Chaotic motion of a nonlinear thermoelastic elliptic plate

In the paper the nonlinear dynamic equation of a harmonically forced elliptic plate is derived, with the effects of large deflection of plate and thermoelasticity taken into account. The Melnikov function method is used to give the critical condition for chaotic motion. A demonstrative example is discussed through the Poincaré mapping, phase portrait and time history. Finally the path to chaotic motion is also discussed. Through the theoretical analysis and numerical computation some beneficial conclusions are obtained. Foundation item: the National natural Science Foundation of China (19672038); the Natural Science Foundation of Shanxi Provence (1880342).     

A Global,Direct Algorithm for Path-Following and Active Static Control of Elastic Bar Structures*

ABSTRACT

A method is presented by means of which the equilibrium path of any elastic bar structure may be traced globally, without applying iteration techniques. The basic idea is that the bar structure is reduced to a set of Initial Value Problems (IVPs) with parameters, and the equilibrium path is piecewise linearly interpolated in the parameter space. The way in which this method is capable of handling problems of active static control is demonstrated. The mathematical basis of this type of method is described by Allgower and Georg [1] as the Piecewise Linear (PL) algorithm. Here it is shown how this algorithm can be applied to problems in structural mechanics.     

Poincaré-cartan integral invariants of Birkhoffian systems

Based on modern differential geometry, the symplectic structure of a Birkhoffian system which is an extension of conservative and nonconservative systems is analyzed. An integral invariant of Poincaré-Cartan’s type is constructed for Birkhoffian systems. Finally, one-dimensional damped vibration is taken as an illustrative example and an integral invariant of Poincaré’s type is found.     

A fractional gradient representation of the Poincaré equations

A gradient representation and a fractional gradient representation of the Poincaré equations are studied. Firstly, the condition presented here for the Poincaré equation can be considered as a gradient system. Then, a condition under which the Poincaré equation can be considered as a fractional gradient system is obtained. Finally, two examples are given to illustrate applications of the result.     

Independent continuous mapping for topological optimization of frame structures

Based on the Independent Continuous Mapping method (ICM), a topological optimization model with continuous topological variables is built by introducing three filter functions for element weight, element allowable stress and element stiffness, which transform the 0-1 type discrete topological variables into continuous topological variables between 0 and 1. Two methods for the filter functions are adopted to avoid the structural singularity and recover falsely deleted elements: the weak material element method and the tiny section element method. Three criteria (no structural singularity, no violated constraints and no change of structural weight) are introduced to judge iteration convergence. These criteria allow finding an appropriate threshold by adjusting a discount factor in the iteration procedure. To improve the efficiency, the original optimization model is transformed into a dual problem according to the dual theory and solved in its dual space. By using MSC/Nastran as the structural solver and MSC/Patran as the developing platform, a topological optimization software of frame structures is accomplished. Numerical examples show that the ICM method is very efficient for the topological optimization of frame structures.The project supported by the National Natural Science Foundation of China (10472003) and Beijing Natural Science Foundation (3042002) The English text was polished by Yunming Chen.     

Dynamical couplings analysis of rigid manipulators

A method for investigation of the dynamical couplings between the manipulator links is presented in this paper. The method is based on quasi velocities introduced originally by Hurtado (2004) and on a diagonalized Poincaré form of the Hamel representation described by Sovinsky et al. (2005). Some observations and a heuristic algorithm are proposed to determine the dynamical couplings between the rigid manipulator links. The presented strategy can be used in the design phase of manipulators and it can be realized via a simulation test. It is validated here on a 3 d.o.f. spatial manipulator.     

Determining Lyapunov spectrum and Lyapunov dimension based on the Poincaré map in a vibro-impact system

We develop a method to compute the Lyapunov spectrum and Lyapunov dimension, which is effective for both symmetric and unsymmetric vibro-impact systems. The Poincaré section is chosen at the moment after impacting, and the six-dimensional Poincaré map is established. The time between two consecutive impacts is determined by the initial conditions and the impact condition, hence the Poincaré map is an implicit map. The Poincaré map is used to calculate all the Lyapunov exponents and the Lyapunov dimension. By numerical simulations, the attractors are represented in the projected Poincaré section, and the Lyapunov spectrum is obtained. The multi-degree-of-freedom vibro-impact system may exhibit complex quasi-periodic attractors, which can be characterized by the Lyapunov dimension.     

The study on the chaotic motion of a nonlinear dynamic system

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Linearized inverse scattering for elastic parameters in a half-space with reflection data

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一类 Duffing 型系统的不动点混沌和Fold/Fold 簇发现象及机理分析

本文主要探究了一类含有两个慢变量的双稳态 Duffing 型系统,通过时间历程图、相图、分岔图等对系统进行数值模拟,然后从理论上分析不同参数下系统的动力学机理. 首先,研究发现当振幅参数取值大于 1 时,系统会表现出不动点混沌现象,并进一步解释了产生不动点混沌的机理. 其次, 介绍了参数空间中的簇发振荡现象,即系统穿过鞍结曲面的一侧到达另一侧所发生的行为,这里也称为鞍结簇发振荡. 事实上,当系统穿过鞍结曲面的时候,它的平衡点个数发生了变化. 然后,使用纵向抛物线路径说明了 Fold/Fold 簇发振荡产生的机理,发现无论常系数项和振幅的取值为多少,只要满足一定的关系,总会产生 Fold/Fold 簇发振荡,之后使用线性路径阐明了新增常系数项会使得系统发生簇发振荡的原因. 并且发现路径与鞍结曲面交点的位置会影响簇发振荡的对称性;路径的跨度会影响簇发振荡的大小. 最后,使用多拐折曲线路径讨论当两个激励项存在 $n$ 倍关系时系统产生的现象. 结果表明当 $n=3$ 时,常系数项的变化会使得系统表现出不同重数的 Fold/Fold 簇发振荡,最高可达到三重簇发振荡. 并且发现在理想状况下如果可以找到一条路径可以分割为 $n$ 段,并且每一段都会与鞍结曲面有交点,那么会产生 $n$ 重 Fold/Fold 簇发振荡.      

三维变系数热传导问题边界元分析中几乎奇异积分计算

在边界积分的数值计算过程中,当源点离积分单元很近时,边界积分就会具有几乎奇异性,此时不能直接用高斯数值积分公式计算几乎奇异积分。本文以三维非均质热传导问题为例,介绍了一种计算几乎奇异边界积分的新方法。首先,采用Newton-Raphson迭代算法确定积分单元上离源点最近的点;然后,将积分单元上任意一点的坐标在最近点处展开成泰勒级数,并计算源点到积分单元任意点的距离;最后,将距离函数代入几乎奇异边界积分中,并运用指数变换方法导出积分单元上几乎奇异积分的计算公式。文中给出了两个非均质热传导问题的算例来验证所述方法的正确性、有效性和稳定性。     

Dynamical behaviors of the periodic parameter-switching system

In this paper, a periodic parameter-switching system about Lorenz oscillators is established. To investigate the bifurcation behavior of this system, Poincaré mapping of the whole system is defined by suitable local sections and local mappings. The location of the fixed point and the parameter values of local bifurcations are calculated by the shooting method and Runge–Kutta method. Then based on the Floquent theory, we conclude that the period-doubling and saddle-node bifurcations play an important role in the generation of various periodic solutions and chaos. Meanwhile, upon the analysis of the equilibrium points of the subsystems, we explore the mechanisms of different periodic switching oscillations.     

FIXED POINT CHAOS AND FOLD/FOLD BURSTING OF A CLASS OF DUFFING SYSTEMS AND THE MECHANISM ANALYSIS 1)

本文主要探究了一类含有两个慢变量的双稳态 Duffing 型系统,通过时间历程图、相图、分岔图等对系统进行数值模拟,然后从理论上分析不同参数下系统的动力学机理. 首先,研究发现当振幅参数取值大于 1 时,系统会表现出不动点混沌现象,并进一步解释了产生不动点混沌的机理. 其次, 介绍了参数空间中的簇发振荡现象,即系统穿过鞍结曲面的一侧到达另一侧所发生的行为,这里也称为鞍结簇发振荡. 事实上,当系统穿过鞍结曲面的时候,它的平衡点个数发生了变化. 然后,使用纵向抛物线路径说明了 Fold/Fold 簇发振荡产生的机理,发现无论常系数项和振幅的取值为多少,只要满足一定的关系,总会产生 Fold/Fold 簇发振荡,之后使用线性路径阐明了新增常系数项会使得系统发生簇发振荡的原因. 并且发现路径与鞍结曲面交点的位置会影响簇发振荡的对称性;路径的跨度会影响簇发振荡的大小. 最后,使用多拐折曲线路径讨论当两个激励项存在 $n$ 倍关系时系统产生的现象. 结果表明当 $n=3$ 时,常系数项的变化会使得系统表现出不同重数的 Fold/Fold 簇发振荡,最高可达到三重簇发振荡. 并且发现在理想状况下如果可以找到一条路径可以分割为 $n$ 段,并且每一段都会与鞍结曲面有交点,那么会产生 $n$ 重 Fold/Fold 簇发振荡.     

Poincaré and the shortcomings of the Hamilton-Jacobi method for classical or quantized mechanics

A great theorem was proven by H. Poincaré in celestial mechanics. It states that, in the most general problems of mechanics, the total energy of the system is the only well behaved first integral of the system, while other so-called integrals cannot be represented by uniform and convergent series. This very important result can be explained and visualized by comparison with standard methods of discussion, as, for example, the Hamilton-Jacobi procedure. The discussion shows that there are serious limitations to the use of this procedure, which collapses in the most general problems (Poincaré theorem) and can be used only for “almost separated” variables. The Poincaré theorem appears to provide the distinction between determinism in mechanics and statistical mechanics according to Boltzmann. The research presented here done under Contract Nonr 266(56) and was first described in a Quarterly Report dated July 31, 1959.     

Structure of saddle-node and cusp bifurcations of periodic orbits near a non-transversal T-point

Non-transversal T-points have been recently found in problems from many different fields: electronic circuits, pendula, and laser problems. In this work, we study a model based on the construction of a Poincaré map that describes the behaviour of curves of saddle-node and cusp bifurcations in the vicinity of such a non-transversal T-point. This model is also able to predict, reproduce, and explain the numerical results previously obtained in Chua’s equation.     

About the basic integral variants of holonomic nonconservative dynamical systems

In this paper, we prove that for holonomic nonconservative dynamical system the Poincaré and Poincaré-Cartan integral invariants do not exist. Instead of them, we introduce the integral variants of Poincaré-Cartan's type and of Poincare's type for holonomic nonconservative dynamical systems, and use these variants to solve the problem of nonlinear vibration. We also prove that the integral invariants introduced in references [1] and [2] are merely the basic integral variants given by this paper. Project supported by National Natural Science Foundation of China     

Detecting unstable periodic orbits and unstable quasiperiodic orbits in vibro-impact systems

In this paper, unstable dynamics is considered for the models of vibro-impact systems with linear differential equations coupled to an impact map. To provide a skeleton for the organization of chaotic attractors, we propose a method for detecting unstable periodic orbits embedded in chaotic attractors through a combination of unconstrained optimization technique and Poincaré map. Three numerical examples from different vibro-impact models demonstrate that the strategy can efficiently detect unstable periodic orbits in chaotic attractors. In order to explore the mechanism responsible for the creation of multi-dimensional tori attractors, we also present another method to detect unstable quasiperiodic orbits embedded multi-dimensional tori attractors by examining a specially transient time series. The upper bound and lower bound of the transient time series (in the Poincaré map) can be obtained by analyzing transient Lyapunov exponent and transient Lyapunov dimension. Some examples verify the effectiveness of the numerical algorithm.     

优化迭代步长的两种改进增量谐波平衡法

增量谐波平衡法(IHB法)是一个半解析半数值的方法, 其最大优点是适合于强非线性系统振动的高精度求解. 然而, IHB法与其他数值方法一样, 也存在如何选择初值的问题, 如初值选择不当, 会存在不收敛的情况. 针对这一问题, 本文提出了两种基于优化算法的IHB法: 一是结合回溯线搜索优化算法(BLS)的改进IHB法(GIHB1), 用来调节IHB法的迭代步长, 使得步长逐渐减小满足收敛条件; 二是引入狗腿算法的思想并结合BLS算法的改进IHB法(GIHB2), 在牛顿-拉弗森(Newton-Raphson)迭代中引入负梯度方向, 并在狗腿算法中引入2个参数来调节BSL搜索方式用于调节迭代的方式, 使迭代方向沿着较快的下降方向, 从而减少迭代的步数, 提升收敛的速度. 最后, 给出的两个算例表明两种改进IHB法在解决初值问题上的有效性.