两自由度非对称三次系统非线性模态的奇异性质

利用非线性模态子空间的不变性和摄动技术,研究两自由度非对称三次系统在奇异条件下系统的性质.重点考虑子系统之间线性耦合退化时的奇异性质.对于非共振情形,所得到的解析结果表明,系统出现单模态运动以及振动局部化现象,这种现象的强弱不但与非线性耦合刚度有关,而且与非对称参数有关.并解析地得到了参数的门槛值;对于1:1共振情形,模态随非线性耦合刚度和非对称参数的变化会出现分岔,得到了参数分岔集以及模态的分岔曲线.     

两自由度非对称三次系统非奇异时的非线性模态及叠加性

本文利用非线性模态子空间的不变性研究两自由度非对称三次系统在非奇异条件下的非线性模态及其模态叠加解有效性，重点考虑这种有效性与模态动力学方程静态分岔之间的关系·大量的数值结果表明，非线性模态解的有效性不仅与其局部性的限制有关，而且与模态动力学方程静态解分岔有关·     

两自由度非线性振动系统主参数激励下的分岔分析

本文给出了参数激励作用下两自由度非线性振动系统,在1:2内共振条件下主参数激励低阶模态的非线性响应.采用多尺度法得到其振幅和相位的调制方程,分析发现平凡解通过树枝分岔产生耦合模态解,采用Melnikov方法研究全局分岔行为,确定了产生Smale马蹄型混沌的参数值.     

新的变系数可积耦合非线性Schr(o)dinger方程及其孤子解

基于延拓结构和Hirota双线性方法研究了广义的变系数耦合非线性Schrdinger方程.首先导出了3组新的变系数可积耦合非线性Schrdinger方程及其线性谱问题(Lax对),然后利用Hirota双线性方法给出了它们的单、双向量孤子解.这些向量孤子解在光孤子通讯中有重要的应用.     

新的变系数可积耦合非线性Schr(o)dinger方程及其孤子解

基于延拓结构和Hirota双线性方法研究了广义的变系数耦合非线性Schr(o)dinger方程.首先导出了3组新的变系数可积耦合非线性Schr(o)dinger方程及其线性谱问题(Lax对),然后利用Hirota双线性方法给出了它们的单、双向量孤子解.这些向量孤子解在光孤子通讯中有重要的应用.     

碰摩转子中弯扭耦合作用的影响分析

建立了一碰摩转子的弯扭耦合数学模型,应用非线性动力学现代理论和一未考虑弯扭耦合的数学模型的动态响应特征进行比较后发现,两类情况下系统都出现了周期、混沌和加周期等非线性动力学现象.虽然两类方程分岔图的变化过程基本相同,但扭转作用的具体影响却不容忽视.这些结果对以后分析转子碰摩现象有一定的参考价值.     

微重力环境下充液球腔非线性耦合动力学研究

采用球坐标系描述球腔中的液体动力学特性并建立一种轴对称贮腔类液刚耦合系统动力学模型．采用模态展开方法分析了微重环境下球形贮箱中的液体晃动问题,给出了球形贮箱内液体晃动速度势函数和波高函数的Gauss超几何级数解析表达式．采用变分原理推导了系统动力学系模型,利用Galerkin 方法对变分方程进行特征频率分析．运用Lagrange方法及非线性动力学方法导出了微重力环境下贮箱中液体与航天器结构耦合的动力学方程组,并对该方程组进行了数值计算,绘出了非线性耦合充液系统自由度随时间的变化历程．     

非线性系统动力分析的模态综合技术

各种模态综合方法已广泛应用于线性结构的动力分析,但是,一般都不适用于非线性系统. 本文基于[20][21]提出的方法,将一种模态综合技术推广到非线性系统的动力分析.该法应用于具有连接件耦合的复杂结构系统,以往把连接件简化为线性弹簧和阻尼器.事实上,这些连接件通常具有非线性弹性和非线性阻尼特性.例如,分段线性弹簧、软特性或硬特性弹簧、库伦阻尼、弹塑性滞后阻尼等.但就各部件而言,仍属线性系统.可以通过计算或试验或兼由两者得到一组各部件的独立的自由界面主模态信息,且只保留低阶主模态.通过连接件的非线性耦合力,集合各部件运动方程而建立成总体的非线性振动方程.这样问题就成为缩减了自由度的非线性求解方程,可以达到节省计算机的存贮和运行时间的目的.对于阶次很高的非线性系统,若能缩减足够的自由度,那么问题就可在普通的计算机上得以解决. 由于一般多自由度非线性振动系统的复杂性,一般而言,这种非线性方程很难找到精确解.因此,对于任意激励下系统的瞬态响应,可以采用数值计算方法求解缩减的非线性方程.     

具有平方、立方非线性项的耦合动力学系统1：2内共振分贫

对一类具有平方,立方非线性项的耦合动力学系统1：2内共振情形进行了研究,首先,用直接方法求出该系统1：2内共振时的Normal Form,该系统的Normal Form中,不仅含有平方非线性项,同时还含有立方非线性项,通过采用适当的变量变换,将4维分贫方程约化成3维,进而得到单变量4次分岔方程,最后用奇异性理论,研究了一类普适开折的分岔特性,该方法可用于4维中心流形上流的强内共振时的分岔行为分析。     

轴向变速运动粘弹性弦线横向振动的复模态Galerkin方法

在考虑初始张力和轴向速度简谐涨落的情况下,利用含预应力三维变形体的运动方程,建立了轴向变速运动弦线横向振动的非线性控制方程,材料的粘弹性行为由Kelvin模型描述．利用匀速运动线性弦线的模态函数构造了变速运动非线性弦线复模态Galerkin方法的基底函数,并借助构造出来的基底函数研究了复模态Galerkin方法在轴向变速运动粘弹性弦线非线性振动分析中的应用．数值结果表明,复模态Galerkin方法相比实模态Galerkin方法对变系数陀螺系统有较高的收敛速度．     

Applied multiscale method to analysis of nonlinear vibration for double-walled carbon nanotubes

The aim of this paper is to show how the concept of nonlinear normal modes (NNMs) can be used to characterize the nonlinear dynamical behaviors of double-walled carbon nanotubes (DWCNTs). DWCNTs are modeled as double simply supported elastic beams with van der Waals (vdW) forces between the inner and outer walls. The multiscale method for deriving the approximate solutions of NNMs is applicable to the nonlinear systems. According to the procedure, the typical features – coaxial and noncoaxial vibrations of the system are exhibited in the literature. Moreover, the case of 1:3 internal resonance is discussed in detail, which can give rise to much more complex phenomenon for DWCNTs systems. Meanwhile, the amplitude-time curves of the nonlinear vibration with different initial conditions are presented, and the amplitude-frequency characteristic curves of the nonlinear vibration are also obtained.     

Experimental study of steady-state localization in coupled beams with active nonlinearities

Summary Steady-state nonlinear motion confinement is experimentally studied in a system of weakly coupled cantilever beams with active stiffness nonlinearities. Quasistatic swept-sine tests are performed by periodically forcing one of the beams at frequencies close to the first two closely spaced modes of the system, and experimental nonlinear frequency response curves for certain nonlinearity levels are generated. Of particular interest is the detection of strongly localized steady-state motions, wherein vibrational energy becomes spatially confined mainly to the directly excited beam. Such motions exist in neighborhoods of strongly localized antiphase nonlinear normal modes (NNMs) which bifurcate from a spatially extended NNM of the system. Steady-state nonlinear motion confinement is an essentially nonlinear phenomenon with no counterpart in linear theory, and can be implemented in vibration and shock isolation designs of mechanical systems.Presently Assistant Professor of Aerospace and Mechanical Engineering, Boston University (from January 1995).     

Comparison of the bifurcation curves of a two-variable and a three-variable circadian rhythm model

Two dynamical systems describing the circadian fluctuation of two proteins (PER and TIM) in cells are compared. A simplified model with two variables has already been investigated. Detailed study of the possible bifurcation has been carried out. Periodic solutions of the differential equations with 24-h period have been obtained numerically. Here the general, more realistic model having three variables is investigated. The possible phase portraits and local bifurcations are studied in detail. The saddle-node and Hopf-bifurcation curves are determined in the plane of two parameters by using the parametric representation method. Using these curves the number and the type of the stationary points can be determined. The relation of the two bifurcation curves and the Takens–Bogdanov bifurcation points are also studied. The bifurcation curves are compared to those obtained for the simplified two-variable system.     

Bifurcation sequences of vibroimpact systems near a 1:2 strong resonance point

Two vibroimpact systems are considered, which can exhibit symmetrical double-impact periodic motions under suitable system parameter conditions. Dynamics of such systems are studied by use of maps derived from the equations of motion, between impacts, supplemented by transition conditions at the instants of impacts. Two-parameter bifurcations of fixed points in the vibroimpact systems, associated with 1:2 strong resonance, are analyzed. Interesting features like Neimark–Sacker bifurcation of period-1 double-impact symmetrical motion, tangent bifurcation of period-2 four-impact motion, period-doubling bifurcation of period-2 four-impact motion and Neimark–Sacker bifurcation of period-4 eight-impact motion, etc., are found to occur near 1:2 resonance point of a vibroimpact system. The quasi-periodic attractor, associated with the fixed point of period-1 double-impact symmetrical motion, is destroyed as a tangent bifurcation of fixed points of period-2 four-impact motion occurs. However, for the other vibroimpact system the quasi-periodic attractor is restored via the collision of stable and unstable fixed points of period-2 four-impact motion. The results mean that there exist possibly more complicated bifurcation sequences of period-two cycle near 1:2 resonance points of non-linear dynamical systems.     

A study on stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point with the moment method

This paper analyzes the stochastic resonance induced by a novel transition of one-dimensional bistable system in the neighborhood of bifurcation point with the method of moment, which refer to the transition of system motion among a potential well of stable fixed point before bifurcation of original system and double-well potential of two coexisting stable fixed points after original system bifurcation at the presence of internal noise. The results show: the semi-analytical result of stochastic resonance of one-dimensional bistable system in the neighborhood of bifurcation point may be obtained, and the semi-analytical result is in accord with the one of Monte Carlo simulation qualitatively, the occurrence of stochastic resonance is related to the bifurcation of noisy nonlinear dynamical system moment equations, which induce the transfer of energy of ensemble average (Ex) of system response in each frequency component and make the energy of ensemble average of system response concentrate on the frequency of input signal, stochastic resonance occurs.     

A digital model of coupled oscillators

A new model of coupled oscillators is proposed and investigated. All phase variables and parameters are integer-valued. The model is shown to exhibit two types of motions, those which involve periodic phase differences, and those which involve drift. Traditional dynamical concepts such as stability, bifurcation and chaos are examined for this class of integer-valued systems. Numerical results are presented for systems of two and three oscillators. This work has application in digital technology.     

Bifurcation diagrams in a class of Kolmogorov systems

We study a class of Kolmogorov systems of dimension two depending on two independent parameters. The local behavior of the system is described in terms of bifurcation diagrams which contain the bifurcation curves separating a node from a focus.     

Stochastic resonance induced by novel random transitions of motion of FitzHugh–Nagumo neuron model

In contrast to the previous studies which have dealt with stochastic resonance induced by random transitions of system motion between two coexisting limit cycle attractors in the FitzHugh–Nagumo (FHN) neuron model after Hopf bifurcation and which have dealt with the phenomenon of stochastic resonance induced by external noise when the model with periodic input has only one attractor before Hopf bifurcation, in this paper we have focused our attention on stochastic resonance (SR) induced by a novel transition behavior, the transitions of motion of the model among one attractor on the left side of bifurcation point and two attractors on the right side of bifurcation point under the perturbation of noise. The results of research show: since one bifurcation of transition from one to two limit cycle attractors and the other bifurcation of transition from two to one limit cycle attractors occur in turn besides Hopf bifurcation, the novel transitions of motion of the model occur when bifurcation parameter is perturbed by weak internal noise; the bifurcation point of the model may stochastically slightly shift to the left or right when FHN neuron model is perturbed by external Gaussian distributed white noise, and then the novel transitions of system motion also occur under the perturbation of external noise; the novel transitions could induce SR alone, and when the novel transitions of motion of the model and the traditional transitions between two coexisting limit cycle attractors after bifurcation occur in the same process the SR also may occur with complicated behaviors types; the mechanism of SR induced by external noise when FHN neuron model with periodic input has only one attractor before Hopf bifurcation is related to this kind of novel transition mentioned above.