Generalization of Seide’s problem by the regulated stochastic linearization technique

In this paper, the nonlinear random vibration of a uniform beam clamped at both of its ends is investigated by the regulated stochastic linearization technique. A closed system of nonlinear algebraic equations for linearization coefficients is obtained using the frequency–response function matrix. Root mean-square response of maximum deflection of the beam that is obtained from the present technique is subsequently compared with that furnished by the conventional linearization. The numerical results show that the regulated stochastic linearization technique constitutes an excellent alternative to the classical linearization scheme for analyzing responses of the clamped–clamped beam.     

STOCHASTIC OPTIMAL CONTROL OF HYSTERETIC SYSTEMS UNDER EXTERNALLY AND PARAMETRICALLY RANDOM EXCITATIONS

A stochastic optimal control method for nonlinear hysteretic systems under exter-nally and/or parametrically random excitations is presented and illustrated with an example ofhysteretic column system.A hysteretic system subject to random excitation is first replaced bya nonlinear non-hysteretic stochastic system.An It stochastic differential equation for the to-tal energy of the system as a one-dimensional controlled diffusion process is derived by usingthe stochastic averaging method of energy envelope.A dynamical programming equation is thenestablished based on the stochastic dynamical programming principle and solved to yield the op-timal control force.Finally,the responses of uncontrolled and controlled systems are evaluatedto determine the control efficacy.It is shown by numerical results that the proposed stochasticoptimal control method is more effective and efficient than other optimal control methods.     

Random responses of a two-degree-of-freedom structure with strongly nonlinear coupling and parametric interaction

This paper investigates the random responses of a TDOF structure with strongly nonlinear coupling and parametric vibration. With the nonlinear coupling of inertia in the equations of motion of the system being removed by successive elimination, the non-Gaussian moment equation method (NGM) is applied and 69 moment equations are integrated with central cumulative truncation technique. The stochastic central difference-cum-statistical linearization method(SCD-SL) and the digital simulation method(DSM) are also used. A comparison of results by different methods are given and the SCD-SL method is the most efficient method. The project supported by National Natural Science Foundation of China     

非线性MEMS微梁的重心有理插值迭代配点法分析

文章利用重心有理插值迭代配点法分析计算非线性MEMS微梁问题。通过处理MEMS微梁的几何通过假设初始函数,将微梁非线性控制方程转换为线性化微分方程,建立逼近非线性微分方程的线性化迭代格式。采用重心有理插值配点法求解线性化微分方程,提出了数值分析MEMS微梁非线性弯曲问题的重心插值迭代配点法。给出了非线性微分方程的直接线性化和Newton线性化计算公式,详细讨论了非线性积分项的计算方法和公式。利用重心有理插值微分矩阵,建立了矩阵-向量化的重心插值迭代配点法的计算公式。数值算例结果表明,重心插值迭代配点法求解微梁非线性弯曲问题,具有计算公式简单、程序实施方便和计算精度高的特点。     

精细积分时域平均法和随机扩阶系统法

讨论含随机参数结构的动力响应的计算问题,发展了精细积分时域平均法（ＴＡＰＩＭ）,它可以用来计算确定性系统受到随机激励时的动力响应;结合随机扩阶系统方法与随机有限元法,将ＴＡＰＩＭ方法应用于计算随机参数结构的动力响应,取得了较好的结果。结出了数值算例,结果表明随机扩阶系统法,随机有限元法与精细积分时域平均法的结合是计算 随机参数结构动力响应的有效方法。     

Stochastic stability for nonlinear systems driven by Lévy noise

This paper is to investigate the stochastic stability for nonlinear systems with Lévy process based on Lyapunov exponents. A method of equivalent linearization is proposed to reduce and simplify the original systems. And the mean square responses are carried out to verify the effectiveness of the proposed approach. Then the Lyapunov exponents will be defined and derived to explore the stochastic stability, and two examples are presented to demonstrate the procedure of equivalent linearization and stochastic stability is considered for these two special examples. The results show that the technique of equivalent linearization can be used to study nonlinear systems excited by Lévy noise.     

Optimal bounded control for nonlinear stochastic smart structure systems based on extended Kalman filter

An optimal bounded control strategy for smart structure systems as controlled Hamiltonian systems with random excitations and noised observations is proposed. The basic dynamic equations for a smart structure system with smart sensors and actuators are firstly given. The nonlinear stochastic control system with noised observations is then obtained from the simplified smart structure system, and the system is expressed by generalized Hamiltonian equations with control, random excitation and dissipative forces. The optimal control problem for nonlinear stochastic systems with noised observations includes two parts: optimal state estimation and optimal response control based on estimated states, which are coupled each other. The probability density of optimally estimated systems has generally infinite dimensions based on the separation theorem. The proposed optimal control strategy gives an approximate separate solution. First, the optimally estimated system state is determined by the observations based on the extended Kalman filter, and the estimated nonlinear system with controls and stochastic excitations is obtained which has finite-dimensional probability density. Second, the dynamical programming equation for the estimated system is determined based on the stochastic dynamical programming principle. The control boundedness due to actuator saturation is considered, and the optimal bounded control law is obtained by the programming equation with the bounded control constraint. The optimal control depends on the estimated system state which is determined by noised observations. The proposed optimal bounded control strategy is finally applied to a single-degree-of-freedom nonlinear stochastic system with control and noised observation. The remarkable vibration control effectiveness is illustrated with numerical results. Thus the proposed optimal bounded control strategy is promising for application to nonlinear stochastic smart structure systems with noised observations.     

Stochastic transient analysis of thermal stresses in solids by explicit time-domain method

Stochastic heat conduction and thermal stress analysis of structures has received considerable attention in recent years. The propagation of uncertain thermal environments will lead to stochastic variations in temperature fields and thermal stresses. Therefore, it is reasonable to consider the variability of thermal environments while conducting thermal analysis. However, for ambient thermal excitations, only stationary random processes have been investigated thus far. In this study, the highly efficient explicit time-domain method(ETDM) is proposed for the analysis of non-stationary stochastic transient heat conduction and thermal stress problems. The explicit time-domain expressions of thermal responses are first constructed for a thermoelastic body. Then the statistical moments of thermal displacements and stresses can be directly obtained based on the explicit expressions of thermal responses. A numerical example involving non-stationary stochastic internal heat generation rate is investigated. The accuracy and efficiency of the proposed method are validated by comparison with the Monte-Carlo simulation.     

Tomographic diagnostics of technical materials and biological tissues using electric current

A mathematical model for impedance computer tomography methods is considered. The continuum formulation of the main problem is studied. Resolving integral equations are derived. A solution algorithm based on the Bubnov—Galerkin method with linearization of nonlinear resolving equations is developed. A numerical example is given, and numerical results are analyzed. Some drawbacks of the model are considered together with methods for avoiding them.     

非Gauss分布随机波浪力对海洋平台的非线性作用

为避免求解决定Maikov过程转移概率密度的Fokker—Planck方程，基于尺度分离的假设导出了一组描述非线性海洋平台受非Gauss分布随机波浪载荷作用所产生响应的矩量的常微分方程组。矩量方程清楚地反映出分别对应随机载荷和结构响应的两种不同统计特性的相互关系。由于矩量方程不依赖载荷的概率分布的具体细节，以它来模拟随机激励作用下的非线性系统将免于Monte Carlo方法所面临的正确模拟载荷概率分布的困难任务。将摄动法用于矩量方程可使线性化不再需要，这样就不会因为线性化而产生不可预料的误差。     

考虑参数不确定性的非线性梁随机振动分析

非线性系统的随机振动分析一直是结构动力学领域中的难点,已有一些研究表明基于矩等效的线性化方法在功率谱预测上会得到不恰当的分析结果;另一方面,由于不确定性在实际工程中普遍存在,如果同时考虑非线性和不确定性,更是显著增加了问题难度。本文以具有非线性非理想边界梁为研究对象,基于梁模型的动力学微分方程推导了对应的广义频响函数,并应用Volterra级数理论建立了非线性系统随机振动的谱分析方法,最后,结合蒙特卡洛抽样方法计算了具有参数不确定性非线性梁响应功率谱的均值和方差,讨论了不确定性对结构随机振动响应统计特征的影响。     

载流圆板在磁场中的随机稳定性分析

本文根据大挠度板壳力学基础理论和电磁弹性力学理论,建立了载流圆板的非线性磁弹性随机振动力学模型,采用伽辽金变分法将其变换成非线性常微分动力学方程．通过拟不可积哈密顿系统的平均理论将该方程等价为一个一维伊藤随机微分方程．通过计算该方程的最大Lyapunov 指数判断该系统的局部随机稳定性,并进一步采用基于随机扩散过程的奇异边界理论判断该系统的全局稳定性．最后通过讨论该系统的稳态概率密度函数图的形状变化讨论了该动力系统的随机Hopf分岔的变化规律,并采用数值模拟对理论分析进行了验证．     

结构动力非线性随机反应的联合概率分布

在密度演化理论基本思想的框架下，对广义密度 演化方程进行推广，导出了结构不同反应量的联合概率密度函数演化方程. 结合确定性结构 非线性动力反应分析与二维偏微分方程求解的有限差分方法，可以获取结构不同反应量的联 合概率密度函数的数值解答. 分析实例表明：结构反应的联合概率密度函数呈丘陵状不规则 分布，而不同反应量之间的相关系数是时变的.     

结构非线性振动及其控制系统的等效力学模型研究

研究了结构非线性振动及其控制系统的等效力学模型建立问题,对结构进行等效线性化处理;对状态变量的观测,仅测量结构的前几个位移。将随机激励和观测噪声等维化处理后,导出了结构非线性振动及其控制系统的等效力学模型。通过等效力学模型的应用研究,得出了多高层结构在风与地震作用下的多维与一维等效力学模型、等效振型力学模型及其线性力学模型,给出了等效力学模型在多高层结构自适应抗风抗震控制中的应用。     

基于时域解耦算法的多液舱浮式结构物运动模拟

对于带有多个晃荡液舱的浮式结构物, 浮体的运动、外场水动力以及各舱内的液体晃荡力会实时相互决定, 发生复杂的耦合作用. 为准确模拟多液舱浮式结构物的运动, 本文引入一种有效的时域解耦算法. 该方法以模态分解法为基础, 通过对浮式结构物所受外域水动力和各液舱内非线性晃荡力进行模态分解, 最终形成时域解耦运动方程, 无需迭代求解过程即可显式计算浮式结构物的瞬时加速度. 该方法可避免传统迭代求解方法在迭代次数、截断误差和收敛特性等方面的不足, 减少解耦过程的计算耗时. 本文进一步结合边界元数值方法, 分别对单液舱浮式结构物和多液舱浮式结构物的工况开展数值模拟研究. 通过与单液舱浮式结构物的实验结果对比, 验证了本文时域解耦算法的有效性. 本文详细分析了晃荡力对单液舱浮式结构物运动的影响, 发现存在一个共振影响区间: 当外场波浪频率在该区间之外时, 可以在时域计算结果中观察到稳定的浮体运动; 在比该区间更低频的波况下, 液舱晃荡力与外场波浪力相位相反甚至可以相互抵消, 此时晃荡液舱的存在可以减弱浮体运动; 在比该区间更高频的波况下, 液舱内晃荡力与外场波浪力可以具有相同相位, 此时晃荡液舱的存在会加剧浮体的运动. 本文进一步研究了四液舱浮式结构物在波浪中的纵荡、垂荡和纵摇运动情况, 发现非线性液舱晃荡可对纵荡和纵摇运动产生影响, 但对垂荡运动影响很小.      

多自由度非线性随机系统的响应与稳定性

响应与稳定性分析一直是随机动力学研究的热点, 发展预测随机响应及判定系统响应性态的方法具有重要的科学意义与广阔的应用前景. 本文综述了有关多自由度非线性随机系统的响应与稳定性的研究. 首先简介用于随机系统响应预测的Fokker-Planck-Kolmogorov方程法、随机平均法、等效线性化法、等效非线性系统法和Monte Carlo模拟法, 评述其优缺点, 进而讨论了多自由度非线性随机系统响应的精确平稳解、近似瞬态解的研究现状. 然后介绍了随机系统稳定性分析的两类方法, 即Lyapunov函数法及Lyapunov指数法,并综述了多自由度非线性随机系统稳定性分析的研究现状. 最后给出几点发展建议.     

滞变智能隔震结构的序列最优控制算法

将地震波简化为一系列脉冲,在每个时间步长上重新构造控制目标函数,建立了一种新的时域内的结构振动最优控制算法,并在滞变模型的智能隔震结构中用状态转移法加以实现.文中采用Bouc-Wen模型描述结构恢复力,并利用等效线性化方法处理非线性运动方程,分别导出了状态反馈和输出加权的两种表达式.文末选用作者承担设计过的实际隔震建筑中的一个单体工程作为算例,比较了输入三种不同地震波时各种算法的控制效果.结果表明,在相同控制能量下,本文算法对滞变结构能有效地削减响应峰值,综合性能优于现有的两种时域内的结构最优控制算法.     

杆系结构非线性损伤随机演化分析

通过研究结构非线性构形状态转移过程考察结构非线性损伤随机演化的思路，建立了作为Markov链的非线性构形状态转移过程的转移速率与屈服应变风险率函数之间的关系。进而，通过力学分析与非线性构形状态的逻辑分析得到转移速率矩阵，从而将一个结构非线性损伤随机演化问题转化为一个非时齐Markov链的分析问题，以三杆桁架结构为例，给出了数值分析结果。     

Effect of damping and wave parameters on offshore structure under random excitation

This paper describes the linearized and nonlinear dynamic response of a tension leg platform (TLP) to random waves and current forces. The forcing term of the equation of motion is inherently nonlinear due to the nonlinear drag force. Two analysis procedures are used: nonlinear time domain analysis and linear frequency domain analysis. For the nonlinear analysis, the random wave particle velocities and accelerations are simulated for a given wave spectrum. The nonlinear equation of motion is then integrated directly to obtain the system response statistics. For the linear frequency domain analysis, the nonlinear drag force is linearized through an introduction of linearization coefficients. The main objective of this paper is to investigate the effect of the structural damping and wave parameters on both nonlinear and linear dynamic response of the TLP by parametric studies. The results of stochastic nonlinear and linear dynamic response of the TLP, with and without the presence of current, are presented and compared.     

Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization

The aim of this work is to study the free dynamic response of thin plates characterized by geometrical nonlinearities. To achieve this task, the equation of motion of the plate is first carried out through modeling by hierarchical finite element method whose interpolating shape functions are sinusoidal. Then, the study of the nonlinear vibrations was carried out by the development of asymptotic linearization and equivalent linearization methods in modal space. The nonlinear angular frequencies are successively deduced by exciting the corresponding vibrating mode of the structure. The confrontation of these results to those obtained by the iterative method in the physical space and to those found in the literature, showed a very good agreement between the various methods. From the elementary nonlinear frequencies we showed that there exists an equivalent linear dynamical system characterized by only one equivalent linear stiffness matrix. Numerical experiments were carried out on beams and thin plates of various dimensions ratios and boundary conditions. These numerical test simulations, whether in time space or frequency space, have showed that the nonlinear elastic energy is restored by the equivalent linear dynamical system. Nevertheless, we have to say that the dynamic effects of modes above the excited one are neglected.