【作者】 黄勇；

【导师】 刘先斌；

【作者基本信息】 南京航空航天大学 ， 一般力学与力学基础， 2012， 博士

【摘要】 气动弹性系统的颤振问题本身是难度很大的科学问题，其理论在航空航天工程、桥梁工程、建筑工程和机械工程等领域有着非常重要的应用。随着研究的深入，人们对于风激振动中普遍存在的随机振动现象无法回避，而随机动力系统理论的发展使得对气动弹性系统的随机颤振问题的研究成为可能，并逐步发展为该领域的研究热点之一。本学位论文基于随机动力系统理论，通过理论分析和数值仿真相结合的方法，研究了气动弹性系统在随机噪声激励下的动气动弹性问题。同时结合各类数据图，试图给出气动弹性系统在随机激励下动态特性的直观描述。论文重点研究了受不同噪声参激的二元机翼颤振系统和黏弹性板颤振系统的随机稳定性问题。为解决高维随机动力系统的降维问题，本文将随机中心流形约化方法推广至高维气弹系统。此外针对工程中希望实现的对随机颤振的有效控制问题，本文分别设计了基于二阶矩和最大Lyapunov指数的控制策略，分析结果表明两者均可以取得很好的控制效果。本文主要包含以下几部分：1）研究了受高斯白噪声参数激励的二元机翼颤振系统响应的统计规律。通过采用Monte Carlo数值仿真，给出系统响应的多样本统计特性以及系统的最大Lyapunov指数，进一步分析发现随机动力系统拥有更为丰富的动力学行为，同时随机颤振点——概率1意义的随机分岔点一般位于确定性颤振点之前。通过最大Lyapunov指数对样本稳定性的判断进一步验证了上述结论；2）研究了宽带噪声和非高斯色噪声作用下黏弹性壁板颤振系统的随机稳定性。通过求解黏弹性壁板颤振系统矩Lyapunov指数的渐近解析表达式，获得各种系统参数与矩稳定和样本稳定之间的关系，对于黏弹性板的随机颤振机理有了更为清晰的认识。通过比较不同类型的噪声对黏弹性壁板颤振系统随机稳定性的影响，认清了不同的噪声类型对此类非线性随机动力系统的不同作用规律；3）研究了宽带噪声作用下二元机翼颤振系统的随机稳定性。通过求解该随机动力系统的矩Lyapunov指数的近似解析表达式，获得了二元机翼的随机稳定性与各系统参数之间的关系，对二元机翼随机颤振系统的样本稳定和矩稳定也有了全面的认识，进一步明确了二元机翼的随机颤振机理；4）研究了突风作用下二元机翼颤振系统的随机中心流形约化方法。通过采用随机中心流形定理的约化方法和规范形理论对系统Fokker-Plank-Komogorov方程进行约化，解出系统的平稳转移概率密度，并给出系统的分岔图。通过上述研究发现，随机中心流形约化方法对于高维随机动力系统的约化是可行的。此外还证明对于噪声外激励的二元机翼颤振系统不存在严格意义上的颤振点——概率1意义的随机分岔点；5）研究了宽带噪声和非高斯色噪声作用下二元机翼颤振系统的反馈控制问题。通过分析发现宽带噪声作用下的二元机翼颤振系统的一、二阶矩方程是闭合的，因此可以依据二阶矩方程进行最优控制策略的设计。非高斯色噪声作用下的二元机翼颤振系统的一、二阶矩并不闭合，因此无法采用二阶矩稳定进行控制，作者通过求解含有反馈控制的系统最大Lyapunov指数的解析表达式，并以此判断系统的样本稳定性，进而对系统实施有效控制。最后，通过对不同噪声作用下具有反馈控制的二元机翼系统分别进行数值仿真，发现这两种控制策略均具有很好的控制效果。更多还原

【Abstract】 The flutter phenomenon of aeroelastic system is difficult scientific problems by itself and has veryimportant applications in aerospace engineering, bridge engineering, construction engineering,mechanical engineering and other fields. With the gradual deepening of the study, the randomvibration in the wind-induced vibration can not be avoided. Then, with the development of thestochastic dynamical system, it becomes possible to study the stochastic flutter of aeroelastic systems,which is one of the research hotspots in this area for the last decade.Based on the stochastic dynamical system theory, the dynamic aeroelastic stabilities are investigatedunder random noise excitation by theoretical analysis and numerical simulation. Then articulating thecomprehensive pictures, the dynamic characteristic of aeroelastic system under random noiseexcitation can be intuitively described.In the present thesis, besides the stochastic stabilization of binary airfoil system and viscoelastic platesystem that are parametrically excited by different kinds of noise，center manifold reduction foraeroelastic system with noise exciting and the effectual stochastic control for stochastic flutter withthe second order moment and the largest Lyapunov exponent are also investigated. The main contentsare as follows：1） The dynamic response statistics of a two-dimensional airfoil system under gaussian white noiseare studied. Based on the Monte carlo numerical simulation, the statistic characteristics ofresponse and the maximum Lyapunov exponents of system are given and rich dynamic behaviorsof stochastic dynamical system are obtained. It’s also found that the stochastic flutter point (thestochastic bifurcation point in probability1sense) is always before the deterministic flutter point.2） The stochastic stabilities of a viscoelastic plate subjected to the excitation of wide band noise ornon-gaussian colored noise are investigated. To understand the shtochastic flutter mechanismmore clearly, the approximate analytic expansions of the moment Lyapunov exponents arederived to get the relationships between all system parameters and the stochastic stabilization ofthe system. At last, by comparing influences of different noise on the stochastic stabilization ofthe visicoelastic plate, the different dynamic behaviors of visicoelastic system driven by differentnoise can be obtained.3） The stochastic stabilization of a binary airfoil subjected to the excitation of wide band noises is discussed. Via deriving the approximate analytic expansion of the moment Lyapunov exponents,the relationships between all system parameters and the stochastic stabilization of binary airfoilsystem are obtained. Then, we have an overall understanding to the shtochastic flutter mechanism,the almost-sure stability and moment stability of two-dimensional airfoil system.4） Center manifold reduction for the flutter of airfoils with gust loading is studied. Via the ideas ofcenter-manifold reduction, normal form and the polar coordinates transformation, an explicitpresentation for the stationary probability density function is found as an approximate analyticalsolution of related Fokker-Plank-Komogorov equation. Eventually, we derive that centermanifold reduction for the high dimensional system is effectual and the D-bifurcation point/theflutter point will disappear under vertical gust.5） Astudy is conducted regarding the stability of a two-dimensional airfoil under different stochasticdisturbances with the feedback control. It’s found that the second order differential momentequations of the airfoil system under wide band noise excited are closed, so the effectual controlparameters can be obtained by searching the second moment system. But the second orderdifferential moment equations of the airfoil system under non-gaussian colored noise excited areunclosed, so the effectual control parameters can’t be obtained by searching the second momentsystem. Then a new stochastic controller is developed by the maximum Lyapunov exponent of thesystem with feedback control. Furthermore, the numerical simulations for the two stochasticcontrollers are included to visualize the good performance in flutter suppression.更多还原