非线性系统模态分叉与模态局部化现象

运用匹配法和多尺度法对一个两自由度非线性系统进行了研究，详细分析了非线性系统的模态分叉和局部化现象。     

一种用于非线性振动系统的模态分析方法

本文提出了一种用于非线性振动系统的模态分析方法,将求解非线性系统模态的问题化为求解非线性特征值、特征向量的问题,利用模态研究系统的响应,文中分析了非线性保守系统、非线性自治系统和非线性非自治系统的线性模态,导出了三个模态包含原理。     

多自由度内共振系统非线性模态的分岔特性

利用多尺度法构造了一个立方非线性1:3内共振系统的内共振非线性模态(NonlinearNormal Modes associated with internal resonance)．研究表明,内共振非线性系统除存在单模态运动外还存在耦合模态运动．耦合内共振模态具有分岔特性．利用奇异性理论对模态分岔方程进行分析发现此类系统的模态存在叉形点分岔和滞后点分岔这两种典型的分岔模式．     

基于假设模态加权的全局模态提取方法及其应用

假设模态法在单一梁、杆、索、板等柔性结构动力学建模中有广泛应用,但在处理组合结构振动问题时,常常因无法反映各部件之间的耦合作用使其应用受限。通过假设模态建立组合结构的近似动力学模型,利用近似模型求得系统的固有频率和相应的特征向量,据此可以有效地获得系统的全局模态。本文以跨中带有多个弹性支撑的简支梁为例,通过假设模态加权来提取系统的全局模态,从而建立系统的动力学模型。对系统进行固有特性分析的结果表明,通过假设模态加权可以方便地获得系统的全局模态;对系统动态响应分析的结果表明,采用本文提出的全局模态建立的非线性动力学模型可以有效地反映系统的非线性动力学特性。     

非线性模态的分类和新的求解方法

引入不可分偶数维不变流形的概念来定义非线性模态．在此基础上，揭示出了一种新的模态——耦合非线性模态，并对实际系统中各种可能的模态进行了分类．这种分类可能是新的构筑非线性模态理论的框架．用此方法构造非线性模态，得到的模态振子具有范式的形式，形式最简、却能反映原系统在平衡点附近的主要动力学行为，且易于得到非线性频率及非线性稳定性等方面的信息．不仅适用于分析一般的多自由度系统，还可用于分析奇数维系统；不仅可构造内共振系统的非耦合模态，还可用于构造内共振耦合模态．从掌握的资料看，以前的方法还不能解决上述所有问题     

非线性模态构造方法与机电耦合系统Hopf分岔

大型汽轮发电机组轴系与电网耦合次同步谐振（ＳＳＲ）现象是在某种参数条件下机电耦合系统产生Ｈｏｐｆ分岔的结果。在文献［１］中，作者提出了分析这种系统Ｈｏｐｆ分岔的非线性模态方法，得出了在固定参数下分岔解的结果。本文针对高维非线性动力系统（包括奇数维），提出新的非线性模态构造方法，并给出了机电耦合次同步振荡系统在辅助参数变化条件下分岔解的的变化规律。     

非线性系统非平稳随机响应的时域模态分析法

采用时域模态分析和统计线性化法，得到了一个计算非线性多自由度系统非平稳随机响应的方法。该方法是基于统计线性化参数在一系列微小时间间隔内保持不变，而在这些微小时间间隔的分界点突然改变的假定。考虑了等效线性化系统的时变性；获得了响应协方差矩阵的递推关系。给出了两个算例，并将计算结果与相应的数字模拟结果进行了比较。结果证明该方法是具有满意精度和有效，而且可用于时变系统     

一类非线性耦合振子的模态分析

本文研究了一非线性耦合振子系统的相似和非相似模态，用模态的方法分析和讨论此系统单振子振动和两振子的同步或反步周期运动，并且给出数值结果，以考察非线性模态的有效性。     

复合柔性结构全局模态函数提取与状态空间模型构建

随着航空航天等领域中实际工程结构的大型化和柔性化,结构的非线性振动和主动振动控制问题越来越凸显.分析和处理此类结构出现的复杂振动问题的关键在于建立系统的非线性动力学模型与状态空间模型.对于由柔性部件、刚体、连接部件构成的复合柔性结构,由于各部件之间的振动耦合效应,单个柔性部件在悬臂、简支和自由等静定边界下的模态与结构的真实模态有较大差异.为此,本文提出复合柔性结构全局模态的解析提取方法,通过全局模态离散得到系统非线性动力学模型,从而构建状态空间模型.该方法采用笛卡尔坐标描述系统的运动,建立系统的运动方程;结合描述柔性部件的偏微分方程、刚体的常微分运动方程、连接界面处力、力矩、位移和转角的匹配条件以及系统的边界条件,利用分离变量法给出统一形式的频率方程,获取系统的固有频率和解析函数表征的全局模态.这里提出的全局模态提取方法不仅便于复合柔性结构固有频率和全局模态的参数化分析,而且为建立复合柔性结构低维非线性动力学模型和状态空间模型提供了有效的途径,对于推进这类结构的非线性动力学分析与主动振动控制研究具有重要意义.      

关于对模态概念的理解

通过分析模态的性质井与复模态和非线性模态比较以加强对模态概念的理解.固有模态的基本性质是模态振动的同频性、对初始条件的不变性、模态的正交性和系统响应的叠加性.复模态仍具有模态振动的同频性,但没有对初始条件的不变性,正交性和叠加性仅在状态空间中成立.非线性模态仅保留了同频性或不变性,不具有正交性和叠加性.     

一类多自由度强非线性振动系统主共振的渐近解法

提出一种渐近方法用来处理一类多自由度强非线性自治振动系统，它是新渐近方法￣［１］的推广。本方法适用于主共振情形，我们建立了振幅和相位所满足的方程。文末用两个例子说明本方法的有效性。     

A very complicated structure of resonances in a nonlinear system with cyclic symmetry: Nonlinear forced localization

The fundamental and subharmonic resonances of a nonlinear cyclic assembly are examined using the asymptotic method of multiple-scales. The system consists of a number of identical cantilever beams coupled by means of weak linear stiffnesses. Assuming beam inextensionality, geometric nonlinearities arise due to longitudinal inertia and the nonlinear relation between beam curvature and transverse displacement. The governing nonlinear partial differential equations are discretized by a Galerkin procedure and the resulting set of coupled ordinary differential equations is solved using an asymptotic analysis. The unforced assembly is known to possess localized nonlinear normal modes, which give rise to a very complicated topological structure of fundamental and subharmonic response curves. In contrast to the linear system which exhibits as many forced resonances as its number of degrees of freedom, the nonlinear system is found to possess a number of additional resonance branches which have no counterparts in linear theory. Some of the additional resonances are spatially localized, corresponding to motions of only a small subset of periodic elements. The analytical results are verified by numerical Poincaré maps, and the forced localization features of the nonlinear assembly are demonstrated by considering its response to impulsive excitations.     

一般叠层圆柱厚壳的非线性动力稳定性分析

基于Ｔｉｍｏｓｈｅｎｋｏ－Ｍｉｎｄｌｉｎ假设及Ｈａｍｉｌｔｏｎ原理，建立了一般纤维叠层圆柱厚壳在参数激励下的非线性振动方程；应用多模态近似和增量谐波平衡法求解了叠层圆柱厚壳的非线性动力稳定性问题。横向剪切变形、端部支承条件等因素的影响被讨论。     

基于摄动原理的复杂土层地震反应分析的子结构法

把约束子结构模态综合法与直接模态摄动法相结合,建立复杂场地三维地震反应等效线性化分析计算方法.应用直接模态摄动原理,可简化各子结构模态分析过程,将特征值求解问题转化为线性代数方程组的求解,从而可有效提高计算效率.算例表明,该方法在提高大规模复杂场地地震反应分析计算效率方面优势明显.     

Singular Characteristics of Nonlinear Normal Modes in a Two Degrees of Freedom Asymmetric Systems with Cubic Nonlinearities

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Localization of compact invariant sets of a new nonlinear finance chaotic system

The application of the nonlinear chaotic dynamic system in economics and finance has expanded rapidly in the last decades. This paper considers the localization of all compact invariant sets of a new three-dimensional autonomous nonlinear finance chaotic system. The boundedness of the new finance chaotic system is the first time being investigated. Based on the iteration theorem and the first-order extremum theorem, a new method is proposed, too. The comparison of our method with the traditional method is presented as well. More specifically, the compact invariant sets are analyzed in three aspects: First of all, a localization of the new finance chaotic system by two frusta and an ellipsoidal used by traditional methods is discussed. Second, a localization of the new finance chaotic system by two frusta and a parabolic cylinder is provided. Third, localization of the new finance chaotic system according to superposition of the ellipsoidal, parabolic cylinder, and two frusta are presented, and the boundedness of the chaotic attracter is smaller than in the classical methods. Numerical simulations are given to indicate the effectiveness of the proposed method.     

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.     

Nonlinear Normal Modes of a Parametrically Excited Cantilever Beam

We investigate theoretically thenonlinear normal modes of a vertical cantilever beam excited by aprincipal parametric resonance. We apply directly the method ofmultiple scales to the governing nonlinear nonautonomousintegral-partial-differential equation and associated boundary conditions.In the absence of damping, it is shown that the system has nonlinear normal modes, as defined by Rosenberg, even in the presence of the parametric excitation.We calculate the spatial correction to the linear mode shapedue to the effects of the inertia and curvature nonlinearities andthe parametric excitation. We compare the result obtained withthe direct approach with that obtained using a single-mode Galerkindiscretization.The deviation between the two predictions increases as the oscillationamplitude increases.     

Damping of Free Elastic Vibrations in Linear Systems

The possibility of using a Mises truss as an absorber of free elastic vibrations in a linear elastic system is examined. The nonlinear normal mode method is used to analyze nonlinear vibrations. A local nonlinear normal mode is shown to be favorable for vibration damping__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 110–117, February 2005.     

Geometric nonlinear formulation and discretization method for a rectangular plate undergoing large overall motions

In the present paper, the geometric nonlinear formulation is developed for dynamic stiffening of a rectangular plate undergoing large overall motions. The dynamic equations, which take into account the stiffening terms, are derived based on the virtual power principle. Finite element method is employed for discretization of the plate. The simulation results of a rotating rectangular plate obtained by using such geometric nonlinear formulation are compared with those obtained by the conventional linear method without consideration of the stiffening effects. The application limit of the conventional linear method is clarified according to the frequency error. Furthermore, the accuracy of the assumed mode method is investigated by comparison of the results obtained by using the present finite element method and those obtained by using the assumed mode method.