非高斯随机激励下非线性系统的随机平均法
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摘要
本论文研究了非高斯随机激励下非线性系统的几种随机平均方法。
在第一章中,介绍了非高斯随机激励下动力学问题的研究概况和随机平均法的近期发展。
在第二章中,介绍了泊松白噪声及其相关概念、随机微分法则和Monte Carlo数值模拟方法。
在第三章中,提出了泊松白噪声激励下拟线性系统的广义FPK方程系数随机平均法,并对四种典型的拟线性系统,即van der Pol振子、Rayleigh振子、能量依赖阻尼系统以及幂律阻尼系统,通过摄动方法求得了平均广义FPK方程的近似平稳解。用Monte Carlo数值模拟方法对摄动解的可靠性进行了验证,发现解析解与数值模拟结果相一致,还发现在预测拟线性系统总能量的概率密度时,激励非高斯特性的影响是不可忽略的。
在第四章中,提出了一种预测拟不可积Hamilton系统在泊松白噪声激励下响应的随机平均法,推导了关于系统Hamilton函数转移概率密度的一维平均广义FPK方程,用摄动方法求解了系统平稳响应的概率密度,并将此方法应用于线性和非线性弹性耦合的van der Pol振子和二自由度碰撞振动系统,验证了方法的有效性。
在第五章中,从运动方程的标准形式出发,推导了非高斯平稳宽带随机激励下n维非线性动力系统的平均广义FPK方程,用级数形式给出了平均广义FPK方程在四阶近似下的系数表达式,并给出了相应的级数收敛条件,而后发现所得平均广义FPK方程可化为Stratonovich得到过的一维非线性动力系统的平均广义FPK方程,并且对泊松白噪声激励下的n维非线性动力系统,将所得平均广义FPK方程与Di Paola和Falsone得到的广义FPK方程的截断形式进行了比较,最后,由van der Pol变换生成的运动方程标准形式及相应的平均广义FPK方程出发,提出了非高斯平稳宽带随机激励下拟线性系统的随机平均法,得到了滤过泊松白噪声激励下拟线性系统响应的平稳概率密度,在算例中通过在不同参数下的Monte Carlo数值模拟验证了相应理论结果的正确性。
最后,在第六章中,对本文进行了总结并指出了后续研究工作的方向。
Several stochastic averaging methods for nonlinear systems subject to non-Gaussian random excitations are investigated in the present dissertation.
In chapter 1, the dynamical problems subject to non-Gaussian random excitations and recent development of stochastic averaging method are introduced.
In chapter 2, Poisson white noise and its related concepts, stochastic differential rules and Monte Carlo simulation methods are introduced.
In chapter 3, stochastic averaging method of coefficients of generalized Fokker-Planck- Kolmogorov (GFPK) equation for quasi linear systems subject to Poisson white noise is proposed. Approximate stationary solutions of the averaged GFPK equations are obtained by using the perturbation method for four typical quasi linear systems, i.e., van der Pol oscillator, Rayleigh oscillator, system with energy-dependent damping, and systems with power law damping. The reliability of the perturbation solution is assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and effect of non-Gaussianity of the excitation process is not negligible for predicting probability density of total energy of quasi linear system in most cases.
In chapter 4, a stochastic averaging method is proposed for predicting the response of multi-degree-of-freedom quasi-noninegrable-Hamiltonian systems subject to Poisson white noises. A one-dimensional averaged GFPK equation for the transition probability density of the Hamiltonian is derived and the probability density of the stationary response of the system is obtained by using the perturbation method. Two examples, two linearly and nonlinearly coupled van der Pol oscillators and 2-degree-of-freedom vibro-impact system, are given to illustrate the application and validity of the proposed method.
In chapter 5, the averaged GFPK equation for response of n-dimensional (n-d) nonlinear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, stochastic averaging method for quasi linear systems subject to non-Gaussian wide-band stationary random excitation is proposed by using the equation of motion generated from van der Pol transformations and its corresponding averaged GFPK equation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for the example are confirmed by using Monte-Carlo simulation for different parameter values.
Finally, in chapter 6, conclusion remarks of the present paper are made and the direction of future research is pointed out.
在第一章中,介绍了非高斯随机激励下动力学问题的研究概况和随机平均法的近期发展。
在第二章中,介绍了泊松白噪声及其相关概念、随机微分法则和Monte Carlo数值模拟方法。
在第三章中,提出了泊松白噪声激励下拟线性系统的广义FPK方程系数随机平均法,并对四种典型的拟线性系统,即van der Pol振子、Rayleigh振子、能量依赖阻尼系统以及幂律阻尼系统,通过摄动方法求得了平均广义FPK方程的近似平稳解。用Monte Carlo数值模拟方法对摄动解的可靠性进行了验证,发现解析解与数值模拟结果相一致,还发现在预测拟线性系统总能量的概率密度时,激励非高斯特性的影响是不可忽略的。
在第四章中,提出了一种预测拟不可积Hamilton系统在泊松白噪声激励下响应的随机平均法,推导了关于系统Hamilton函数转移概率密度的一维平均广义FPK方程,用摄动方法求解了系统平稳响应的概率密度,并将此方法应用于线性和非线性弹性耦合的van der Pol振子和二自由度碰撞振动系统,验证了方法的有效性。
在第五章中,从运动方程的标准形式出发,推导了非高斯平稳宽带随机激励下n维非线性动力系统的平均广义FPK方程,用级数形式给出了平均广义FPK方程在四阶近似下的系数表达式,并给出了相应的级数收敛条件,而后发现所得平均广义FPK方程可化为Stratonovich得到过的一维非线性动力系统的平均广义FPK方程,并且对泊松白噪声激励下的n维非线性动力系统,将所得平均广义FPK方程与Di Paola和Falsone得到的广义FPK方程的截断形式进行了比较,最后,由van der Pol变换生成的运动方程标准形式及相应的平均广义FPK方程出发,提出了非高斯平稳宽带随机激励下拟线性系统的随机平均法,得到了滤过泊松白噪声激励下拟线性系统响应的平稳概率密度,在算例中通过在不同参数下的Monte Carlo数值模拟验证了相应理论结果的正确性。
最后,在第六章中,对本文进行了总结并指出了后续研究工作的方向。
Several stochastic averaging methods for nonlinear systems subject to non-Gaussian random excitations are investigated in the present dissertation.
In chapter 1, the dynamical problems subject to non-Gaussian random excitations and recent development of stochastic averaging method are introduced.
In chapter 2, Poisson white noise and its related concepts, stochastic differential rules and Monte Carlo simulation methods are introduced.
In chapter 3, stochastic averaging method of coefficients of generalized Fokker-Planck- Kolmogorov (GFPK) equation for quasi linear systems subject to Poisson white noise is proposed. Approximate stationary solutions of the averaged GFPK equations are obtained by using the perturbation method for four typical quasi linear systems, i.e., van der Pol oscillator, Rayleigh oscillator, system with energy-dependent damping, and systems with power law damping. The reliability of the perturbation solution is assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and effect of non-Gaussianity of the excitation process is not negligible for predicting probability density of total energy of quasi linear system in most cases.
In chapter 4, a stochastic averaging method is proposed for predicting the response of multi-degree-of-freedom quasi-noninegrable-Hamiltonian systems subject to Poisson white noises. A one-dimensional averaged GFPK equation for the transition probability density of the Hamiltonian is derived and the probability density of the stationary response of the system is obtained by using the perturbation method. Two examples, two linearly and nonlinearly coupled van der Pol oscillators and 2-degree-of-freedom vibro-impact system, are given to illustrate the application and validity of the proposed method.
In chapter 5, the averaged GFPK equation for response of n-dimensional (n-d) nonlinear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, stochastic averaging method for quasi linear systems subject to non-Gaussian wide-band stationary random excitation is proposed by using the equation of motion generated from van der Pol transformations and its corresponding averaged GFPK equation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for the example are confirmed by using Monte-Carlo simulation for different parameter values.
Finally, in chapter 6, conclusion remarks of the present paper are made and the direction of future research is pointed out.
引文
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