非线性实噪声系统随机稳定性和随机分岔研究

发布时间：2018-01-25 02:46

本文关键词： 随机稳定性随机颤振矩Lyapunov指数特征谱展开式 L.Arnold摄动法分数阶　出处：《南京航空航天大学》2016年博士论文　论文类型：学位论文

【摘要】：随机动力系统的稳定性一直是随机动力学理论研究的焦点问题之一,在航空航天工程、船舶工程、车辆工程、工业与民用建筑工程和国防工程中均有着广泛的应用。本学位论文分别对三维中心流形上的余维二分岔系统、二元机翼、含有分数阶阻尼的单自由度线性振子与粘弹性壁板在高斯实噪声参数激励下的随机稳定性——矩Lyapunov指数和随机分岔行为进行了研究,其主要内容如下:首先对三维中心流形上的余维二分岔系统在遍历实噪声参数激励下的矩Lyapunov指数进行了研究。其中遍历实噪声被设定为一以n维Ornstein-Uhlenbeck(O-U)过程为变量的可积标量函数。为了使噪声模型具有更好的一般性,本文放松了噪声所需满足的条件——噪声的强混条件(strong mixing condition)和细致平衡条件(detailed balance condition)。基于Fokker-Planck算子以及其伴随算子的特征谱展开式和L.Arnold摄动法得到系统有限的p阶矩Lyapunov指数渐近的近似解。进一步,将此近似解与数值仿真结果进行了对比并证实了方法的有效性。最终,通过p阶矩Lyapunov指数的近似解析解对系统各个参数的变化以及不同的噪声对随机稳定性的影响进行了分析。其次,研究了随机气流作用下二元机翼的随机颤振行为。其中,湍流扰动被模拟为遍历的实噪声,二元机翼在随机气流作用下的随机颤振行为模型化为四维系统的随机分岔和随机稳定性问题。通过使用L.Arnold摄动法与两类算子的特征谱展开式,得到了系统矩Lyapunov指数的渐近近似解,并与数值仿真结果对比以证实了此方法的有效性。最终,基于矩Lyapunov指数的解析结果,对二元机翼的各个系统参数对随机稳定性的影响进行了分析,并指出在一定情况下,噪声可增强系统的稳定性。第四章研究了一个受实噪声以及谐和周期载荷联合参数激励的单自由度分数阶阻尼线性振子的随机稳定性问题。首先通过三角变换,对分数阶导数项进行了近似,从而使得变换后的代数式不再含有分数阶导数。进一步应用L.Arnold摄动法以及对Fokker-Planck算子以及其伴随算子的特征谱展开式,得到系统随机稳定性的两个重要指标:矩Lyapunov指数和最大Lyapunov指数的渐近近似解。基于这两个指标的结果,对分数阶导数的阶数?对系统随机稳定性的影响进行了分析与对比,最终发现:由于分数阶的导数的引入,系统的固有频率?也会对系统的随机稳定性产生重要影响。本学位论文的第五章研究了粘弹性壁板的随机颤振问题。由于气流力学特征的分散性,常常被当成随机激励来处理。此外,本文中壁板的粘弹性物理特征由分数阶Kelvin-Voigt本构关系来描述。首先,应用活塞理论得到粘弹性壁板在随机激励下的前两阶模态,并因此得到四维系统控制方程——随机微分方程。再应用两类算子的特征谱展开式与L.Arnold摄动法分别得到系统在非共振情况下与共振情况下的矩Lyapunov指数与最大Lyapunov指数。基于此,最终对该系统的随机稳定性进行了系统的分析,并对分数阶Kelvin-Voigt本构关系的引入而对系统随机稳定性的影响进行了详细的讨论。
[Abstract]:The stability of stochastic dynamical systems has always been a focus of research on the theory of stochastic dynamics, in aerospace engineering, marine engineering, vehicle engineering, industrial and civil engineering and national defense engineering is widely used. This thesis of the three-dimensional flow center of codimension two bifurcation system on the form, two yuan wing, single degree of freedom linear vibrator and viscoelastic panel with fractional damping in Gauss noise parameters under the excitation of stochastic stability - moment Lyapunov index and stochastic bifurcation behavior are studied, the main contents are as follows: first, for the moment Lyapunov index in the real noise excitation parameters of traverse of codimension two bifurcation system three-dimensional center manifold was studied. Which traverses the real noise is set to an n-dimensional Ornstein-Uhlenbeck (O-U) process for variable integrable scalar function. In order to make the model has better noise In general, the relaxation required for noise conditions, noise strong mixing conditions (strong mixing condition) and the detailed balance condition (detailed balance condition). Fokker-Planck operator and its adjoint operator feature spectrum expansion and L.Arnold perturbation approximate solution method to get the P moment Lyapunov index system based on the asymptotic finite further, the approximate solutions and the numerical simulation results were compared and confirmed the effectiveness of the method. Finally, the P moment Lyapunov index of the approximate analytical solution of changes of system parameters and different noise on stochastic stability are analyzed. Secondly, the random behavior of flutter wing two yuan random under the action of air flow. The turbulence is modeled as the real noise ergodic, stochastic flutter behavior model of two yuan in the wing under the action of random airflow into four-dimensional system Stochastic bifurcation and stochastic stability problem. Features by using the L.Arnold perturbation method and two kinds of operator spectrum expansion, asymptotic system moment Lyapunov index approximation, and compared with the numerical simulation results to demonstrate the effectiveness of this method. Finally, the analytical results of moment based on Lyapunov index, effects of various parameters on the system two Yuan wing the stochastic stability are analyzed, and pointed out that in certain circumstances, the noise can enhance the stability of the system. The fourth chapter studies the stochastic stability problem of a real noise and harmonic periodic load combined with parametric excitation of single degree of freedom linear fractional order damping oscillator. Firstly by trigonometric transform, approximate to the fractional order derivative, which makes the algebraic transformation after no fractional derivative. Further application of L.Arnold perturbation method and the Fokker-Planck operator and its adjoint Operator characteristic spectrum expansion, two important indexes system of stochastic stability obtained: asymptotic moment Lyapunov index and Lyapunov index of the approximate solution. The two indexes based on the results, the order of fractional derivative? Influence on the system of stochastic stability analysis and comparison, it was found that due to the introduction of fractional derivative order of the natural frequency of the system? Will also have an important impact on the stability of stochastic system. In this thesis, the fifth chapter studies the random flutter of viscoelastic panel. The dispersion due to airflow mechanics characteristics, are often considered to deal with random excitation. In addition, the physical characteristics of the viscoelastic panel by fractional in order to describe the constitutive relations of Kelvin-Voigt. Firstly, using piston theory get viscoelastic panel under random excitation of the first two order modes, and thus get a four-dimensional system - random control equation Differential equation. Feature and application of two types of operator spectrum expansion and L.Arnold perturbation method in the system under the condition of non resonance and resonance case moment Lyapunov index and Lyapunov index were obtained. Based on this, the end of the system stochastic stability analysis of the system, and the fractional Kelvin-Voigt constitutive relation introduction and influence on the system of stochastic stability are discussed in detail.

【学位授予单位】：南京航空航天大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O175

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