1):78―84.Li Jie.Research development for the orthogonal expansion method of stochastic structural analysis[J].Journal of Vibration Engineering,1999,12(1):78―84.(in Chinese)
[16]冷小磊.线性随机系统演变随机响应问题研究及随机Duffing系统中分叉与混沌初探[D].西安:西北工业大学,2002.Leng Xiaolei.Study on evolutionary random response problems of stochastic linear systems and elementary study on bifurcation and chaos of stochastic duffing system[D].Xi’an:Northwest Ploytechnical University,2002.(in Chinese)
[17]Fang T,Leng X,Ma X.λ-PDF and Gegenbauer polynomial approximation for dynamic response problems of random structure[J].Acta Mechanica Sinica,2004,20(3):292―298.
[18]Schu Ller G I,Pradlwarter H J,Schenk C A.Non-stationary response of large linear FE models under stochastic loading[J].Computers and Structures,2003,81(8):937―947.
[19]Schenk C A,Pradlwarter H J,Schueller G I.On the dynamic stochastic response of FE models[J].Probabilistic Engineering Mechanics,2004,19(1/2):161―170.
[20]Schenk C A,Pradlwarter H J,Schueller G I.Non-stationary response of large,non-linear finite element systems under stochastic loading[J].Computers and Structures,2005,83(14):1086―1102.
[21]吴存利.随机结构与随机动力系统中的响应,分岔与混沌及其控制[D].西安:西北工业大学,2007.Wu Cunli.Response,bifurcation and chaos and their control in stochastic structure and stochastic dynamical system[D].Xi’an:Northwest Ploytechnical University,2007.(in Chinese)
[22]王竹溪,郭敦仁.特殊函数概论[M].北京:科学出版社,1979.Wang Zhuxi,Guo Dunren.Introduction to special functions[M].Beijing:Science Press,1979.(in Chinese)
[23]朱位秋.随机振动[M].北京:科学出版社,1992.Zhu Weiqiu.Random vibration[M].Beijing:Science Press,1992.(in Chinese)
[24]Zhong W.On precise integration method[J].Journal of Computational and Applied Mathematics,2004,163(1):59―78.
[25]De La Fuente E.An efficient procedure to obtain exact solutions in random vibration analysis of linear structures[J].Engineering Structures,2008,30(11):2981―2990.
[26]林家浩,沈为平.结构非平稳随机响应的混合型精细时程积分[J].振动工程学报,1995,8(2):127―135.Lin Jiahao,Shen Weiping.High-precision integration of mixed type for analysis of non-stationary random response[J].Journal of Vibration Engineering,1995,8(2):127―135.(in Chinese)
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