矩形薄板振动的随机分岔和可靠性研究
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摘要
矩形薄板是一种在各工程领域中广泛应用的工程材料,它具有大变形,易发生颤振的特点。虽然国内外学者对矩形薄板的振动及动力学特性进行了大量的研究,但其研究主要集中在确定系统中。而薄板受到的外激励中有很多随机因素,因此用随机非线性动力学理论研究矩形薄板的非线性运行规律及动力学特性和控制策略是十分必要的。本文主要完成了以下工作:
1、考虑外界随机激励对矩形薄板振动系统的影响,将外激励简化为高斯白噪声,利用弹性薄板理论和Galerkin方法建立了矩形薄板系统的随机非线性动力学模型,对该弱阻尼、弱激励的拟不可积Hamilton系统,首次运用拟不可积Hamilton理论和乘积遍历性定理计算了模型的Lyapunov指数,分析了系统的局部稳定性,通过对一维扩散过程的边界分析,得出该系统的全局稳定性条件;根据系统响应的联合概率密度和平稳概率密度以及不同参数条件,研究了该矩形薄板随机振动的随机Hopf分岔行为,并对分岔参数进行了分析,还通过数值仿真进行了验证。
2、运用随机平均法,将Hamilton函数表示为一维扩散过程,建立了可靠性函数和“首次穿越”时间的概率密度所满足的BK方程。结合初始条件和边界条件得到了数值结果,还分析了“首次穿越”现象对系统性态的影响,给出了该系统发生首次穿越现象后的动力学行为。
3、利用随机动态规划原理及随机平均法首次给出了以最大可靠性为目标的随机最优控制策略,说明了当控制力为有界函数时,随机最优控制即是Bang-Bang控制。并采用有限差分法对受控系统的可靠性函数、“首次穿越”损坏的概率密度函数所满足的偏微分方程进行了数值仿真。数值结果表明,随着控制约束力的增大,系统的安全性得到了增强,系统被破坏的可能性将会降低。
4、综合应用随机平均法、随机稳定性及可靠性理论研究开了具有摩擦边界的矩形薄板的随机振动问题。
Rectengular thin plate is a kind of basic structure which is widely used in many engneering fields. Thin plate is easy to have big deformations and obvious vibration. Lots of researches have been done at home and abroad, but most of them focus on certainty system. It is necessary to discuss the character of rectengular thin plate vibration system in the frame of stochastic nonlinear dynamics theory for the external excitation is obviously stochastic excitation. This dissertation studies the complex nonlinear phenomenon in the semi-active suspension system and the control strategy using stochastic nonlinear dynamics theory. The main content is as follows:
1. Applying the stochastic nonlinear dynamics theory to rectengular thin plate vibration system considering the impact of random factors. Simplifying in plate excitation as gauss white noise, establish v rectengular thin plate vibration system model based on stochastic nonlinear dynamics theory and Galerkin method, considering the law of force and acceleration. The max Lyapunov exponent is calculated by quasi non-integrable Hamiltonian theory and Oseledec multiplicative ergodic theory, the local stability conditions have been obtained; the global stability conditions have also been obtained by judging the modality of the singular boundary; the stochastic Hopf bifurcation is analyzed from the sharp change of stable and joint probability densities, and the parameter condition of stochastic Hopf bifurcation have been discussed through the numerical simulation.
2. A two-dimension stochastic nonlinear dynamical model of rectengular thin plate vibration system has been presented considering the stochastic factor. The Hamilton function is described as one dimension diffusion process by using stochastic average method, the global stability conditions is also obtained by judging the modality of the singular boundary; the backward Kolmogorov equation for reliability function and the generalized Pontryagin equation for conditional moment of the first-passage time have been established, the numerical results are given according to the classification of boundary conditions and initial conditions of these two equations. At last, the charactor of first-passage on system had been analyzed.
3. The optimal control strategy aimed to obtain the maximization of reliable function has been accessed by dynamic programming principles. The optimal control laws are“bang-bang”controls which are derived from the finit control force. Numerical simulations have been done with the backward Kolmogorov equation for reliability function and the generalized Pontryagin equation for conditional moment of the first-passage time which is under control by using finite difference method. The numerical results indicated that the security enhanced when the constrained control force increasd.
1、考虑外界随机激励对矩形薄板振动系统的影响,将外激励简化为高斯白噪声,利用弹性薄板理论和Galerkin方法建立了矩形薄板系统的随机非线性动力学模型,对该弱阻尼、弱激励的拟不可积Hamilton系统,首次运用拟不可积Hamilton理论和乘积遍历性定理计算了模型的Lyapunov指数,分析了系统的局部稳定性,通过对一维扩散过程的边界分析,得出该系统的全局稳定性条件;根据系统响应的联合概率密度和平稳概率密度以及不同参数条件,研究了该矩形薄板随机振动的随机Hopf分岔行为,并对分岔参数进行了分析,还通过数值仿真进行了验证。
2、运用随机平均法,将Hamilton函数表示为一维扩散过程,建立了可靠性函数和“首次穿越”时间的概率密度所满足的BK方程。结合初始条件和边界条件得到了数值结果,还分析了“首次穿越”现象对系统性态的影响,给出了该系统发生首次穿越现象后的动力学行为。
3、利用随机动态规划原理及随机平均法首次给出了以最大可靠性为目标的随机最优控制策略,说明了当控制力为有界函数时,随机最优控制即是Bang-Bang控制。并采用有限差分法对受控系统的可靠性函数、“首次穿越”损坏的概率密度函数所满足的偏微分方程进行了数值仿真。数值结果表明,随着控制约束力的增大,系统的安全性得到了增强,系统被破坏的可能性将会降低。
4、综合应用随机平均法、随机稳定性及可靠性理论研究开了具有摩擦边界的矩形薄板的随机振动问题。
Rectengular thin plate is a kind of basic structure which is widely used in many engneering fields. Thin plate is easy to have big deformations and obvious vibration. Lots of researches have been done at home and abroad, but most of them focus on certainty system. It is necessary to discuss the character of rectengular thin plate vibration system in the frame of stochastic nonlinear dynamics theory for the external excitation is obviously stochastic excitation. This dissertation studies the complex nonlinear phenomenon in the semi-active suspension system and the control strategy using stochastic nonlinear dynamics theory. The main content is as follows:
1. Applying the stochastic nonlinear dynamics theory to rectengular thin plate vibration system considering the impact of random factors. Simplifying in plate excitation as gauss white noise, establish v rectengular thin plate vibration system model based on stochastic nonlinear dynamics theory and Galerkin method, considering the law of force and acceleration. The max Lyapunov exponent is calculated by quasi non-integrable Hamiltonian theory and Oseledec multiplicative ergodic theory, the local stability conditions have been obtained; the global stability conditions have also been obtained by judging the modality of the singular boundary; the stochastic Hopf bifurcation is analyzed from the sharp change of stable and joint probability densities, and the parameter condition of stochastic Hopf bifurcation have been discussed through the numerical simulation.
2. A two-dimension stochastic nonlinear dynamical model of rectengular thin plate vibration system has been presented considering the stochastic factor. The Hamilton function is described as one dimension diffusion process by using stochastic average method, the global stability conditions is also obtained by judging the modality of the singular boundary; the backward Kolmogorov equation for reliability function and the generalized Pontryagin equation for conditional moment of the first-passage time have been established, the numerical results are given according to the classification of boundary conditions and initial conditions of these two equations. At last, the charactor of first-passage on system had been analyzed.
3. The optimal control strategy aimed to obtain the maximization of reliable function has been accessed by dynamic programming principles. The optimal control laws are“bang-bang”controls which are derived from the finit control force. Numerical simulations have been done with the backward Kolmogorov equation for reliability function and the generalized Pontryagin equation for conditional moment of the first-passage time which is under control by using finite difference method. The numerical results indicated that the security enhanced when the constrained control force increasd.
引文
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