横向磁场中大挠度金属薄板的混沌振动

利用Karman关于板的大挠度理论,考虑涡电流在板中引起的Lorentz力,导出了在横向磁场和横向载荷共同作用下薄板的非线性运动方程.借助Bubnov-Galerkin法将非线性偏微分方程转化为含三次非线性项的常微分方程.在定性分析的基础上,利用次谐轨道的Melnikov函数给出了发生Smale马蹄型混沌运动的阈值条件,进而数值计算了系统的分岔图、相应的相图、Poincaré映射和时程曲线,给出了混沌运动的数字特征.分析结果表明:磁感应强度和外载荷都会影响系统的振动特性. 关键词： 金属薄板 大挠度 横向磁场 混沌振动     

非线性电路中的混沌现象

混沌来自非线性．非线性电路中有十分丰富的分叉和混沌现象．虽然早在１９２７年Ｖａｎ ｄｅｒ Ｐｏｌ就发现了“不规则的噪音”，但当时尚未意识到这就是现在所称的混沌．只是直到前几年Ｌ．Ｏ．Ｃｈｕａ（蔡少棠）教授设想出蔡氏电路后，混沌电路的研究才在全世界普遍开展．本文主要介绍蔡氏电路的结构，非线性电阻器的实现，以及电路混沌的实验研究方法．     

复摆振动中的混沌现象

利用组合物理实验仪,研究了复摆振动中的混沌现象,分析了产生混沌的物理条件,并给出不同条件下的混沌图形.     

非线性音乐物理学与混沌

爱因斯坦早就预言:“由于物理学的基本方程都是非线性的,因此,所有的数学物理都必须从头研究”。果然,当代“非线性科学”正在全世界迅速崛起和发展,它不只是向数学物理,而是几乎向所有的学科和。领域提出了挑战。人们称它是20世纪物理学继相对论和量子力学之后的第三次革命。目前,它是众多学科和领域共同关注的前沿研究课题之一。音乐物理学也不例外。在音乐物理学中,几乎处处遇到非线性现象。 人的声带发声,气流从平流转变成湍流时才能激发声带的明显振动。管乐器发声的第一原理就是湍流。例如最简单的单簧管,其管身为线性元件而簧为非线性元件。演奏者控制气流,发生不同程度的湍流,从而改变音色和音高。唢呐的哨片便是非线性元件,其非线性效应极强,在弦乐器中,一般弦上的作用力都是非线性的。例如钢琴,演奏者击键作用力一般都超过线性阻尼振动范围。拉弦乐器的弓-弦作用一般也都是非线性的。小提琴如此,中国的京胡更明显,其松香颗粒极大地增加了弓-弦摩擦阻尼,使其发出非线性效应极强的特殊音色。至于打击乐器,其非线性特征就更明显。例如中国大锣、钹、镲等等,其振动一般都是非线性振动。此外,在音乐厅建筑声学中也必须考虑非线性因素,因为真实的边界条件一般都是非线性的。     

非线性串联电路中的拟张弛振动

由于电阻温升的改变滞后于电流的改变,在非线性串联电路中可能引起电流或电压振幅的周期性突变——拟张弛振动。本文分析了产生这种拟张弛振动的充要条件,以及当参数变化时系统的性态等问题。     

延迟-非线性反馈控制混沌

提出了基于稳定性准则的延迟非线性反馈控制混沌的方法，即SC延迟非线性反馈控制法. 通过对混沌系统的适当分离，得到一个特殊的非线性函数，并利用混沌输出信号与其延迟信号的非线性函数的差，构造了连续反馈输入干扰，以控制混沌轨到某一期望的不稳周期轨上. 该方法继承了延迟反馈控制方法的优点，实现了自-控制过程. 另外由于该方法基于线性系统的稳定性准则，保证了控制的有效性. 控制过程可随时开始，具有简便、灵活性. 给出耦合Duffing振子的例子，数值模拟结果显示了SC延迟反馈方法控制的有效性. 关键词： 稳定性准则 混沌控制 延迟反馈 干扰     

组合线性弹簧振子中的非线性振动

从拉格朗日方程出发,分析了几种常见的线性弹簧组合,对作非线性振动弹簧振子进行了数值求解.当作微小振动时,正好是几种典型的非线性振动.通过计算得出解析解并与数值解进行了对比.     

自治混沌系统的线性和非线性广义同步

研究了自治混沌系统的广义同步问题.基于改进的状态观测器方法和极点配置技术，提出了一种新的广义同步方案，扩展了混沌广义同步的适用范围，并用该方法实现了自治混沌系统的线性及非线性广义同步.根据状态观测器理论，给出了驱动-响应系统获得全局渐进广义同步的充分条件.数值仿真实验进一步验证了所提方法的有效性.     

NON-LINEAR TRANSVERSE VIBRATIONS OF A SIMPLY SUPPORTED BEAM CARRYING CONCENTRATED MASSES

An Euler-Bernoulli beam carrying concentrated masses is considered to be a beam-mass system. The beam is simply supported at both ends. The non-linear equations of motion are derived including stretching due to immovable end conditions. The stretching introduces cubic non-linearities into the equations. Forcing and damping terms are also included. Exact solutions for the natural frequencies are given for the linear problem. For the non-linear problem, an approximate solution using a perturbation method is searched. Non-linear terms of the perturbation series appear as corrections to the linear problem. Amplitude and phase modulation equations are obtained. Non-linear free and forced vibrations are investigated in detail. The effect of the positions, magnitudes and number of the masses are investigated.     

用激光束演示振动合成

本文给出一种用激光束演示振动合成的新的实验方法，用两金属片作振源来源示振动合成，比《大学物理》１９８４年第４期上曾报道的用两个小型扬声器作振源，演示效果更好，制作更简便。     

非线性环形光学腔中分岔混沌的新行为

利用数值模拟方法研究了非线性环形光学腔中的Ikeda方程,发现:长延时下,分岔图的拓扑结构、分岔方式、收敛速率以及周期窗口的结构都随参数B,φ₀的取值不同而异。短延时下,通过适当近似,将Ikeda方程化为带有一定记亿效应的迭代映象,所得到的近似系统显示出非倍周期分岔行为,并伴有一些奇特的周期窗口,混沌区中的无限嵌套的自相似结构变得模糊,不易分辨。 关键词：     

FREE VIBRATIONS OF SIMPLY SUPPORTED AND MULTILAYERED MAGNETO-ELECTRO-ELASTIC PLATES

Analytical solutions are derived for free vibrations of three-dimensional, linear anisotropic, magneto-electro-elastic, and multilayered rectangular plates under simply supported edge conditions. For any homogeneous layer, we construct the general solution in terms of a simple formalism that resembles the Stroh formalism, from which any physical quantities can be solved for given boundary conditions. In particular, the dispersion equation that characterizes the relationship between the natural frequency and wavenumber can be obtained in a simple form. For multilayered plates, we derive the dispersion relation in terms of the propagator matrices. The present solution includes all previous solutions, such as piezoelectric, piezomagnetic, and purely elastic solutions as special cases, and can serve as benchmarks to various thick plate theories and numerical methods used for the modelling of layered composite structures. Typical natural frequencies and mode shapes are presented for sandwich piezoelectric/piezomagnetic plates. It is shown clearly that some of the modes are purely elastic while others are fully coupled with piezoelectric/piezomagnetic quantities, with the latter depending strongly upon the material property and stacking sequence. These frequency and mode shape features could be of particular interest to the analysis and design of various “smart” sensors/actuators constructed from magneto-electro-elastic composite laminates.     

BIFURCATIONS AND CHAOS OF AN IMMERSED CANTILEVER BEAM IN A FLUID AND CARRYING AN INTERMEDIATE MASS

The concern of this work is the local stability and period-doubling bifurcations of the response to a transverse harmonic excitation of a slender cantilever beam partially immersed in a fluid and carrying an intermediate lumped mass. The unimodal form of the non-linear dynamic model describing the beam-mass in-plane large-amplitude flexural vibration, which accounts for axial inertia, non-linear curvature and inextensibility condition, developed in Al-Qaisia et al. (2000Shock and Vibration7 , 179-194), is analyzed and studied for the resonance responses of the first three modes of vibration, using two-term harmonic balance method. Then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict the zones of symmetry breaking leading to period-doubling bifurcation and chaos on the resonance response curves. The results of the present work are verified for selected physical system parameters by numerical simulations using methods of the qualitative theory, and good agreement was obtained between the analytical and numerical results. Also, analytical prediction of the period-doubling bifurcation and chaos boundaries obtained using a period-doubling bifurcation criterion proposed in Al-Qaisia and Hamdan (2001 Journal of Sound and Vibration244, 453-479) are compared with those of computer simulations. In addition, results of the effect of fluid density, fluid depth, mass ratio, mass position and damping on the period-doubling bifurcation diagrams are studies and presented.     

含二次非线性项受迫振动系统中的分岔与混沌现象

用数值积分及二维叠代对含二次非线性项受迫振动系统x+kx+ω₀²x-dx²=μcosωt中的分岔与混沌现象进行了研究。控制参数μ的一系列临界值与α成反比。 关键词：     

OBSERVATIONS OF HOMOCLINIC CHAOS THROUGH ALTERNATING PERIODIC-CHAOTIC SEQUENCES IN A NONLINEAR CIRCUIT

Homoclinic chaos in the alternating periodic-chaotic sequences is observed in a nonlinear circuit with sinusoidal driving force. In particular, a complete Alternating Periodic-Chaotic sequence is recorded with a high-resolution up to P⁽⁸⁾ state. The experimental results, analyzed by constructing the time of flight and the next maximal amplitude return maps, are in good agreement with the scenario described by Shilnikov. The underlying dynamics of homoclinic chaos is determined from the next amplitude return map, to be that of a unimodal map and thus a strong dissipation case.     

一种基于系统变量的线性和非线性变换实现混沌控制的方法

提出了一种通过系统变量的线性和非线性变换实现混沌控制的方法,以Henon映象和Lorenz系统作为两个典型的例子,验证了这种控制方法的有效性.结果表明,通过改变系统变量之间的线性变换矩阵,可以实现混沌系统中各种不稳定周期轨道的稳定控制.将这种方法推广到非线性(神经网络)变换,可以获得稳定的新的动力学行为.同时,从信息熵的角度,讨论了这种混沌控制方法的物理机制. 关键词：