随机与谐和激励联合作用下非线性系统的有限差分解
|
摘要
研究了高斯白噪声与谐和激励联合作用下非线性系统对应FPK方程的瞬态解。基于九点隐式有限差分格式,给出FPK方程的差分数值解,并应用于4类不同的非线性振子,求得了相应的瞬态解,并研究了边缘概率密度函数和联合概率密度函数随时间的演化历程。
In this paper,the transient probability density function of nonlinear system subjected to harmonic and stochastic excitations is investigated. Based on the implicit finite difference method,the transient probability density function of nonlinear system is proposed. The non-stationary response of four types of nonlinear oscillators are employed. The evolution of marginal probability density function and joint probability density function is discussed.
In this paper,the transient probability density function of nonlinear system subjected to harmonic and stochastic excitations is investigated. Based on the implicit finite difference method,the transient probability density function of nonlinear system is proposed. The non-stationary response of four types of nonlinear oscillators are employed. The evolution of marginal probability density function and joint probability density function is discussed.
引文
[1]朱位秋.非线性随机动力学与控制-Hamilton理论体系框架[M].北京:科学出版社,2003.
[2]Langley R S.A finite element method for the statistics of non-linear random vibration[J].J Sound Vib,1985,101:41-54.
[3]Sun J Q,Hsu C S.First-passage time probability of nonlinear stochastic systems by generalized cell mapping method[J].J Sound Vib,1988,124:233-248.
[4]Sun J Q,Hsu C S.The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation[J].ASME J Appl Mech,1990,57:1018-1025.
[5]Yu J S,Cai G Q,Lin Y K.A new path integration procedure based on Gauss-Legendre scheme[J].Int J Non-Linear Mech,1997,32:759-768.
[6]Xie W X,Xu W,Cai L.Study of the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations by path integration[J].Appl Math Comput,2006,172:1212-1224.
[7]Benaroya H,Rehak M.Finite element methods in probabilistic structural analysis:A selectiwe rewiew[J].Appl Mech Rev,1988,41:201-213.
[8]Zorzano M P,Mais H,Vazquez L.Numerical solution for Fokker-Planck equations in accelerators[J].Physica D,1998,113:379-381.
[9]Zorzano M P,Mais H,Vazquez L.Numerical solution of two dimensional Fokker-Planck equations[J].Appl Math Comp,1999,98:109-117.
[10]Kumar P,Narayanan S.Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems[J].Sādhanā,2006,31:445-461.
[11]黄志龙,张丽强.用差分法与超松弛迭代法求高维平稳FPK方程的解[J].计算力学学报,2008,25:177-182.
[12]王文杰,徐伟.随机外激非线性系统FPK方程的四阶中心C-N型隐式差分解[J].动力学与控制学报,2011(9):139-142.
[13]Sepehriana B,Radpoorb M K.Numerical solution of non-linear Fokker Planck equation using finite differences method and the cubic spline functions[J].Appl Math Comp,2015,262:187-190.
[14]孙鹏,刘建华,姜文安,等.二阶FPK方程的概率密度演化分析[J].江西科学,2018,36(5):709-715.
[2]Langley R S.A finite element method for the statistics of non-linear random vibration[J].J Sound Vib,1985,101:41-54.
[3]Sun J Q,Hsu C S.First-passage time probability of nonlinear stochastic systems by generalized cell mapping method[J].J Sound Vib,1988,124:233-248.
[4]Sun J Q,Hsu C S.The generalized cell mapping method in nonlinear random vibration based upon short-time Gaussian approximation[J].ASME J Appl Mech,1990,57:1018-1025.
[5]Yu J S,Cai G Q,Lin Y K.A new path integration procedure based on Gauss-Legendre scheme[J].Int J Non-Linear Mech,1997,32:759-768.
[6]Xie W X,Xu W,Cai L.Study of the Duffing-Rayleigh oscillator subject to harmonic and stochastic excitations by path integration[J].Appl Math Comput,2006,172:1212-1224.
[7]Benaroya H,Rehak M.Finite element methods in probabilistic structural analysis:A selectiwe rewiew[J].Appl Mech Rev,1988,41:201-213.
[8]Zorzano M P,Mais H,Vazquez L.Numerical solution for Fokker-Planck equations in accelerators[J].Physica D,1998,113:379-381.
[9]Zorzano M P,Mais H,Vazquez L.Numerical solution of two dimensional Fokker-Planck equations[J].Appl Math Comp,1999,98:109-117.
[10]Kumar P,Narayanan S.Solution of Fokker-Planck equation by finite element and finite difference methods for nonlinear systems[J].Sādhanā,2006,31:445-461.
[11]黄志龙,张丽强.用差分法与超松弛迭代法求高维平稳FPK方程的解[J].计算力学学报,2008,25:177-182.
[12]王文杰,徐伟.随机外激非线性系统FPK方程的四阶中心C-N型隐式差分解[J].动力学与控制学报,2011(9):139-142.
[13]Sepehriana B,Radpoorb M K.Numerical solution of non-linear Fokker Planck equation using finite differences method and the cubic spline functions[J].Appl Math Comp,2015,262:187-190.
[14]孙鹏,刘建华,姜文安,等.二阶FPK方程的概率密度演化分析[J].江西科学,2018,36(5):709-715.