一种非高斯随机振动过程数值模拟方法
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摘要
目的考虑真实随机振动的非高斯特性,提出一种根据已知信息生成与其相符的非高斯随机振动过程的数值模拟方法。方法基于均值、方差、偏斜度、峭度及功率谱密度函数(或自相关函数)等约束条件,对非高斯随机振动进行模拟。根据功率谱获取非高斯过程的自相关矩阵;通过Hermite多项式的正交性质和多项式混沌展开方法推导出的公式,构造满足标准正态分布随机过程的协方差矩阵,并对其进行谱分解和主成分分析;最后,利用Karhunen-Loeve展开和多项式混沌展开来表示所模拟的非高斯振动过程。结果随着采样点个数的增加,实测数据与模拟数据之间的误差越来越小,该方法具有较好的模拟精度。结论应用多项式混沌展开、Karhunen-Loeve展开以及蒙特卡洛等方法,可生成非高斯随机振动过程,并得到准确有效的各项统计参数模拟值。
The work aims to propose an appropriate numerical simulation method for non-Gaussian random vibration processes generated according to the known information when considering non-Gaussian property of the actual random vibration. Based on such constraint conditions as mean, variance, skewness, kurtosis and PSD function(or autocorrelation function), the non-Gaussian random vibration was simulated. The autocorrelation matrix of the non-Gaussian process was obtained from the PSD. Through the formula derived from Hermite polynomial orthogonal property and polynomial chaos expansion, the covariance matrix of the standard normal distribution random process was constructed, and the spectral decomposition and principal component analysis were also performed. Finally, the simulated non-Gaussian vibration process was represented by Karhunen-Loeve expansion and polynomial chaos expansion. As the number of sampling points increased, the error between measured data and simulated data became smaller and smaller, besides, the proposed method had good simulation accuracy. Combined with polynomial chaos expansion, Karhunen-Loeve expansion and Monte Carlo method, the non-Gaussian random vibration process can be generated and accurate and effective simulation values of various statistical parameters can be obtained.
The work aims to propose an appropriate numerical simulation method for non-Gaussian random vibration processes generated according to the known information when considering non-Gaussian property of the actual random vibration. Based on such constraint conditions as mean, variance, skewness, kurtosis and PSD function(or autocorrelation function), the non-Gaussian random vibration was simulated. The autocorrelation matrix of the non-Gaussian process was obtained from the PSD. Through the formula derived from Hermite polynomial orthogonal property and polynomial chaos expansion, the covariance matrix of the standard normal distribution random process was constructed, and the spectral decomposition and principal component analysis were also performed. Finally, the simulated non-Gaussian vibration process was represented by Karhunen-Loeve expansion and polynomial chaos expansion. As the number of sampling points increased, the error between measured data and simulated data became smaller and smaller, besides, the proposed method had good simulation accuracy. Combined with polynomial chaos expansion, Karhunen-Loeve expansion and Monte Carlo method, the non-Gaussian random vibration process can be generated and accurate and effective simulation values of various statistical parameters can be obtained.
引文
[1]程红伟.非高斯随机载荷作用下结构疲劳寿命及可靠性研究[D].长沙:国防科学技术大学,2014.CHENG Hong-wei.Research on the Fatigue Life and Reliability of Mechanical Structure under NonGaussian Random Loading[D].Changsha:National University of Defense Technology,2014.
[2]李锦华,陈水生.非高斯随机过程模拟与预测的研究进展[J].华东交通大学学报,2011,28(6):1-6.LI Jin-hua,CHEN Shui-sheng.Advances in Simulation and Prediction of Non-Gaussian Stochastic Processes[J].Journal of East China Jiaotong University,2011,28(6):1-6.
[3]STEINWOLF A.Shaker Random Testing with Low Kurtosis:Review of the Methods and Application for Sigma Limiting[J].Shock and Vibration,2010,17(2):219-231.
[4]ROUILLARD V.Quantifying the Non-stationarity of Vehicle Vibrations with the Run Rest[J].Packaging Technology and Science,2014,27(3):203-219.
[5]RIZZI S A,PRZEKP A,TURNER T.On the Response of a Nonlinear Structure to High Kurtosis Non-Gaussian Random Loadings[R].NASA Technical Report,2013.
[6]SEONG S H,PETERKA J A.Computer Simulation of Non-Gaussian Multiple Wind Pressure Time Series[J].Journal of Wind Engineering&Industrial Aerodynamics,1997,72(1):95-105.
[7]SEONGS H,PETERKA J A.Digital Generation of Surface-pressure Fluctuations with Spiky Features[J].Journal of Wind Engineering&Industrial Aerodynamics,1998,73(2):181-192.
[8]蒋瑜,陶俊勇,王得志,等.一种新的非高斯随机振动数值模拟方法[J].振动与冲击,2012,31(19):169-173.JIANG Yu,TAO Jun-yong,WANG De-zhi,et al.ANovel Approach for Numerical Simulation of a Non-Gaussian Random Vibration[J].Journal of Vibration and Shock,2012,31(19):169-173.
[9]夏静,袁宏杰,徐如远.一种新的非高斯随机振动信号的模拟方法[J].北京航空航天大学学报,2019,45(2):366-372.XIA Jing,YUAN Hong-jie,XU Ru-yuan.A New Simulation Method of Non-Gaussian Random Vibration Signals[J].Journal of Beijing University of Aeronautics and Astronautics,2019,45(2):366-372.
[10]黄铭枫,孙轩涛,冯鹤,等.基于风压谱和Hermite模型的大跨干煤棚风压场数值模拟研究[J].振动与冲击,2018,37(23):111-119.HUANG Ming-feng,SUN Xuan-tao,FENG He,et al.Numerical Simulation of Wind Pressure Field of Long-span Lattice Structures Based on Wind Pressure Spectra and Hermite Model[J].Journal of Vibration and Shock,2018,37(23):111-119.
[11]徐飞,李传日,姜同敏,等.非高斯随机振动的模拟方法[J].北京航空航天大学学报,2014,40(9):1239-1244.XU Fei,LI Chuan-ri,JIANG Tong-min,et al.Simulation of Non-Gaussian Random Vibration[J].Journal of Beijing University of Aeronautics and Astronautics,2014,40(9):1239-1244.
[12]朱大鹏,李明月.铁路非高斯随机振动的数字模拟与包装件响应分析[J].包装工程,2016,37(1):1-5.ZHU Da-peng,LI Ming-yue.Digital Simulation of Non-Gaussian Random Vibration of Railway and Packaging System Response Analysis[J].Packaging Engineering,2016,37(1):1-5.
[13]蒋瑜,陈循,陶俊勇,等.指定功率谱密度、偏斜度和峭度值下的非高斯随机过程数字模拟[J].系统仿真学报,2006,18(5):1127-1130.JIANG Yu,CHEN Xun,TAO Jun-yong,et al.Numerically Simulating Non-Gaussian Random Processes with Specified PSD,Skewness and Kurtosis[J].Journal of System Simulation,2006,18(5):1127-1130.
[14]李继根,张新发.矩阵分析与计算[M].武汉:武汉大学出版社,2013.LI Ji-gen,ZHANG Xin-fa.Matrix Analysis and Computation[M].Wuhan:Wuhan University Press,2013.
[15]PHOON K K,HUANG S P,QUEK S T.Simulation of Second-order Processes Using Karhunen-loeve Expansion[J].Computers&Structures,2002,80(12):1049-1060.
[16]MOURELATOS Z P,MAJCHER M,PANDEY V,et al.Time-dependent Reliability Analysis Using the Total Probability Theorem[J].Journal of Mechanical Design2015,137(3):031405.
[17]XIU D,KARNIADAKIS G.The Wiener-askey Polynomial Chaos for Stochastic Differential Equations[J].SIAM Journal on Scientific Computing,2002,24(2):619-644.
[18]PUIG B,AKIAN J L.Non-Gaussian Simulation Using Hermite Polynomials Expansion and Maximum Entropy Principle[J].Probabilistic Engineering Mechanics,2004,19(4):293-305.
[19]李春明.优化方法[M].南京:东南大学出版社,2009.LI Chun-ming.Optimization Method[M].Nanjing:Southeast University Press,2009.
[20]肖柳青,周石鹏.随机模拟方法与应用[M].北京:北京大学出版社,2014.XIAO Liu-qing,ZHOU Shi-peng.Stochastic Simulation Methods and Its Applications[M].Beijing:Peking University Press,2014.
[2]李锦华,陈水生.非高斯随机过程模拟与预测的研究进展[J].华东交通大学学报,2011,28(6):1-6.LI Jin-hua,CHEN Shui-sheng.Advances in Simulation and Prediction of Non-Gaussian Stochastic Processes[J].Journal of East China Jiaotong University,2011,28(6):1-6.
[3]STEINWOLF A.Shaker Random Testing with Low Kurtosis:Review of the Methods and Application for Sigma Limiting[J].Shock and Vibration,2010,17(2):219-231.
[4]ROUILLARD V.Quantifying the Non-stationarity of Vehicle Vibrations with the Run Rest[J].Packaging Technology and Science,2014,27(3):203-219.
[5]RIZZI S A,PRZEKP A,TURNER T.On the Response of a Nonlinear Structure to High Kurtosis Non-Gaussian Random Loadings[R].NASA Technical Report,2013.
[6]SEONG S H,PETERKA J A.Computer Simulation of Non-Gaussian Multiple Wind Pressure Time Series[J].Journal of Wind Engineering&Industrial Aerodynamics,1997,72(1):95-105.
[7]SEONGS H,PETERKA J A.Digital Generation of Surface-pressure Fluctuations with Spiky Features[J].Journal of Wind Engineering&Industrial Aerodynamics,1998,73(2):181-192.
[8]蒋瑜,陶俊勇,王得志,等.一种新的非高斯随机振动数值模拟方法[J].振动与冲击,2012,31(19):169-173.JIANG Yu,TAO Jun-yong,WANG De-zhi,et al.ANovel Approach for Numerical Simulation of a Non-Gaussian Random Vibration[J].Journal of Vibration and Shock,2012,31(19):169-173.
[9]夏静,袁宏杰,徐如远.一种新的非高斯随机振动信号的模拟方法[J].北京航空航天大学学报,2019,45(2):366-372.XIA Jing,YUAN Hong-jie,XU Ru-yuan.A New Simulation Method of Non-Gaussian Random Vibration Signals[J].Journal of Beijing University of Aeronautics and Astronautics,2019,45(2):366-372.
[10]黄铭枫,孙轩涛,冯鹤,等.基于风压谱和Hermite模型的大跨干煤棚风压场数值模拟研究[J].振动与冲击,2018,37(23):111-119.HUANG Ming-feng,SUN Xuan-tao,FENG He,et al.Numerical Simulation of Wind Pressure Field of Long-span Lattice Structures Based on Wind Pressure Spectra and Hermite Model[J].Journal of Vibration and Shock,2018,37(23):111-119.
[11]徐飞,李传日,姜同敏,等.非高斯随机振动的模拟方法[J].北京航空航天大学学报,2014,40(9):1239-1244.XU Fei,LI Chuan-ri,JIANG Tong-min,et al.Simulation of Non-Gaussian Random Vibration[J].Journal of Beijing University of Aeronautics and Astronautics,2014,40(9):1239-1244.
[12]朱大鹏,李明月.铁路非高斯随机振动的数字模拟与包装件响应分析[J].包装工程,2016,37(1):1-5.ZHU Da-peng,LI Ming-yue.Digital Simulation of Non-Gaussian Random Vibration of Railway and Packaging System Response Analysis[J].Packaging Engineering,2016,37(1):1-5.
[13]蒋瑜,陈循,陶俊勇,等.指定功率谱密度、偏斜度和峭度值下的非高斯随机过程数字模拟[J].系统仿真学报,2006,18(5):1127-1130.JIANG Yu,CHEN Xun,TAO Jun-yong,et al.Numerically Simulating Non-Gaussian Random Processes with Specified PSD,Skewness and Kurtosis[J].Journal of System Simulation,2006,18(5):1127-1130.
[14]李继根,张新发.矩阵分析与计算[M].武汉:武汉大学出版社,2013.LI Ji-gen,ZHANG Xin-fa.Matrix Analysis and Computation[M].Wuhan:Wuhan University Press,2013.
[15]PHOON K K,HUANG S P,QUEK S T.Simulation of Second-order Processes Using Karhunen-loeve Expansion[J].Computers&Structures,2002,80(12):1049-1060.
[16]MOURELATOS Z P,MAJCHER M,PANDEY V,et al.Time-dependent Reliability Analysis Using the Total Probability Theorem[J].Journal of Mechanical Design2015,137(3):031405.
[17]XIU D,KARNIADAKIS G.The Wiener-askey Polynomial Chaos for Stochastic Differential Equations[J].SIAM Journal on Scientific Computing,2002,24(2):619-644.
[18]PUIG B,AKIAN J L.Non-Gaussian Simulation Using Hermite Polynomials Expansion and Maximum Entropy Principle[J].Probabilistic Engineering Mechanics,2004,19(4):293-305.
[19]李春明.优化方法[M].南京:东南大学出版社,2009.LI Chun-ming.Optimization Method[M].Nanjing:Southeast University Press,2009.
[20]肖柳青,周石鹏.随机模拟方法与应用[M].北京:北京大学出版社,2014.XIAO Liu-qing,ZHOU Shi-peng.Stochastic Simulation Methods and Its Applications[M].Beijing:Peking University Press,2014.