给出不确定性激励下的动态响应边界——一种非随机振动分析方法
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摘要
提出了一种非随机振动分析方法,可给出系统在不确定性激励下的动态响应边界,从而为实验信息相对缺乏的不确定性振动分析及未来的可靠性设计提供一种新的计算工具.采用非概率凸模型过程而非传统的随机过程描述不确定性动态激励,仅需知道激励在任意时刻点的边界信息而非精确概率分布,从而有效降低对大样本量的依赖性.针对单自由度和多自由度系统,建立了相应的非随机振动分析算法,以求解系统在不确定性动态激励下的响应区间;另外,也给出了蒙特卡罗仿真方法,为非随机振动提供一种最为一般的分析工具.最后,通过3个数值算例验证了本文方法的有效性.非随机振动分析方法可以作为传统随机振动理论的补充,在工程不确定性结构动力学分析及结构可靠性设计领域发挥作用.
A non-random vibration analysis method is proposed in this paper, which calculates the dynamic response bounds of vibrational systems under time-variant uncertain excitations. It provides a prominsing alternative computational tool for uncertain vibration analysis in case of lack of experimental information and the corresponding reliability design in the future. The non-probabilistic convex model process, rather than traditional stochastic process, is used to describe uncertain dynamic excitations because the former needs only the bound information instead of precise probability distribution at any time point and therefore dependence on large sample size is weakened effectively. Based on the convex model process, non-random vibration analysis algorithms are formulated to obtain dynamic response bounds of SDOF system and MDOF system under time-variant uncertain excitations, respectively. Besides, corresponding Monte Carlo method is proposed to verify accuracy of the response bounds calculated and provide a general analytical tool for non-random vibration analysis. Finally, the feasibility of the non-random vibration analysis method is validated by several numerical examples. The proposed non-random vibration analysis method could provide a promising supplementfor random vibration theory, and thereby plays an important role in structural uncertain dynamic analysis and reliability design.for engineering problems.
A non-random vibration analysis method is proposed in this paper, which calculates the dynamic response bounds of vibrational systems under time-variant uncertain excitations. It provides a prominsing alternative computational tool for uncertain vibration analysis in case of lack of experimental information and the corresponding reliability design in the future. The non-probabilistic convex model process, rather than traditional stochastic process, is used to describe uncertain dynamic excitations because the former needs only the bound information instead of precise probability distribution at any time point and therefore dependence on large sample size is weakened effectively. Based on the convex model process, non-random vibration analysis algorithms are formulated to obtain dynamic response bounds of SDOF system and MDOF system under time-variant uncertain excitations, respectively. Besides, corresponding Monte Carlo method is proposed to verify accuracy of the response bounds calculated and provide a general analytical tool for non-random vibration analysis. Finally, the feasibility of the non-random vibration analysis method is validated by several numerical examples. The proposed non-random vibration analysis method could provide a promising supplementfor random vibration theory, and thereby plays an important role in structural uncertain dynamic analysis and reliability design.for engineering problems.
引文
1徐全智.随机过程及应用.北京:高等教育出版社,2013.6(Xu Quanzhi.Stochastical Process and Applications.Beijing:Higher Education Press,2013.6(in Chinese))
2 Crandall SH.Random Vibration.Cambridge:The MIT Press,1958
3 Lin YK.Probabilistic Theory of Structural Dynamics.New York:Mc Graw-Hill,1967
4 Morrow CT,Muchmore RB.Shortcomings of present methods of measuring and simulating vibration environments.ASME J Appl Mech,1955,22:367-371
5 Kiureghian AD.A response spectrum method for random vibration analysis of MDF systems.Earthquake Engineering&Structural Dynamics,1981,9(5):419-435
6林家浩,张亚辉.随机振动的虚拟激励法.北京:科学出版社,2004(Lin Jiahao,Zhang Yahui.Pseudo Excitation Method of Random Vibration.Beijing:Science Press,2004(in Chinese))
7 Caprani CC.Application of the pseudo-excitation method to assessment of walking variability on footbridge vibration.Comput Struct,2014,132:43-54
8 Zhang YW,Zhao Y,Zhang YH,et al.Riding comfort optimization of railway trains based on pseudo-excitation method and symplectic method.J Sound Vib,2013,332(21):5255-5270
9 Caughey TK.Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation.Journal of the Acoustical Society of America,1963,35(11):1683
10 Fuller AT.Analysis of nonlinear stochastic systems by means of the Fokker-Planck equation.International Journal of Control,1969,9(6):603-655
11 Crandall SH.Perturbation techniques for random vibration of nonlinear systems.J Acoust Soc Am,1963,35(11):1700-1705
12 Roberts JB,Spanos PD.Stochastic averaging:an approximate method of solving random vibration problems.International Journal of Non-Linear Mechanics,1986,21(2):111-134.
13 Zhu WQ.Stochastic averaging methods in random vibration.Appl.Mech.Rev,1988,41(5):189-199.
14 Zhu WQ.Recent developments and applications of the stochastic averaging method in random vibration.Appl Mech Rev,1996,49(10S):S72-S80
15 Iwan WD,Mason AB.Equivalent linearization for systems subjected to non-stationary random excitation.International Journal of Non-Linear Mechanics,1980,15(2):71-82.
16 He JH.Some asympotic methods for strongly nonlinear equations.International Journal of Modern Physics B,2012,20(10):1141-1199
17 Andrieu-Renaud C,Sudret B,Lemaire M.The PHI2 method:a way to compute time-variant reliability.Reliab Eng Syst Safety,2004,84(1):75-86
18 Zhang YW,Wen BC,Liu QL.First passage of uncertain single degree-of-freedom nonlinear oscillators.Comput Meth Appl Mech Eng,1998,165(1):223-231
19 Beck AT.The random barrier-crossing problem.Probabilist Eng Mech,2008,23:134-145
20 Conte JP,Peng BF.Fully nonstationary analytical earthquake ground-motion model.ASCE J Engng Mech,1997,123:15-24
21李杰,陈建兵.随机结构动力反应分析的概率密度演化方法.力学学报,2003,35(4):437-442(Li Jie,Chen Jianbing.The probability density evolution method for the analysis of stochastic structural dynamic response.Chinese Journal of Theoretical and Applied Mechanics,2003,35(4):437-442(in Chinese))
22李杰,陈建兵.随机结构动力可靠度分析的概率密度演化方法.振动工程学报,2004,17(2):121-125(Li Jie,Chen Jianbing.The probability density evolution method for the analysis of stochastic structural dynamic reliability.Journal of Vibration Engineering,2004,17(2):121-125(in Chinese))
23 Jiang C,Huang XP,Han X,et al.A time-variant reliability analysis method based on stochastic process discretization.Journal of Mechanical Design,2014,136(9):543-552
24 Ben-Haim Y,Elishakoff I.Convex Models of Uncertainties in Applied Mechanics.Amsterdam:Elsevier Science Publisher,1990
25 Jiang C,Ni BY,Han X,et al.Non-probabilistic convex model process:A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems.Comput Methods Appl Mech Engrg,2014,268:656-676
26 Jiang C,Han X,Lu GY,et al.Correlation analysis of nonprobabilistic convex model and corresponding structural reliability technique.Comput Methods Appl Mech Engrg,2011,200(33-36):2528-2546
27 Jiang C,Ni BY,Liu NY,et al.A non-random vibration analysis method based on an interval process model,submitted
28 Wang XJ,Wang L,Elishakoff I,et al.Probability and convexity concepts are not antagonistic.Acta Mech,2011,219:45-64
29 Kang Z,Luo YJ.Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex model.Computer Methods in Applied Mechanics&Engineering,2009,198:3228-3238
30 Newmark NM.A method of computation for structural dynamics.Journal of the Engineering Mechanics Division,1959,85(3):67-94
31 Bathe KJ,Wilson EL.Stability and accuracy analysis of direct integration methods.Earthquake Engineering&Structural Dynamics,1972,1(3):283-291
32 Bathe KJ,Wilson EL.Numerical Methods in Finite Element Analysis.New Jersey:New Jersey Prentice Hall,1976
33 Everett Iii H.Generalized lagrange multiplier method for solving problems of optimum allocation of resources.Operations Research,1963,11(3):399-417
34余志生.汽车理论(第二版).北京:机械工业出版社,1989:169-215(Yu Zhisheng.Automobile Theory(2nd edn).Beijing:China Machine Press,1989:169-215(in Chinese))
35 Long XY,Elishakoff I,Jiang C,et al.Notes on random vibration of a vehicle model and other discrete systems possessing repeated natural frequencies.Archive of Applied Mechanics,2014,84(8):1091-1101
2 Crandall SH.Random Vibration.Cambridge:The MIT Press,1958
3 Lin YK.Probabilistic Theory of Structural Dynamics.New York:Mc Graw-Hill,1967
4 Morrow CT,Muchmore RB.Shortcomings of present methods of measuring and simulating vibration environments.ASME J Appl Mech,1955,22:367-371
5 Kiureghian AD.A response spectrum method for random vibration analysis of MDF systems.Earthquake Engineering&Structural Dynamics,1981,9(5):419-435
6林家浩,张亚辉.随机振动的虚拟激励法.北京:科学出版社,2004(Lin Jiahao,Zhang Yahui.Pseudo Excitation Method of Random Vibration.Beijing:Science Press,2004(in Chinese))
7 Caprani CC.Application of the pseudo-excitation method to assessment of walking variability on footbridge vibration.Comput Struct,2014,132:43-54
8 Zhang YW,Zhao Y,Zhang YH,et al.Riding comfort optimization of railway trains based on pseudo-excitation method and symplectic method.J Sound Vib,2013,332(21):5255-5270
9 Caughey TK.Derivation and application of the Fokker-Planck equation to discrete nonlinear dynamic systems subjected to white random excitation.Journal of the Acoustical Society of America,1963,35(11):1683
10 Fuller AT.Analysis of nonlinear stochastic systems by means of the Fokker-Planck equation.International Journal of Control,1969,9(6):603-655
11 Crandall SH.Perturbation techniques for random vibration of nonlinear systems.J Acoust Soc Am,1963,35(11):1700-1705
12 Roberts JB,Spanos PD.Stochastic averaging:an approximate method of solving random vibration problems.International Journal of Non-Linear Mechanics,1986,21(2):111-134.
13 Zhu WQ.Stochastic averaging methods in random vibration.Appl.Mech.Rev,1988,41(5):189-199.
14 Zhu WQ.Recent developments and applications of the stochastic averaging method in random vibration.Appl Mech Rev,1996,49(10S):S72-S80
15 Iwan WD,Mason AB.Equivalent linearization for systems subjected to non-stationary random excitation.International Journal of Non-Linear Mechanics,1980,15(2):71-82.
16 He JH.Some asympotic methods for strongly nonlinear equations.International Journal of Modern Physics B,2012,20(10):1141-1199
17 Andrieu-Renaud C,Sudret B,Lemaire M.The PHI2 method:a way to compute time-variant reliability.Reliab Eng Syst Safety,2004,84(1):75-86
18 Zhang YW,Wen BC,Liu QL.First passage of uncertain single degree-of-freedom nonlinear oscillators.Comput Meth Appl Mech Eng,1998,165(1):223-231
19 Beck AT.The random barrier-crossing problem.Probabilist Eng Mech,2008,23:134-145
20 Conte JP,Peng BF.Fully nonstationary analytical earthquake ground-motion model.ASCE J Engng Mech,1997,123:15-24
21李杰,陈建兵.随机结构动力反应分析的概率密度演化方法.力学学报,2003,35(4):437-442(Li Jie,Chen Jianbing.The probability density evolution method for the analysis of stochastic structural dynamic response.Chinese Journal of Theoretical and Applied Mechanics,2003,35(4):437-442(in Chinese))
22李杰,陈建兵.随机结构动力可靠度分析的概率密度演化方法.振动工程学报,2004,17(2):121-125(Li Jie,Chen Jianbing.The probability density evolution method for the analysis of stochastic structural dynamic reliability.Journal of Vibration Engineering,2004,17(2):121-125(in Chinese))
23 Jiang C,Huang XP,Han X,et al.A time-variant reliability analysis method based on stochastic process discretization.Journal of Mechanical Design,2014,136(9):543-552
24 Ben-Haim Y,Elishakoff I.Convex Models of Uncertainties in Applied Mechanics.Amsterdam:Elsevier Science Publisher,1990
25 Jiang C,Ni BY,Han X,et al.Non-probabilistic convex model process:A new method of time-variant uncertainty analysis and its application to structural dynamic reliability problems.Comput Methods Appl Mech Engrg,2014,268:656-676
26 Jiang C,Han X,Lu GY,et al.Correlation analysis of nonprobabilistic convex model and corresponding structural reliability technique.Comput Methods Appl Mech Engrg,2011,200(33-36):2528-2546
27 Jiang C,Ni BY,Liu NY,et al.A non-random vibration analysis method based on an interval process model,submitted
28 Wang XJ,Wang L,Elishakoff I,et al.Probability and convexity concepts are not antagonistic.Acta Mech,2011,219:45-64
29 Kang Z,Luo YJ.Non-probabilistic reliability-based topology optimization of geometrically nonlinear structures using convex model.Computer Methods in Applied Mechanics&Engineering,2009,198:3228-3238
30 Newmark NM.A method of computation for structural dynamics.Journal of the Engineering Mechanics Division,1959,85(3):67-94
31 Bathe KJ,Wilson EL.Stability and accuracy analysis of direct integration methods.Earthquake Engineering&Structural Dynamics,1972,1(3):283-291
32 Bathe KJ,Wilson EL.Numerical Methods in Finite Element Analysis.New Jersey:New Jersey Prentice Hall,1976
33 Everett Iii H.Generalized lagrange multiplier method for solving problems of optimum allocation of resources.Operations Research,1963,11(3):399-417
34余志生.汽车理论(第二版).北京:机械工业出版社,1989:169-215(Yu Zhisheng.Automobile Theory(2nd edn).Beijing:China Machine Press,1989:169-215(in Chinese))
35 Long XY,Elishakoff I,Jiang C,et al.Notes on random vibration of a vehicle model and other discrete systems possessing repeated natural frequencies.Archive of Applied Mechanics,2014,84(8):1091-1101