不规则区域平面弹性问题的正则区域重心插值配点法

将不规则区域嵌入到规则的矩形区域,在矩形区域上将弹性平面问题的控制方程采用重心Lagrange插值离散,得到控制方程矩阵形式的离散表达式。在边界节点上利用重心插值离散边界条件,规则区域采用置换法施加边界条件,不规则区域采用附加法施加边界条件,得到求解平面弹性问题的过约束线性代数方程组,采用最小二乘法进行求解,得到整个规则区域上的位移数值解。利用重心插值计算得到不规则区域内任意节点的位移值,计算精度可到10^-14以上。数值算例验证了所建立方法的有效性和计算精度。     

On statistical quasi-linearization

In carrying out the statistical linearization procedure to a non-linear system subjected to an external random excitation, a Gaussian probability distribution is assumed for the system response. If the random excitation is non-Gaussian, however, the procedure may lead to a large error since the response of bother the original non-linear system and the replacement linear system are not Gaussian distributed. It is found that in some cases such a system can be transformed to one under parametric excitations of Gaussian white noises. Then the quasi-linearization procedure, proposed originally for non-linear systems under both external and parametric excitations of Gaussian white noises, can be applied to these cases. In the procedure, exact statistical moments of the replacing quasi-linear system are used to calculate the linearization parameters. Since the assumption of a Gaussian probability distribution is avoided, the accuracy of the approximation method is improved. The approach is applied to non-linear systems under two types of non-Gaussian excitations: randomized sinusoidal process and polynomials of a filtered process. Numerical examples are investigated, and the calculated results show that the proposed method has higher accuracy than the conventional linearization, as compared with the results obtained from Monte Carlo simulations.     

A probabilistic linearization method for non-linear systems subjected to additive and multiplicative excitations

A general method to obtain approximate solutions for the random response of non-linear systems subjected to both additive and multiplicative Gaussian white noises is presented. Starting from the concept of linearization, the proposed method of “Probabilistic Linearization” (PL) is based on the replacement of the Fokker–Planck equation of the original non-linear system with an equivalent one relative to a linear system subjected to additive excitation only. By means of the general scheme of the weighted residuals, the unknown coefficients of the equivalent system are determined. Assuming a Gaussian probability density function of the response process and by choosing the weighting functions in a suitable way, the equivalence of the proposed method, called “Gaussian Probabilistic Linearization” (GPL), with the “Gaussian Stochastic Linearization” (GSL) applied to the coefficients of the Itô differential rule is evidenced. In addition, the generalization of the proposed method, called “Generalized Gaussian Probabilistic Linearization” (GGPL), is presented. Numerical applications show as, varying the choice of the weighting functions, it is possible to obtain different linearizations, with a variable degree of accuracy. For the two examples considered, different suitable combinations of the weighting functions lead to different equivalent linear systems, all characterized by the exact solution in terms of variance.     

Fluid flow of incompressible viscous fluid through a non-linear elastic tube

The study of viscous flow in tubes with deformable walls is of specific interest in industry and biomedical technology and in understanding various phenomena in medicine and biology (atherosclerosis, artery replacement by a graft, etc) as well. The present work describes numerically the behavior of a viscous incompressible fluid through a tube with a non-linear elastic membrane insertion. The membrane insertion in the solid tube is composed by non-linear elastic material, following Fung’s (Biomechanics: mechanical properties of living tissue, 2nd edn. Springer, New York, 1993) type strain–energy density function. The fluid is described through a Navier–Stokes code coupled with a system of non linear equations, governing the interaction with the membrane deformation. The objective of this work is the study of the deformation of a non-linear elastic membrane insertion interacting with the fluid flow. The case of the linear elastic material of the membrane is also considered. These two cases are compared and the results are evaluated. The advantages of considering membrane nonlinear elastic material are well established. Finally, the case of an axisymmetric elastic tube with variable stiffness along the tube and membrane sections is studied, trying to substitute the solid tube with a membrane of high stiffness, exhibiting more realistic response.     

The relationship between the stability and the optimality of linear systems (II)

Different from the inverse problem put forward by R.E.Kalman, another kind of inverse problem of linear optimal control is proposed and discussed in [1] as follows: Given an asymptotically stable linear constant system and a nonnegative quadratic performance index, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended. Similar conclusions are obtained for linear discrete systems and linear time-variable systems. According to these conclusions we can say that the correspondence between the asymptotically stable system and the optimal feedback system is the inherent character of all kinds of linear systems.     

Equation governing the probability density evolution of multi-dimensional linear fractional differential systems subject to Gaussian white noise

Stochastic fractional differential systems are important and useful in the mathematics, physics, and engineering fields. However, the determination of their probabilistic responses is difficult due to their non-Markovian property. The recently developed globally-evolving-based generalized density evolution equation (GE-GDEE), which is a unified partial differential equation (PDE) governing the transient probability density function (PDF) of a generic path-continuous process, including non-Markovian ones, provides a feasible tool to solve this problem. In the paper, the GE-GDEE for multi-dimensional linear fractional differential systems subject to Gaussian white noise is established. In particular, it is proved that in the GE-GDEE corresponding to the state-quantities of interest, the intrinsic drift coefficient is a time-varying linear function, and can be analytically determined. In this sense, an alternative low-dimensional equivalent linear integer-order differential system with exact closed-form coefficients for the original high-dimensional linear fractional differential system can be constructed such that their transient PDFs are identical. Specifically, for a multi-dimensional linear fractional differential system, if only one or two quantities are of interest, GE-GDEE is only in one or two dimensions, and the surrogate system would be a one- or two-dimensional linear integer-order system. Several examples are studied to assess the merit of the proposed method. Though presently the closed-form intrinsic drift coefficient is only available for linear stochastic fractional differential systems, the findings in the present paper provide a remarkable demonstration on the existence and eligibility of GE-GDEE for the case that the original high-dimensional system itself is non-Markovian, and provide insights for the physical-mechanism-informed determination of intrinsic drift and diffusion coefficients of GE-GDEE of more generic complex nonlinear systems.     

Fracture behavior of a bonded magneto-electro-elastic rectangular plate with an interface crack

In this paper the anti-plane problem for an interface crack between two dissimilar magneto-electro-elastic plates subjected to anti-plane mechanical and in-plane magneto-electrical loads is investigated. The interface crack is assumed to be either magneto-electrically impermeable or permeable, and the position of the interface crack is arbitrary. The finite Fourier transform method is employed to reduce the mixed boundary-value problem to triple trigonometric series equations. The dislocation density functions and proper replacement of the variables are introduced to reduce these series equations to a standard Cauchy singular integral equation of the first kind. The resulting integral equation together with the corresponding single-valued condition is approximated as a system of linear algebra equations which can be easily solved. Field intensity factors and energy release rates are determined numerically and discussed in detail. Numerical results show the effects of crack configuration and loading combination parameters on the fracture behaviors of crack tips according to energy release rate criterion. The study of this problem is expected to have applications to the investigation of dynamic fracture properties of magneto-electro-elastic materials with cracks.     

THE RELATIONSHIP BETWEEN THE STABILITY AND THE OPTIMALITY OF LINEAR SYSTEMS(Ⅱ)

Different from the inverse problem put forward by R.E.Kalman, another kind ofinverse problem of linear optimal control is proposed and discussed in[1] as follows:Givenan asymptotically stable linear constant system and a nonnegative quadratic performanceindex, when can a state-feedback be separated from the stable system so that this state-feedback control law is optimal for the given index? In this paper this problem is extended.Similar conclusions are obtained for linear discrete systems and linear time-variablesystems. According to these conclusions we can say that the correspondence between theasymptotically stable system and the optimal feedback system is the inherent character ofall kinds of linear systems.     

Vibration reduction evaluation of a linear system with a nonlinear energy sink under a harmonic and random excitation

The nonlinear behaviors and vibration reduction of a linear system with a nonlinear energy sink(NES) are investigated. The linear system is excited by a harmonic and random base excitation, consisting of a mass block, a linear spring, and a linear viscous damper. The NES is composed of a mass block, a linear viscous damper, and a spring with ideal cubic nonlinear stiffness. Based on the generalized harmonic function method,the steady-state Fokker-Planck-Kolmogorov equation is presented to reveal...     

Symmetry Breaking for a System of Two Linear Second-Order Ordinary Differential Equations

A new canonical form for a system of two linear second-orderordinary differential equations (odes) is obtained. The latter isdecisive in unravelling symmetry structure of a system of two linearsecond-order odes. Namely we establish that the point symmetry Liealgebra of a system of two linear second-order odes can be5-, 6-, 7-, 8- or 15-dimensional. This result enhances both the richness andthe complexity of the symmetry structure of linear systems.     

四维超混沌系统Hopf分岔分析与反控制

对超混沌系统进行分岔反控制的研究已成为当前一个重要研究方向,常采用线性控制器实现反控制。首先,对一个四维超混沌系统的Hopf分岔特性进行了分析,利用高维分岔理论推导出分岔特性与参数之间的关系式,以此判断系统的分岔类型。然后,设计一个由线性与非线性组合成的混合控制器对系统进行分岔反控制,控制参数取值不同时,系统会呈现出不同的分岔特性。通过分析得出,调控线性控制器参数可以使系统Hopf分岔提前或延迟发生;同时,调控混合控制器的两个控制参数,可以改变系统Hopf分岔特性,实现分岔反控制。     

On the stability of systems having several equilibrium states

A frequency domain theorem is given for the stability analysis of systems that have several equilibrium states. Under consideration are closed loop systems containing a single instantaneous nonlinearity in an otherwise linear system. A condition is derived, involving the frequency response of the linear part of the system, which guarantees that all bounded solutions approach an equilibrium state, and a geometrical interpretation of the results is found.     

大型非比例阻尼线性系统的地震反应复振型分析方法

对于大型非比例阻尼线性系统,当采用基于复振型的振型叠加方法进行动力反应分析时,按照通用算法(例如雅克比法)求解结构全部振型向量的计算工作量很大,甚至是不现实的.本文将经典阻尼系统中行之有效的Lanczos法和子空同迭代法加以推广和改进,给出了一组可对原来的方程进行自由度缩减的实向量基,然后与Foss变换相结合,得到一组实用的复向量基,使之适用于求解复杂非比例阻尼线性系统的任意低阶复振型和相应的复特征值,适用于任意扰力作用下的动力反应分析.理论推导和实例计算表明,本文所给出的复向量基概念清晰,计算效率高,能够适应对具有非比例阻尼特性的大型复杂结构进行动力分析的实际需要,其中包括地震作用下的时程分析和反应谱振型叠加分析.     

NGPG-Stability of Linear Multistep Methods for Systems of Generalized Neutral Delay Differential Equations

The stability analysis of linear multistep methods for the numerical solutions of the systems of generalized neutral delay differential equations is discussed. The stability behaviour of linear multistep methods was analysed for the solution of the generalized system of linear neutral test equations. After the establishment of a sufficient condition for asymptotic stability of the solutions of the generalized system, it is shown that a linear multistep method is NGP _G -stable if and only if it is A-stable.     

动力吸振器复合非线性能量阱对线性镗杆系统的振动控制

研究了线性动力吸振器复合非线性能量阱对线性镗杆在外部简谐激励下的振动控制. 忽略镗杆系统中的非线性因素, 建立了附加线性动力吸振器和非线性能量阱的镗杆系统的三自由度运动方程, 研究了附加复合式动力吸振器的镗杆系统的受迫振动. 通过平均法得到了附加复合式动力吸振器的镗杆系统的近似解析解, 并利用数值解验证了近似解析解的准确性, 两者具有很好的一致性. 利用近似解析解详细分析了线性动力吸振器和非线性能量阱的参数对镗杆振动抑制性能的影响. 对给定质量的复合式动力吸振器进行了参数优化, 其中线性动力吸振器参数采用H_∞优化方法的近似解析解进行了优化, 非线性能量阱的阻尼利用系统的近似解析解进行了优化. 分析结果表明, 线性动力吸振器与非线性能量阱组合可以有效抑制线性镗杆系统的振动, 而且采用参数优化后的复合式动力吸振器可以获得更好的减振效果. 通过附加非线性能量阱, 不但可以提高线性动力吸振器的振动抑制效果, 而且还可以提高振动控制系统的鲁棒性.      

车载DR系统自适应扩展卡尔曼滤波模型的建立及仿真研究

提出了车载DR系统的自适应扩展卡尔曼滤波模型及其滤波算法，计算机仿真结果表明，应用该模型和算法与简单航位推算方法相比，可以明显提高DR系统的定位精度。     

Localization of energy in nonlinear systems with two degrees of freedom

Energy pumping in a two-degrees-of-freedom system with linear and essentially nonlinear oscillators is studied. The kinetic energy envelopes of the linear and nonlinear subsystems are chosen to be the main characteristics of the process under consideration. A criterion that the energy of the linear oscillator excited at time zero is completely pumped over into the nonlinear oscillator is established together with an additional condition whereby the energy does not return to the linear subsystem. Optimal energy pumping mode is established using global optimization. The effect of the parameters of the system on the main characteristics is assessed __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 115–125, May 2007.     

NGP_G-STABILITY OF LINEAR MULTISTEP METHODS FOR SYSTEMS OF GENERALIZED NEUTRAL DELAY DIFFERENTIAL EQUATIONS

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基于区域分解的结构动力学系统首次穿越失效

提出了高斯白噪声激励的线性及非线性结构动力学系统的首次穿越失效概率的估计方法. 对于线性结构动力学系统,失效区域被分解为互斥的基本失效域之和,每个基本失效域可用其设计点完全描述,并以正态分布代替卡方分布估计失效概率中的参数. 对于非线性结构动力学系统,基于Rice穿越理论,将非线性方程转化为与之具有相同平均上穿率的线性化方程,然后利用文中方法对等效线性化方程估计首穿失效概率. 最后给出了线性及非线性结构动力学系统的数值例子,并将所提方法与蒙特卡罗法及重要样本法相比较,模拟结果显示了方法的正确性与有效性.