高维非光滑动力系统的周期响应数值分析

本文研究以复杂弹塑性结构，含间隙或干摩擦的机械等为背景的高维非光滑动力系统周期响应的数值解法。提出了一种对短时间历程动响应进行曲线拟合后外推周期响应的迭代格式，改变了现有方法未充分利用动力系统内在特性及中间计算结果所含信息的不足，使计算效率有了阶次性的提高且收敛性能亦大为改善。     

A new method for the periodic solution of strongly nonlinear systems

A new method for the periodic solution of strongly nonlinear system is given. By using this method, the existance and stability of the periodic solution can be decided, and the approximate expression of the periodic solution can also be found. The project supported by the National Natural Science Foundation of China     

RESEARCH OF THE PERIODIC MOTION AND STABILITY OF TWO-DEGREE-OF-FREEDOM NONLINEAR OSCILLATING SYSTEMS

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｔｈｅｓｙｓｔｅｍｏｆｎｏｎｌｉｎｅａｒｏｓｃｉｌｌａｔｉｎｇ ,ｐｅｒｉｏｄｉｃｍｏｔｉｏｎｉｓｏｆｐｒｉｍｅｉｍｐｏｒｔａｎｃｅ .Ｂｕｔｅｘｉｓｔｅｎｃｅｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｓｉｓａｖｅｒｙｄｉｆｆｉｃｕｌｔｑｕｅｓｔｉｏｎ .Ｌｕｃｋｉｌｙｔｈｅｒｅｅｘｉｓｔｓｏｍｅｋｉｎｄｓｏｆｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｉｎａｃｔｕａｌｐｈｙｓｉｃａｌｓｙｓｔｅｍｓ .Ｔｈｅｒｅｆｏｒｅ ,ｗｅｕｓｕａｌｌｙｃｏｎｃｅｎｔｒａｔｅｄｏｕｒａｔｔｅｎｔｉｏｎｏｎｔ…     

弹塑性接触问题的非光滑非线性方程组方法

将求解三维弹性摩擦接触问题的非光滑非线性方程组方法推广到弹塑性(Mises材料)情形，提出了两种应用方法：一种是将非光滑非线性方程组方法和求解弹塑性问题常用的Newton—Raphson迭代方法结合起来；另一种是将问题写成统一的非光滑非线性方程组，直接求解。数值算例验证了两种方法的有效性，并进行了结果比较。     

Stability analysis for multi-parameter linear periodic systems

Summary This paper is devoted to stability analysis of general linear periodic systems depending on real parameters. The Floquet method and perturbation technique are the basis of the development. We start out with the first and higher-order derivatives of the Floquet matrix with respect to problem parameters. Then the behaviour of simple and multiple multipliers of the system with a change of parameters is studied. Weak and strong interactions of multipliers in the complex plane are treated separately. The presented theory is exemplified and discussed. Received 27 April 1998; accepted for publication 5 November 1998     

Symplectic analysis for wave propagation in one-dimensional nonlinear periodic structures

The wave propagation problem in the nonlinear periodic mass-spring structure chain is analyzed using the symplectic mathematical method. The energy method is used to construct the dynamic equation, and the nonlinear dynamic equation is linearized using the small parameter perturbation method. Eigen-solutions of the symplectic matrix are used to analyze the wave propagation problem in nonlinear periodic lattices. Nonlinearity in the mass-spring chain, arising from the nonlinear spring stiffness effect, has profound effects on the overall transmission of the chain. The wave propagation characteristics are altered due to nonlinearity, and related to the incident wave intensity, which is a genuine nonlinear effect not present in the corresponding linear model. Numerical results show how the increase of nonlinearity or incident wave amplitude leads to closing of transmitting gaps. Comparison with the normal recursive approach shows effectiveness and superiority of the symplectic method for the wave propagation problem in nonlinear periodic structures.     

Numerical analysis of bifurcation buckling for rotationally periodic structures under rotationally periodic loads

By considering the characteristics of deformation of rotationally periodic structures under rotationally periodic loads, the periodic structure is divided into some identical substructures in this study. The degrees-of-freedom (DOFs) of joint nodes between the neighboring substructures are classified as master and slave ones. The stress and strain conditions of the whole structure are obtained by solving the elastic static equations for only one substructure by introducing the displacement constraints between master and slave DOFs. The complex constraint method is used to get the bifurcation buckling load and mode for the whole rotationally periodic structure by solving the eigenvalue problem for only one substructure without introducing any additional approximation. The finite element (FE) formulation of shell element of relative degrees of freedom (SERDF) in the buckling analysis is derived. Different measures of tackling internal degrees of freedom for different kinds of buckling problems and different stages of numerical analysis are presented. Some numerical examples are given to illustrate the high efficiency and validity of this method.     

Dynamical response of hyper-elastic cylindrical shells under periodic load

Dynamical responses, such as motion and destruction of hyper-elastic cylindrical shells subject to periodic or suddenly applied constant load on the inner surface, are studied within a framework of finite elasto-dynamics. By numerical computation and dynamic qualitative analysis of the nonlinear differential equation, it is shown that there exists a certain critical value for the internal load describing motion of the inner surface of the shell. Motion of the shell is nonlinear periodic or quasi-periodic oscillation when the average load of the periodic load or the constant load is less than its critical value. However, the shell will be destroyed when the load exceeds the critical value. Solution to the static equilibrium problem is a fixed point for the dynamical response of the corresponding system under a suddenly applied constant load. The property of fixed point is related to the property of the dynamical solution and motion of the shell. The effects of thickness and load parameters on the critical value and oscillation of the shell are discussed.     

不确定非线性结构动力响应的区间分析方法

研究多自由度非线性不确定参数系统的动力响应问题. 以区间数学为基础,将不确定 性参数用区间进行定量化,借助一阶Taylor级数,给出了近似估计非线性振动系统动力响 应范围的区间分析方法. 从数学证明和数值算例两方面,将其与概率摄动有限元法进行了比 较,结果显示区间分析方法对不确定参数先验信息具有要求较少、精度较高的优点.     

随机结构非线性动力响应的概率密度演化分析

提出了随机结构非线性动力响应分析的概率密度演化方法．根据结构动力响应的随机状态方程，利用概率守恒原理，建立了随机结构非线性动力响应的概率密度演化方程．结合Newmark-Beta时程积分方法与Lax-Wendroff差分格式，提出了概率密度演化方程的数值分析方法．通过与Monte Carlo分析方法对比，表明所给出的概率密度演化方法具有良好的计算精度和较小的计算工作量．研究表明：随机结构非线性动力响应概率密度具有典型的演化特征，随着时间增长，概率密度曲线分布趋于复杂．     

复合材料层合板壳的非线性热动态响应分析

本文基于高阶剪切变形理论，考虑同天应变和横向剪切应变的影响，对受热复合材料层合板壳的非线性热动态响应进行分析，计及了转动惯量的影响，给出了通用性较好的Ｃ＾０类有限元公式，文中数值算同现有文献和三维有限元计算结果进行了比较，证明了本文方法的精确、有效性，文中还就层合板的边界条件、纵厚比及铺设角度对非线性热动态响应的影响进行了分析。     

On the inverse optimization problem for periodic systems

Consideration is given to the problem inverse to the problem of designing the optimal (in the sense of minimization of a quadratic functional) controller for a linear periodic system. Problem statement: given matrices describing the dynamics of the system and a control matrix, determine the weighting matrices of the quadratic functional. To solve this problem, an algorithm based on linear matrix inequalities is proposed __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 12, pp. 100–106, December 2005.     

Numerical method in dynamic response of nonlinear systems

A kind of modal synthesis techniques, which is applicable to vibration analysis for linear substructures with nonlinear coupling attachments, has been extended to nonlinear dynamic analysis of large complex structural systems^[9]. In this paper, a process is suggested to dynamic analysis of large complex structural systems with nonlinear characteristics of each substructure. At the end of this paper, an example shows the defendable accuracy of the results and high efficiency of this process.This paper was collected in Proceedings of Symposium on Nonlinear Mechanics, Vol. IV, 1982, Wuxi, China (in Chinese).     

基于state-space-split法的滞回非线性系统随机动力响应分析

采用SSS(state-space-split)法,建立了引入Bouc-Wen滞回模型的杜芬非线性系统在高斯白噪声激励下的概率密度函数(PDF)的近似求解方法,分析了其随机动力响应变化规律.首先,将Bouc-Wen滞回模型引入杜芬非线性系统,分别考虑非线性系统中的几何非线性和材料非线性对动力响应的影响.随后,对该模型进...     

A modal analysis for the dynamic response of fluid-structure systems

In this paper a theory of modal analysis for the dynamic response of fluid-structure systems is presented. A pair of generalized eigenvalue equations with three real symmetric matrices and their relationships are derived from the finite element equations in the form of structural displacements and fluid velocity potential. Generalized orthogonality relations of modal vectors are then developed. The response of the system to external excitation is derived in closed form by modal expansion. Two examples of the solution are given for illustration.     

STABILITY ANALYSIS OF LINEAR AND NONLINEAR PERIODIC CONVECTION IN THERMOHALINE DOUBLE－DIFFUSIVE SYSTEMS

ＳＴＡＢＩＬＩＴＹＡＮＡＬＹＳＩＳＯＦＬＩＮＥＡＲＡＮＤＮＯＮＬＩＮＥＡＲＰＥＲＩＯＤＩＣＣＯＮＶＥＣＴＩＯＮＩＮＴＨＥＲＭＯＨＡＬＩＮＥＤＯＵＢＬＥ－ＤＩＦＦＵＳＩＶＥＳＹＳＴＥＭＳＺｈａｎｇＤｉｍｉｎｇ（张涤明）;ＬｉＬｉｎ（李琳）;ＨｕａｎｇＨ...     

On the modelling of dynamic problems for plates with a periodic structure

Summary The subject of analysis is the bending of elastic plates exhibiting a nonhomogeneous periodic structure and/or a periodically variable thickness in a certain direction parallel to the plate's midplane. The fundamental modelling problem is how to obtain an effective 2D-model of a plate under consideration, i.e., a 2D-model represented by PDEs with constant coefficients. This problem for periodic plates has been solved independently in [5] and [10], using asymptotic homogenization. However, homogenization neglects dynamic phenomena related to the plate's rotational inertia and cannot be applied to the analysis of higher-order vibration frequences. The main aim of this contribution is to formulate a new non-asymptotic effective 2D-model of a periodic plate which is free from the mentioned drawbacks and describes the dynamic behaviour of plates having the thickness of the order of the period length. The proposed model is applied to the analysis of some vibration problems.     

基于动力学响应的多组件系统布局优化方法

针对圆形安装板上多组件布局问题,以组件系统静平衡性为约束,以系统在随机振动下的动力学响应为优化目标,建立了多组件系统的动力学布局优化模型。针对所建立的优化模型,提出采用序列二次规划方法和改进遗传算法相结合的策略进行多约束条件下的组件布局优化。最后,对两个典型算例进行了动力学优化计算,结果表明,本文所建立的优化模型在多约束的情况下可提高组件系统的动力学性能,满足静力学和动力学性能要求。     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.