加性非平稳激励下结构速度响应的FPK方程降维
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摘要
结构在随机激励下的非线性响应分析是具有高度挑战性的困难问题.对于白噪声或过滤白噪声激励,求解FPK方程将获得结构响应的精确解.遗憾的是,对于非线性多自由度系统,FPK方程难以直接求解.事实上,其数值解法严重受限于方程维度,而解析求解则仅适用于少数特定的系统,且多是稳态解.因此,将FPK方程进行降维,是求解高维随机动力响应分析问题的重要途径.本文针对幅值调制的加性白噪声激励下多自由度非线性结构的非平稳随机响应分析问题,将联合概率密度函数满足的高维FPK方程进行降维.针对结构速度响应概率密度函数求解,通过引入等价漂移系数,原FPK方程可转化为一维FPK型方程.建议了构造等价漂移系数的条件均值函数方法.进而,采用路径积分方法求解降维FPK型方程,得到速度概率密度函数的数值解答.结合单自由度Rayleigh振子、十层线性剪切型框架和非线性剪切型框架结构在幅值调制的加性白噪声激励下的非平稳速度响应求解,讨论了本文方法的精度和效率,验证了其有效性.
The nonlinear response analysis of structures subjected to random excitation is a highly challenging problem.The solution of FPK equation provides exact solutions to these problems. However, for nonlinear multi-degree-of-freedom systems, the direct solutions of the FPK equation is prohibitively difficult. Actually, the numerical solutions are strictly limited by the dimension of the equation, while the analytical solutions are available only for very few specific systems,and most of them are steady-state solution. Therefore, reducing the dimension of the FPK equation is an important way to solve the high-dimensional nonlinear dynamic response analysis problem. In the present paper, for the nonstationary response analysis of multi-degree-of-freedom nonlinear structures subjected to amplitude-modulated additive white noise,the high-dimensional FPK equation in terms of the joint probability density function is reduced in dimension. For the probability density function of velocity response, it is converted to a one-dimensional FPK-like equation through introducing the equivalent drift coefficient. The method of conditional mean function estimate is suggested to construct the equivalent drift coefficient. Afterwards, the numerical results of probability density function of velocity can be obtained by applying the path integration method to solve the dimension-reduced FPK-like equation. The accuracy and efficiency of the proposed method are discussed and verified through the numerical examples, including the non-stationary response analysis of velocity of a single-degree-of-freedom Rayleigh oscillator, a ten-story linear shear frame structure and a nonlinear shear frame structure subjected to amplitude-modulated additive white noise.
The nonlinear response analysis of structures subjected to random excitation is a highly challenging problem.The solution of FPK equation provides exact solutions to these problems. However, for nonlinear multi-degree-of-freedom systems, the direct solutions of the FPK equation is prohibitively difficult. Actually, the numerical solutions are strictly limited by the dimension of the equation, while the analytical solutions are available only for very few specific systems,and most of them are steady-state solution. Therefore, reducing the dimension of the FPK equation is an important way to solve the high-dimensional nonlinear dynamic response analysis problem. In the present paper, for the nonstationary response analysis of multi-degree-of-freedom nonlinear structures subjected to amplitude-modulated additive white noise,the high-dimensional FPK equation in terms of the joint probability density function is reduced in dimension. For the probability density function of velocity response, it is converted to a one-dimensional FPK-like equation through introducing the equivalent drift coefficient. The method of conditional mean function estimate is suggested to construct the equivalent drift coefficient. Afterwards, the numerical results of probability density function of velocity can be obtained by applying the path integration method to solve the dimension-reduced FPK-like equation. The accuracy and efficiency of the proposed method are discussed and verified through the numerical examples, including the non-stationary response analysis of velocity of a single-degree-of-freedom Rayleigh oscillator, a ten-story linear shear frame structure and a nonlinear shear frame structure subjected to amplitude-modulated additive white noise.
引文
1 Lin YK.Probabilistic Theory of Structural Dynamics.New York:McGraw-Hill Book Company,1967
2 Cukier RI,Lakatos-Lindenberg K,Shuler KE.Orthogonal polynomial solutions of the Fokker-Planck equation.Journal of Statistical Physics,1973,9(2):137-144
3 Lutes LD,Sarkani S.Random Vibrations:Analysis of Structural and Mechanical Systems.Oxford:Elsevier,2004
4 Li J,Chen JB.Stochastic Dynamics of Structures.Singapore:John Wiley&Sons,2009
5 Roberts JB,Spanos PD.Random Vibration and Statistical Linearization.Chichester:John Wiley&Sons Ltd,1990
6朱位秋.随机振动.北京:科学出版社,1992(Zhu Weiqiu.Random Vibration.Beijing:Science Press,1992(in Chinese))
7 Naess A,Moan T.Stochastic Dynamics of Marine Structures.Cambridge University Press,2013
8 Er GK.An improved closure method for analysis of nonlinear stochastic systems.Nonlinear Dynamics,1998,17(3):285-297
9 Risken H.The Fokker-Planck Equation:Methods of Solution and Applications.2nd ed.Berlin:Springer-Verlag,1989
10 Caughey TK,Ma F.The exact steady-state solution of a class of non-linear stochastic systems.International Journal of Non-Linear Mechanics,1982,17(3):137-142
11 Soize C.The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions.Singapore:World Scientific,1994
12朱位秋.非线性随机动力学与控制:Hamilton理论体系框架.北京:科学出版社,2003(Zhu Weiqiu.Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation.Beijing:Science Press,2003(in Chinese))
13刘俊,陈林聪,孙建桥.随机激励下滞迟系统的稳态响应闭合解.力学学报,2017,49(3):685-692(Liu Jun,Chen Lincong,Sun JianQiao.The closed-form solution of steady state response of hysteretic system under stochastic excitation.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):685-692(in Chinese))
14 Park BT,Petrosian V.Fokker-Planck equations of stochastic acceleration:A study of numerical methods.The Astrophysical Journal Supplement Series,1996,103:255
15 Spencer BF,Bergman LA.On the numerical solution of the FokkerPlanck equation for nonlinear stochastic systems.Nonlinear Dynamics,1993,4,357-72
16 Naess A,Hegstad BK.Response statistics of van der Pol oscillators excited by white noise.Nonlinear Dynamics,1994,5(3):287-297
17 Yu JS,Cai GQ,Lin YK.A new path integration procedure based on Gauss-Legendre scheme.International Journal of Non-Linear Mechanics,1997,32(4):759-768
18 Naess A,Moe V.Efficient path integration methods for nonlinear dynamic systems.Probabilistic Engineering Mechanics,2000,15(2):221-231
19 Kougioumtzoglou IA,Spanos PD.An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators.Probabilistic Engineering Mechanics,2012,28:125-131
20徐伟.非线性随机动力学的若干数值方法及应用.北京:科学出版社,2013(Xu Wei.Numerical Analysis Methods for Stochastic Dynamical System.Beijing:Science Press,2013(in Chinese))
21 Kougioumtzoglou IA,Di Matteo A,Spanos PD,et al.An efficient Wiener path integral technique formulation for stochastic response determination of nonlinear MDOF systems.Journal of Applied Mechanics,2015,82(10):101005
22 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
23 Huang ZL,Zhu WQ.Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations.Probabilistic Engineering Mechanics,2004,19(3):219-228
24 Er GK.Methodology for the solutions of some reduced FokkerPlanck equations in high dimensions.Annals of Physics,2011,523(3):247-258
25陈建兵,李杰.结构随机地震反应与可靠度的概率密度演化分析研究进展.工程力学,2014(4):1-10(Chen Jianbing,Li Jie.Probability density evolution method for stochastic seismic response and reliability of structures.Engineering Mechanics,2014(4):1-10(in Chinese))
26 Li J,Chen JB.The principle of preservation of probability and the generalized density evolution equation.Structural Safety,2008,30(1):65-77
27李杰,陈建兵.概率密度演化理论的若干研究进展.应用数学和力学,2017,38(1),32-43(Li Jie,Chen Jianbing.Some new advances in the probability density evolution method.Applied Mathematics and Mechanics,2017,38(1),32-43(in Chinese))
28 Chen JB,Yuan SR.Dimension reduction of the FPK equation via an equivalence of probability flux for additively excited systems.Journal of Engineering Mechanics,2014,140(11):04014088
29 Chen JB,Yuan SR.PDEM-based dimension-reduction of FPK equation for additively excited hysteretic nonlinear systems.Probabilistic Engineering Mechanics,2014,38:111-118
30 Chen JB,Lin PH.Dimension-reduction of FPK equation via equivalent drift coefficient.Theoretical and Applied Mechanics Letters,2014,4(1):013002
31 Chen JB,Rui ZM.Dimension-reduced FPK equation for additive white-noise excited nonlinear structures.Probabilistic Engineering Mechanics,2018,53:1-13
32刘晶波,谭辉,宝鑫等.土-结构动力相互作用分析中基于人工边界子结构的地震波动输入方法.力学学报,2018,50(1):32-43(Liu Jingbo,Tan Hui,Bao Xin,et al.The seismic wave input method for soil-structure dynamic interaction analysis based on the substructure of artificial boundaries.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):32-43(in Chinese))
33 Gardiner CW.Handbook of Stochastic Methods.Berlin,Springer,1985
34 Shinozuka M,Sato Y.Simulation of nonstationary random process.Journal of Engineering Mechanics,1967,91:11-40
35 Kwuimy CAK,Nbendjo BRN.Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection.Physics Letters A,2011,375(39):3442-3449
36 Wen YK.Approximate method for nonlinear random vibration.Journal of the Engineering Mechanics Division,1975,101(4):389-401
37公徐路,许鹏飞.含时滞反馈与涨落质量的记忆阻尼系统的随机共振.力学学报,2018,50(4):880-889(Gong Xulu,Xu Pengfei.Stochastic resonance of a memorial-damped system with time delay feedback and fluctuating mass.Chinese Journal of Theoretical and Applied Mechanics,2018,50(4):880-889(in Chinese))
2 Cukier RI,Lakatos-Lindenberg K,Shuler KE.Orthogonal polynomial solutions of the Fokker-Planck equation.Journal of Statistical Physics,1973,9(2):137-144
3 Lutes LD,Sarkani S.Random Vibrations:Analysis of Structural and Mechanical Systems.Oxford:Elsevier,2004
4 Li J,Chen JB.Stochastic Dynamics of Structures.Singapore:John Wiley&Sons,2009
5 Roberts JB,Spanos PD.Random Vibration and Statistical Linearization.Chichester:John Wiley&Sons Ltd,1990
6朱位秋.随机振动.北京:科学出版社,1992(Zhu Weiqiu.Random Vibration.Beijing:Science Press,1992(in Chinese))
7 Naess A,Moan T.Stochastic Dynamics of Marine Structures.Cambridge University Press,2013
8 Er GK.An improved closure method for analysis of nonlinear stochastic systems.Nonlinear Dynamics,1998,17(3):285-297
9 Risken H.The Fokker-Planck Equation:Methods of Solution and Applications.2nd ed.Berlin:Springer-Verlag,1989
10 Caughey TK,Ma F.The exact steady-state solution of a class of non-linear stochastic systems.International Journal of Non-Linear Mechanics,1982,17(3):137-142
11 Soize C.The Fokker-Planck Equation for Stochastic Dynamical Systems and Its Explicit Steady State Solutions.Singapore:World Scientific,1994
12朱位秋.非线性随机动力学与控制:Hamilton理论体系框架.北京:科学出版社,2003(Zhu Weiqiu.Nonlinear Stochastic Dynamics and Control in Hamiltonian Formulation.Beijing:Science Press,2003(in Chinese))
13刘俊,陈林聪,孙建桥.随机激励下滞迟系统的稳态响应闭合解.力学学报,2017,49(3):685-692(Liu Jun,Chen Lincong,Sun JianQiao.The closed-form solution of steady state response of hysteretic system under stochastic excitation.Chinese Journal of Theoretical and Applied Mechanics,2017,49(3):685-692(in Chinese))
14 Park BT,Petrosian V.Fokker-Planck equations of stochastic acceleration:A study of numerical methods.The Astrophysical Journal Supplement Series,1996,103:255
15 Spencer BF,Bergman LA.On the numerical solution of the FokkerPlanck equation for nonlinear stochastic systems.Nonlinear Dynamics,1993,4,357-72
16 Naess A,Hegstad BK.Response statistics of van der Pol oscillators excited by white noise.Nonlinear Dynamics,1994,5(3):287-297
17 Yu JS,Cai GQ,Lin YK.A new path integration procedure based on Gauss-Legendre scheme.International Journal of Non-Linear Mechanics,1997,32(4):759-768
18 Naess A,Moe V.Efficient path integration methods for nonlinear dynamic systems.Probabilistic Engineering Mechanics,2000,15(2):221-231
19 Kougioumtzoglou IA,Spanos PD.An analytical Wiener path integral technique for non-stationary response determination of nonlinear oscillators.Probabilistic Engineering Mechanics,2012,28:125-131
20徐伟.非线性随机动力学的若干数值方法及应用.北京:科学出版社,2013(Xu Wei.Numerical Analysis Methods for Stochastic Dynamical System.Beijing:Science Press,2013(in Chinese))
21 Kougioumtzoglou IA,Di Matteo A,Spanos PD,et al.An efficient Wiener path integral technique formulation for stochastic response determination of nonlinear MDOF systems.Journal of Applied Mechanics,2015,82(10):101005
22 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
23 Huang ZL,Zhu WQ.Stochastic averaging of quasi-integrable Hamiltonian systems under bounded noise excitations.Probabilistic Engineering Mechanics,2004,19(3):219-228
24 Er GK.Methodology for the solutions of some reduced FokkerPlanck equations in high dimensions.Annals of Physics,2011,523(3):247-258
25陈建兵,李杰.结构随机地震反应与可靠度的概率密度演化分析研究进展.工程力学,2014(4):1-10(Chen Jianbing,Li Jie.Probability density evolution method for stochastic seismic response and reliability of structures.Engineering Mechanics,2014(4):1-10(in Chinese))
26 Li J,Chen JB.The principle of preservation of probability and the generalized density evolution equation.Structural Safety,2008,30(1):65-77
27李杰,陈建兵.概率密度演化理论的若干研究进展.应用数学和力学,2017,38(1),32-43(Li Jie,Chen Jianbing.Some new advances in the probability density evolution method.Applied Mathematics and Mechanics,2017,38(1),32-43(in Chinese))
28 Chen JB,Yuan SR.Dimension reduction of the FPK equation via an equivalence of probability flux for additively excited systems.Journal of Engineering Mechanics,2014,140(11):04014088
29 Chen JB,Yuan SR.PDEM-based dimension-reduction of FPK equation for additively excited hysteretic nonlinear systems.Probabilistic Engineering Mechanics,2014,38:111-118
30 Chen JB,Lin PH.Dimension-reduction of FPK equation via equivalent drift coefficient.Theoretical and Applied Mechanics Letters,2014,4(1):013002
31 Chen JB,Rui ZM.Dimension-reduced FPK equation for additive white-noise excited nonlinear structures.Probabilistic Engineering Mechanics,2018,53:1-13
32刘晶波,谭辉,宝鑫等.土-结构动力相互作用分析中基于人工边界子结构的地震波动输入方法.力学学报,2018,50(1):32-43(Liu Jingbo,Tan Hui,Bao Xin,et al.The seismic wave input method for soil-structure dynamic interaction analysis based on the substructure of artificial boundaries.Chinese Journal of Theoretical and Applied Mechanics,2018,50(1):32-43(in Chinese))
33 Gardiner CW.Handbook of Stochastic Methods.Berlin,Springer,1985
34 Shinozuka M,Sato Y.Simulation of nonstationary random process.Journal of Engineering Mechanics,1967,91:11-40
35 Kwuimy CAK,Nbendjo BRN.Active control of horseshoes chaos in a driven Rayleigh oscillator with fractional order deflection.Physics Letters A,2011,375(39):3442-3449
36 Wen YK.Approximate method for nonlinear random vibration.Journal of the Engineering Mechanics Division,1975,101(4):389-401
37公徐路,许鹏飞.含时滞反馈与涨落质量的记忆阻尼系统的随机共振.力学学报,2018,50(4):880-889(Gong Xulu,Xu Pengfei.Stochastic resonance of a memorial-damped system with time delay feedback and fluctuating mass.Chinese Journal of Theoretical and Applied Mechanics,2018,50(4):880-889(in Chinese))