Invariant Manifolds for Nonautonomous Systems with Application to One-Step Methods

In this paper we study the existence of invariant manifolds for a special type of nonautonomous systems which arise in the study of discretization methods. According to [10], a one-step scheme of step-size for an autonomous system can be interpreted as the -flow of a perturbed nonautonomous system. The perturbation is rapidly forced in the sense that it is periodic with respect to time with period . Assuming a saddle node for the autonomous system, we prove that these rapidly forced perturbations have center manifolds which exist in a uniform neighborhood and which converge to a center manifold of the autonomous system as tends to zero. Our results are applied to obtain a smooth continuation as well as estimates of the well known center manifolds for one-step schemes. They also form the basis for studying saddle-node homoclinic orbits under discretization.     

Deformation of an elastic helix in contact with a rigid cylinder

Summary The deformation of a short helix in contact with a rigid cylinder is investigated. Deformations occur due to bending, torsion and longitudinal elasticity of the helix. Shear deformation is neglected. Some of the equations describing the problem have been given already in Love's Treatise on the Mathematical Theory of Elasticity, in terms of curvature changes. Nevertheless, the equations for small deformations have to be reformulated in terms of displacements and rotations, because contact constraints cannot be expressed in terms of curvature. Friction is neglected, thus the problem is symmetric, and it is sufficient to determine its solution for one half of the helix.Without friction between the cylinder and the helix, the contact problem arises only for a helix longer than one length of twist. For a shorter helix, the global equilibrium conditions cannot be satisfied for nonvanishing contact forces. For the minimum length, there are two noncontact zones, and the helix is in contact with the cylinder only at three points: at the ends and in the middle. For a slightly longer helix, four contact points and three noncontact regions are found. The dependence of the noncontact zones and the contact forces, which are of practical interest, can be calculated as a function of the length of the helix and its geometrical parameters. The case of a very long helix with more than four contact points remains unsolved.     

一个新的八节点非协调曲边四边形平面单元

本文在文[1]和文[2]的基础上,提出构造非协调有限元的新方法。该方法不用一般的变分原理,可适用任意变系数正定和非正定偏微分方程。利用这一方法得到一个新的八节点四边形平面应力单元。与一般有限元相比,位移和应力可提高一阶收敛精度。形成单刚矩阵时,不需要进行数值积分。单元之间的协调条件容易满足,文中给出收敛性证明。文末给出数值算例,表明利用本文的方法,应力和位移均可获得满意的数值精度。     

Sine-Gordon方程的截断系统的同宿轨道

研究Ｓｉｎｅ Ｇｏｒｄｏｎ方程的广义渐近惯性流形上的常微分方程组，证实了在一定参数条件下存在Ｗｉｇｇｉｎｓ［１］意义下的同宿轨道．计算表明，与Ｂｉｓｈｏｐ［２］用数值计算得到的Ｓｉｎｅ Ｇｏｒｄｏｎ方程产生混沌的参数值尚有差别，考虑到同宿出现参数值往往低于混沌出现参数值，故结果在定性上正确，而且改进了文［１］中的结果．     

组合壳体的弹塑性有限元分析

本文采用文[1]提出的曲壳单元,根据Prandtl-Reuss塑性流动理论和Mises等向强化屈服准则,建立了壳体的弹塑性有限元格式,同时按照罚单元原理建立了组合壳体的连接条件,编制了相应的计算程序,具体计算了带接管球壳和等径三通等算例,取得了较好的结果。     

Strain determination on curved surfaces by objective white light speckles

A close-range objective white light speckle method has been used for strain determination on simply curved surfaces. The speckle field is not the artificially printed random dots but only the radiance distribution of object surface which was slightly treated before the test to produce fine structure of optical details. The holographic film, instead of holographic plate, is attatched to the curved surface when illuminated by a flash light. Whole field fringe patterns are obtained with high sensitivities and large adjustable range. There is no longer a requirement to derive the speckle movement on the image plane from the object surface. Defocusing problem has been avoided. It can be practically applied to engineering problems with considerable convenience because of the very simple recording system and little demand for environmental stability.     

Deterministic and stochastic analyses of chaotic and overturning responses of a slender rocking object

The relationship between chaos and overturning in the rocking response of a rigid object under periodic excitation is examined from both deterministic and stochastic points of view. A stochastie extension of the deterministic Melnikov function (employed to provide a lower bound for the possible chaotic domain in parameter space) is derived by taking into account the presence of random noise. The associated Fokker-Planck equation is derived to obtain the joint probability density functions in state space. It is shown that global behavior of the rocking motion can be effectively studied via the evolution of the joint probability density function. A mean Poincaré mapping technique is developed to average out noise effects on the chaotic response to reconstruct the embedded strange attractor on the Poincaré section. The close relationship between chaos and overturning is demonstrated by examining the structure of the invariant manifolds. It is found that the presence of noise enlarges the boundary of possible chaotic domains in parameter space and bridges the domains of attraction of coexisting responses. Numerical results consistent with the Foguel alternative theorem, which discerns asymptotic stabilities of responses, indicate that the overturning attracting domain is of the greatest strength. The presence of an embedded strange attractor (reconstructed using the mean Poincaré mapping technique) indicates the existence of transient chaotic rocking response.     

A Hartman-Grobman theorem for the Cahn-Hilliard and phase-field equations

In this paper, we study the structural stability of the Cahn-Hilliard equation and the phase-field equations. We show that the Cahn-Hilliard equation and the phase-field equations are topologically conjugate to a decoupled system of a linear equation of infinite dimension and an ordinary differential equation which is the reduced equation on the inertial manifold; particularly, the flow nearby hyperbolic stationary solutions is structurally stable.     

圆截面杆在曲线井中的屈曲问题

本文研究了线弹性的圆截面压杆在刚性井壁的曲线井中的屈曲问题。压杆两端为铰支，杆和井壁之间的摩擦力忽略不计，失稳前假设压杆位于曲线井的较低侧。分析了压杆屈曲时的几何变形条件，导出了压杆的曲率，变形能和各项外力势的表达式，用最小势能原理导出压杆失稳的微分方程和边界条件。引入梁单元，用有限单元法求得了压杆失稳时的临界压力和杆轴线状态。结果显示，临界压力随着压杆自重和曲线井半径的增加而增加。自重为零时，杆轴线为半个正弦曲线；自重不为零时，杆轴线为振荡的曲线，且振荡主要集中在压杆顶部。     

新的圆柱曲板剪切屈曲实验的研究

提出一个用两块曲板以单向加载进行的Ｒ１０００ｍｍ的圆柱曲板剪切屈曲新实验方案。其原理清楚、装置简单紧凑、节省试材、实验进行方便和不需扭转设备等优点，而实验结果和成熟的工作符合良好．