复合随机振动分析的自适应回归算法
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摘要
基于虚拟激励法(PEM)和广义多项式混沌展开(g PC)提出一种求解复合随机振动问题的自适应回归算法。通过求解随机系统在虚拟激励下的运动方程得到本文所关注的随机物理响应,并将其在以不确定参数为自变量的正交多项式函数空间内展开,应用自适应采样与自适应基函数筛选相结合的回归算法确定多项式基函数系数,进而给出随机响应的概率特征。本文方法是一种非介入算法,不需要改变控制方程的求解维度,便于使用既有的求解程序进行分析。数值算例中,对具有不确定参数的车轨耦合系统在随机轨道不平顺激励下的随机振动响应进行分析,将计算结果与50000样本Monte Carlo法进行了比对验证,相对误差不足1%,表明了本文方法具有很好的工程应用前景。
An adaptive regression algorithm for double random vibration analysis is proposed based on pseudo excitation method(PEM) and generalized polynomial chaos expansion method(g PC). The motion equations of random system due to pseudo excitations are established, and the physical response of concern are expanded in an orthogonal polynomial space. The coefficients of the polynomial basis functions are solved by the regression algorithm consisting of the adaptive sampling and the adaptive selection of the basis functions, and then the probability characteristics of the response can be obtained. The proposed method is a non-instrusive algorithm, which does not need to change the dimension of the control equations and the existing deterministic solution programs can be used. In the numerical example, the random vibration analysis for the vehicle-track coupled systems with uncertain parameters subjected to random track irregularities is implemented. Results obtained arecompared to 50000 samples Monte-Carlo simulations and the relative errors are less than 1% which shows that the proposed method is very effective for engineering applications.
An adaptive regression algorithm for double random vibration analysis is proposed based on pseudo excitation method(PEM) and generalized polynomial chaos expansion method(g PC). The motion equations of random system due to pseudo excitations are established, and the physical response of concern are expanded in an orthogonal polynomial space. The coefficients of the polynomial basis functions are solved by the regression algorithm consisting of the adaptive sampling and the adaptive selection of the basis functions, and then the probability characteristics of the response can be obtained. The proposed method is a non-instrusive algorithm, which does not need to change the dimension of the control equations and the existing deterministic solution programs can be used. In the numerical example, the random vibration analysis for the vehicle-track coupled systems with uncertain parameters subjected to random track irregularities is implemented. Results obtained arecompared to 50000 samples Monte-Carlo simulations and the relative errors are less than 1% which shows that the proposed method is very effective for engineering applications.
引文
[1]乔红威,吕震宙.随机结构随机激励下的响应灵敏度分析[J].振动与冲击,2008,27(3):60-62.(Qiao Hongwei,Lv Zhenzhou.Response sensitivity analysis of stochastic structures under random excitation[J].Journal of Vibration and Shock,2008,27(3):60-62(in Chinese)).
[2]Chen S H,Liu Z S,Zhang Z F.Random vibration analysis for large-scale structures with random parameters[J].Computers&Structures,1992,43(4):681-685.
[3]林家浩,易平.线性随机结构的平稳随机响应[J].计算力学学报,2001,18(4):402-408.(Lin Jia Hao,Yi Ping.Stationary random responses of structures with stochastic parameters[J].Chinese Journal of Computational Mechanics,2001,18(4):402-408(in Chinese)).
[4]范么清.非线性复合随机振动方法研究及其工程应用[D].上海:同济大学,2007.(Fan Meqing.Study on methods for non-linear compound stochastic vibration and their applications in engineering[D].Shanghai:Tongji University,2007(in Chinese)).
[5]Wiener N.The homogeneous chaos[J].American Journal of Mathematics,1938,60(4):897-936.
[6]Ghanem R G,Spanos P D.Stochastic finite elements:a spectral approach[M].New York:Springer-Verlag,1991.
[7]Xiu D,Karniadakis G E.The Wiener-Askey polynomial chaos for stochastic differential equations[J].SIAM Journal on Scientific Computing,2002,24(2):619-644.
[8]Xiu D.Numerical methods for stochastic computations:a spectral method approach[M].Princeton:Princeton University Press,2010.
[9]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 radom vibration analysis[J].Acta Mechanica Sinica,1996,28(1):66-75(in Chinese)).
[10]Keshavarzzadeh V,Ghanem R G,Masri S F,et al.Convergence acceleration of polynomial chaos solutions via sequence transformation[J].Computer Methods in Applied Mechanics and Engineering,2014,271(2):167-184.
[11]Choi S K,Grandhi R V,Canfield R A.Structural reliability under non-Gaussian stochastic behavior[J].Computers&Structures,2004,82(13):1113-1121.
[12]Xiong F F,Chen S S,Xiong Y.Dynamic system uncertainty propagation using polynomial chaos[J].Chinese Journal of Aeronautics,2014,27(5):1156-1170.
[13]林家浩,张亚辉.随机振动的虚拟激励法[M].北京:科学出版社,2004.(Lin Jiahao,Zhang Yahui.Pseudo excitation method in random vibration[M].Beijing:Science Press,2004(in Chinese)).
[14]楼梦麟,王凤武.多自由度线性结构复合随机振动分析的计算方法[C]//2005年上海市国际工业博览会第三届上海市“工程与振动”科技论坛论文集.上海:同济大学电子音像出版社,2005.(Lou Menglin,Wang Fengwu.Analysis approach for double random vibration of linear multi degree of freedom structure[C]//Shanghai International Industry Fair Third‘Engineering and Vibration’Technology Forum Collection.Shanghai:Tongji University Electronics Audio-video Press,2005(in Chinese)).
[15]赵岩,项盼,张有为,等.不确定车轨耦合系统辛随机振动分析[J].力学学报,2012,44(4):769-778.(Zhao Yan,Xiang Pan,Zhang Youwei,et al.Symplectic random vibration analysis for coupled vehicle-track systems with parameter uncertainties[J].Chinese Journal of Theoretical and Applied Mechanics,2012,44(4):769-778(in Chinese)).
[16]项盼,赵岩,林家浩.复合随机振动分析的混合PC-PEM方法[J].振动与冲击,2015,34(4):35-39.(Xiang Pan,Zhao Yan,Lin Jiahao.Hybrid PC-PEM for complex random vibration analysis[J].Journal of Vibration and Shock,2015,34(4):35-39(in Chinese)).
[17]Caprani C C.Application of the pseudo-excitation method to assessment of walking variability on footbridge vibration[J].Computers&Structures,2014,132(1):43-54.
[18]De Rosa S,Franco F,Ciappi E.A simplified method for the analysis of the stochastic response in discrete coordinates[J].Journal of Sound and Vibration,2015,339:359-375.
[19]陈贤川,赵阳,董石麟.基于虚拟激励法的空间网格结构风致抖振响应分析[J].计算力学学报,2006,23(6):684-689.(Chen Xianchuan,Zhao Yang,Dong Shilin.Buffeting analysis of space lattice structures based on pseudo excitation method[J].Chinese Journal of Computational Mechanics,2006,23(6):684-689(in Chinese)).
[20]杨新文,翟婉明,和振兴.高速列车通过钢轨焊接接头时引起的轮轨冲击噪声的研究[J].振动与冲击,2011,30(7):143-147.(Yang Xinwen,Zhai Wanming,He Zhenxing.Wheel/rail impact noise generation due to high speed train passing over rail weld joints[J].Journal of Vibration and Shock,2011,30(7):143-147(in Chinese)).
[21]廖俊.基于正交分解的随机振动响应分析与随机载荷识别研究[D].哈尔滨:哈尔滨工业大学,2011.(Liao Jun.Research on response analysis of structure random vibration and random loading identification based on orthogonal decomposition[D].Harbin:Harbin Institute of Technology,2011(in Chinese)).
[22]Wang G G.Adaptive response surface method using inherited latin hypercube design points[J].Journal of Mechanical Design,2003,125(2):210-220.
[23]袁志发,宋世德.多元统计分析[M].北京:科学出版社,2009.(Yuan Zhifa,Song Shide.Multivariate statistical analysis[M].Beijing:Science Press,2009(in Chinese)).
[24]Blatman G,Sudret B.An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis[J].Probabilistic Engineering Mechanics,2010,25(2):183-197.
[2]Chen S H,Liu Z S,Zhang Z F.Random vibration analysis for large-scale structures with random parameters[J].Computers&Structures,1992,43(4):681-685.
[3]林家浩,易平.线性随机结构的平稳随机响应[J].计算力学学报,2001,18(4):402-408.(Lin Jia Hao,Yi Ping.Stationary random responses of structures with stochastic parameters[J].Chinese Journal of Computational Mechanics,2001,18(4):402-408(in Chinese)).
[4]范么清.非线性复合随机振动方法研究及其工程应用[D].上海:同济大学,2007.(Fan Meqing.Study on methods for non-linear compound stochastic vibration and their applications in engineering[D].Shanghai:Tongji University,2007(in Chinese)).
[5]Wiener N.The homogeneous chaos[J].American Journal of Mathematics,1938,60(4):897-936.
[6]Ghanem R G,Spanos P D.Stochastic finite elements:a spectral approach[M].New York:Springer-Verlag,1991.
[7]Xiu D,Karniadakis G E.The Wiener-Askey polynomial chaos for stochastic differential equations[J].SIAM Journal on Scientific Computing,2002,24(2):619-644.
[8]Xiu D.Numerical methods for stochastic computations:a spectral method approach[M].Princeton:Princeton University Press,2010.
[9]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 radom vibration analysis[J].Acta Mechanica Sinica,1996,28(1):66-75(in Chinese)).
[10]Keshavarzzadeh V,Ghanem R G,Masri S F,et al.Convergence acceleration of polynomial chaos solutions via sequence transformation[J].Computer Methods in Applied Mechanics and Engineering,2014,271(2):167-184.
[11]Choi S K,Grandhi R V,Canfield R A.Structural reliability under non-Gaussian stochastic behavior[J].Computers&Structures,2004,82(13):1113-1121.
[12]Xiong F F,Chen S S,Xiong Y.Dynamic system uncertainty propagation using polynomial chaos[J].Chinese Journal of Aeronautics,2014,27(5):1156-1170.
[13]林家浩,张亚辉.随机振动的虚拟激励法[M].北京:科学出版社,2004.(Lin Jiahao,Zhang Yahui.Pseudo excitation method in random vibration[M].Beijing:Science Press,2004(in Chinese)).
[14]楼梦麟,王凤武.多自由度线性结构复合随机振动分析的计算方法[C]//2005年上海市国际工业博览会第三届上海市“工程与振动”科技论坛论文集.上海:同济大学电子音像出版社,2005.(Lou Menglin,Wang Fengwu.Analysis approach for double random vibration of linear multi degree of freedom structure[C]//Shanghai International Industry Fair Third‘Engineering and Vibration’Technology Forum Collection.Shanghai:Tongji University Electronics Audio-video Press,2005(in Chinese)).
[15]赵岩,项盼,张有为,等.不确定车轨耦合系统辛随机振动分析[J].力学学报,2012,44(4):769-778.(Zhao Yan,Xiang Pan,Zhang Youwei,et al.Symplectic random vibration analysis for coupled vehicle-track systems with parameter uncertainties[J].Chinese Journal of Theoretical and Applied Mechanics,2012,44(4):769-778(in Chinese)).
[16]项盼,赵岩,林家浩.复合随机振动分析的混合PC-PEM方法[J].振动与冲击,2015,34(4):35-39.(Xiang Pan,Zhao Yan,Lin Jiahao.Hybrid PC-PEM for complex random vibration analysis[J].Journal of Vibration and Shock,2015,34(4):35-39(in Chinese)).
[17]Caprani C C.Application of the pseudo-excitation method to assessment of walking variability on footbridge vibration[J].Computers&Structures,2014,132(1):43-54.
[18]De Rosa S,Franco F,Ciappi E.A simplified method for the analysis of the stochastic response in discrete coordinates[J].Journal of Sound and Vibration,2015,339:359-375.
[19]陈贤川,赵阳,董石麟.基于虚拟激励法的空间网格结构风致抖振响应分析[J].计算力学学报,2006,23(6):684-689.(Chen Xianchuan,Zhao Yang,Dong Shilin.Buffeting analysis of space lattice structures based on pseudo excitation method[J].Chinese Journal of Computational Mechanics,2006,23(6):684-689(in Chinese)).
[20]杨新文,翟婉明,和振兴.高速列车通过钢轨焊接接头时引起的轮轨冲击噪声的研究[J].振动与冲击,2011,30(7):143-147.(Yang Xinwen,Zhai Wanming,He Zhenxing.Wheel/rail impact noise generation due to high speed train passing over rail weld joints[J].Journal of Vibration and Shock,2011,30(7):143-147(in Chinese)).
[21]廖俊.基于正交分解的随机振动响应分析与随机载荷识别研究[D].哈尔滨:哈尔滨工业大学,2011.(Liao Jun.Research on response analysis of structure random vibration and random loading identification based on orthogonal decomposition[D].Harbin:Harbin Institute of Technology,2011(in Chinese)).
[22]Wang G G.Adaptive response surface method using inherited latin hypercube design points[J].Journal of Mechanical Design,2003,125(2):210-220.
[23]袁志发,宋世德.多元统计分析[M].北京:科学出版社,2009.(Yuan Zhifa,Song Shide.Multivariate statistical analysis[M].Beijing:Science Press,2009(in Chinese)).
[24]Blatman G,Sudret B.An adaptive algorithm to build up sparse polynomial chaos expansions for stochastic finite element analysis[J].Probabilistic Engineering Mechanics,2010,25(2):183-197.