考虑损伤效应的悬索非线性共振响应分析

悬索在其施工、运营和维护阶段会不可避免地遭受损伤,导致振动特性发生改变。本文基于哈密顿变分原理,引入与损伤程度、范围和位置相关的三个无量纲参数,建立损伤效应影响下悬索面内动力学模型,并推导其无穷维的非线性动力学微分方程。利用高阶多尺度法得到系统发生主共振响应时的幅频响应方程及稳态解。数值算例表明,悬索线性和非线性共振响应特性与损伤效应密切相关。悬索一旦发生损伤,其张力减小,垂跨比增加,将形成新的静力构形。受损悬索的固有频率将下降,且随着损伤程度增加而进一步减小。损伤会导致悬索正/反对称模态频率的交点发生偏移,影响系统内共振响应特性;损伤会引发系统振动特性发生明显定量和定性改变,但是垂跨比不同,其共振响应特性受损伤影响会有明显区别;损伤甚至会直接改变系统稳态响应幅值以及稳定解的数量,导致系统产生明显大幅振动,影响结构安全。     

温度变化对悬索模态耦合共振特性影响

悬索是一种典型的大跨度低阻尼柔性系统,其包含平方和立方非线性特征,从而呈现出各种非线性动力学行为,尤其是在不同模态之间发生的耦合共振响应。此外实际工程中悬索受气温、太阳辐射、风等因素影响,周围温度场变化明显,而悬索线性和非线性振动特性对于温度变化较为敏感。本研究以悬索同时发生主共振和3∶1内共振为例,将之前忽略模态耦合的单自由度模型扩展到两自由度模型,并利用多尺度法求得系统直角坐标下的平均方程。基于所绘制的系统各类响应曲线,对温度变化下悬索模态耦合振动特性开展详细论述。数值算例结果表明：温度下降(上升)时,Irvine参数更大(更小)的悬索容易发生3∶1内共振;在内共振的区间,低阶模态响应幅值受温度变化的影响大于高阶模态的响应幅值;霍普夫分岔对于温度变化的敏感程度要高于鞍结点分岔;在耦合共振区间,系统周期运动对温度变化较为敏感,温度变化有可能导致系统的周期运动变为非周期。     

悬索的多重内共振研究

本文对谐波激励的悬索的非线性响应进行了研究,同时考虑了如下问题(1):面内第三阶对称模态的主共振:(2):面内第一阶、第三阶对称模态和面外第五阶模态之间的内共振.本方首先针对考虑大变形的悬索动力学方程,由线性理论求得各阶频率,考察可能出现的内共振.然后利用直接法对悬索的运动学方程和边界条件进行非线性求解.由多尺度法得到系统的平均方程和悬索响应的二阶近似解.随后利用Newton-Raphson 方法和弧长法对特定张拉索进行数值仿真计算,得到面内第一阶对称模态、面内第三阶对称模态和面外第五阶模态的稳态解,并分析了解的稳定性.绘制幅频响应曲线,发现了关于悬索响应的多种分叉现象,并且对各种分叉现象周期解、混沌解进行了讨论.     

受损悬索对称性破缺下非线性耦合振动研究

对称性是振动理论中5大美学特征之一,然而对称性破缺又难以避免.本文以工程中常见的易损结构—悬索为例,探究当该系统遭遇非对称性损伤时,对称性破缺对其面内耦合振动特性影响.首先建立受损悬索面内非线性动力学模型,并采用Galerkin法得到离散的无穷维微分方程.利用多尺度法计算该非线性系统发生面内耦合共振响应的调谐方程.截取...     

悬索在考虑1:3内共振情况下的动力学行为

研究了悬索在受到外激励作用下考虑1:3内共振情况下的两模态非线性响应.对于一定范围内悬索的弹性-几何参数而言,悬索的第三阶面内对称模态的固有频率接近于第一阶面内对称模态固有频率的三倍,从而导致1:3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动得到主共振情况下的平均方程.接下来对平均方程的稳态解、周期解以及混沌解进行了研究.最后利用Runge-Kutta法研究了悬索两自由度离散模型的非线性响应.     

三阶模态耦合效应下悬索的1:1主共振与稳定性分析

针对悬索的振动,研究了模态耦合效应对悬索振动特征的影响。首先基于哈密顿原理推导了考虑抗弯刚度影响的悬索的偏微分振动方程,采用Galerkin方法得到了悬索的前三阶模态耦合振动常微分方程组。采用多尺度法分析了悬索的一阶、二阶和三阶主共振,得到了一阶、二阶和三阶主共振的幅-频响应方程,接着基于Lyapunov稳定性理论进行了稳定性分析,最后进行了数值算例分析。算例分析表明,当1:1主共振发生时,一阶主共振产生的幅值远大于二阶和三阶主共振产生的幅值,即当悬索振动时,能量主要以一阶模态幅值的形式散发;在同阶次幅值-σ曲线中,随着F的增加,1:1主共振产生的幅值有所增加;在幅值-V曲线中,随着σ的增加,临界跳跃点有向右偏移的趋势,σ增加会导致幅值增加;档距越大,一阶、二阶和三阶1:1主共振产生的幅值越大,但一阶主共振产生的幅值增加最为明显。     

悬索在外激励作用下的1∶3内共振分析Ⅰ∶离散法

研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应。对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在。首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程。     

悬索在外激励作用下的1∶3内共振分析I∶离散法

研究了悬索在受到外激励作用和考虑1∶3内共振情况下的两模态非线性响应.对于一定范围内的悬索弹性-几何参数而言,悬索第三阶面内对称模态的固有频率接近于第一阶面内对称模态的固有频率的3倍,从而导致1∶3内共振的存在.首先利用Galerkin方法把悬索的面内运动方程进行离散,然后利用多尺度法对离散的运动方程进行摄动,可得到两组不同主共振情况下的平均方程.     

悬索主共振响应的多尺度解与同伦分析解

利用哈密顿变分原理,引入拟静态假设,建立了悬索面内非线性运动方程,并采用Galerkin方法对其进行离散。接着运用多尺度法和同伦分析法得到了悬索前两阶模态主共振响应的近似解。为验证这两种分析方法的适用性,同时采用龙格-库塔法对方程直接进行了数值积分。数值计算结果表明,随着悬索垂跨比以及振幅的增加,由多尺度法与同伦分析法得到的幅频响应曲线存在明显的定性与定量的差别,而同伦分析法结果与数值法的结果更加接近。最后比较了两种分析方法得到的位移场与索力时程响应曲线。     

轴向移动局部浸液单向板的1:3内共振分析

考虑单向板的轴向速度、轴向张力、流固耦合作用以及阻尼等因素, 基于由 von Kármán薄板大挠度方程得到的轴向移动局部浸液单向板的非线性振动方程, 研究了外激励作用下单向板在1:3内共振情况时的非线性振动特性. 首先利用Galerkin法对非线性振动方程离散化, 然后分别应用数值法和近似解析法对离散后模态方程组进行求解, 获得了系统内共振情况下复杂的幅频特性曲线, 并讨论了周期解的稳定性. 最后研究了1:3内共振系统平均方程组的运动分岔现象.     

Resonance and Bifurcation in a Nonlinear Duffing System with Cubic Coupled Terms

The dynamic behaviors of two-degree-of-freedom Duffing system with cubic coupled terms are studied. First, the steady-state responses in principal resonance and internal resonance of the system are analyzed by the multiple scales method. Then, the bifurcation structure is investigated as a function of the strength of the driving force F. In addition to the familiar routes to chaos already encountered in unidimensional Duffing oscillators, this model exhibits symmetry-breaking, period-doubling of both types and a great deal of highly periodic motion and Hopf bifurcation, many of which occur more than once. We explore the chaotic behaviors of our model using three indicators, namely the top Lyapunov exponent, Poincaré cross-section and phase portrait, which are plotted to show the manifestation of coexisting periodic and chaotic attractors.     

Super-Harmonic and Internal Resonances of a Suspended Cable with Nearly Commensurable Natural Frequencies

In this paper, super-harmonic and internal resonance characteristics ofa viscously damped cable with nearly commensurable natural frequenciesare investigated by use of a novel method. The proposed frequency-domainsolution method is based on the combined use of a three-dimensionalnonlinear finite element approach and the incremental harmonic balancetechnique. It is an accurate algorithm in the sense that it accommodatesmulti-harmonic components and no mode-based model reduction is utilizedin the solution process. The alternating frequency/amplitude-controlledalgorithm enables complete solution to the frequency-response curvesincluding unstable branches, sub- and super-harmonic resonance andinternal resonance. A suspended cable paradigm under internal resonancecondition is studied using the proposed method. Nonlinear response andmodal interaction characteristics of the cable at different frequencyregions are identified from analysis of response profiles and harmoniccomponent features. The super-harmonic and internal resonance responsesare respectively characterized based on the harmonic distribution. Underan in-plane harmonic excitation, the two-to-one internal resonancebetween the in-plane and out-of-plane modes and the super-harmonicresonance around the second symmetric in-plane mode are revealed. Strongnonlinear interaction among different modes in the parameter spaceranging from primary resonance to super-harmonic resonance is observed.     

考虑弯曲刚度斜拉索面内面外内共振分析

本文研究桥梁工程中含弯曲刚度斜拉索的面内面外内共振问题．描述了工程中斜拉索变形的三种状态,考虑弯曲刚度、大变形及垂度等因素,忽略斜拉索纵向惯性力的影响,运用Hamilton变分原理建立了含弯曲刚度的斜拉索面内面外耦合偏微分控制方程,采用Galerkin方法对偏微分方程离散,并运用多尺度摄动方法进行了求解,获得了斜拉索可能存在的内共振模式,以工程中一根斜拉索为例,运用有限元法对其进行动力特性分析,列出了斜拉索前10阶面内面外振动频率,找出面内面外可能产生内共振的模态,分别研究了主共振条件下斜拉索面内和面外1:1、2:1内共振情形,获得了有意义的结论．     

NONLINEAR VIBRATION OF THE COUPLED STRUCTURE OF SUSPENDED-CABLE-STAYED BEAM-1:2 INTERNAL RESONANCE

Through the Galerkin method the nonlinear ordinary differential equations （ODEs） in time are obtained from the nonlinear partial differential equations （PDEs） to describe the mo- tion of the coupled structure of a suspended-cable-stayed beam. In the PDEs, the curvature of main cables and the deformation of cable stays are taken into account. The dynamics of the struc- ture is investigated based on the ODEs when the structure is subjected to a harmonic excitation in the presence of both high-frequency principle resonance and 1：2 internal resonance. It is found that there are typical jumps and saturation phenomena of the vibration amplitude in the struc- ture. And the structure may present quasi-periodic vibration or chaos, if the stiffness of the cable stays membrane and frequency of external excitation are disturbed.     

Nonlinear Dynamics of a Cantilever Beam Carrying an Attached Mass with 1:3:9 Internal Resonances

In this paper the nonlinear response of a base-excited slender beam carrying an attached mass is investigated with 1:3:9 internal resonances for principal and combinationparametric resonances. Here the method of normal forms is used to reduce the second order nonlinear temporal differential equation of motion of the system to a set offirst order nonlinear differential equations which are used to find the fixed-point, periodic, quasi-periodic and chaotic responses of the system.Stability and bifurcation analysis of the responses are carried out and bifurcation sets are plotted. Many chaotic phenomena are reported in this paper.     

Steady state motions of shallow arch under periodic force with 1∶2 internal resonance on the plane of physical parameters

The bifurcation dynamics of shallow arch which possesses initial deflection under periodic excitation for the case of 1∶2 internal resonance is studied in this paper. The whole parametric plane is divided into several different regions according to the types of motions; then the distribution of steady state motions of shallow arch on the plane of physical parameters is obtained. Combining with numerical method, the dynamics of the system in different regions, especially in the Hopf bifurcation region, is studied in detail. The rule of the mode interaction and the route to chaos of the system is also analysed at the end. Project supported by National Natural Science Foundation and National Youth Science Foundation of China     

STEADY STATE MOTIONS OF SHALLOW ARCH UNDER PERIODIC FORCE WITH 1:2 INTERNAL RESONANCE ON THE PLANE OF PHYSICAL PARAMETERS

I.IntroductionAlthoughalotofpapersaboutthedynamicalstudyofshallowarchunderperiodicforcehavebeenpresentedl"ZI,theseresultshavebeenobtainedunderthehypothesisofonemode.Whenthearchhasinitialstaticdeflection,thevibrationfrequenciesof'thefirstorderandthesecondordermodeswillbeinresonance.Atthistimethehypothesisofonemodecannotmeetthenecessityofthedynamicalstudyoftheshallowarch.Thetwomodesofthesystem'willreacttoeachotherandtheenergycanbetransferredbetweenthevariousmodes.Inordertoanalysetheruleofthemo…     

Free Vibration in a Class of Self-Excited Oscillators with 1:3 Internal Resonance

Free vibration of a two degree of freedom weakly nonlinear oscillator is investigated. The type of nonlinearity considered is symmetric, it involves displacement as well as velocity terms and gives rise to self-excited oscillations in many engineering applications. After presenting the equations of motion in a general form, a perturbation methodology is applied for the case of 1:3 internal resonance. This yields a set of four slow-flow nonlinear equations, governing the amplitudes and phases of approximate motions of the system. It is then shown that these equations possess three distinct types of solutions, corresponding to trivial, single-mode and mixed-mode response of the system. The stability analysis of all these solutions is also performed. Next, numerical results are presented by applying this analysis to a specific practical example. Response diagrams are obtained for various combinations of the system parameters, in an effort to provide a complete picture of the dynamics and understand the transition from conditions of 1:3 internal resonance to non-resonant response. Emphasis is placed on identifying the effect of the linear damping, the frequency detuning and the stiffness nonlinearity parameters. Finally, the predictions of the approximate analysis are confirmed and extended further by direct integration of the averaged equations. This reveals the existence of other regular and irregular motions and illustrates the transition from phase-locked to drift response, which takes place through a Hopf bifurcation and a homoclinic explosion of the averaged equations.     

1:2 Internal Resonance of Coupled Dynamic System with Quadratic and Cubic Nonlinearities

The1:2 internal resonance of coupled dynamic system with quadratic and cubic nonlinearities is studied. The normal forms of this system in1:2 internal resonance were derived by using the direct method of normal form. In the normal forms, quadratic and cubic nonlinearities were remained. Based on a new convenient transformation technique, the4-dimension bifurcation equations were reduced to3-dimension. A bifurcation equation with one-dimension was obtained. Then the bifurcation behaviors of a universal unfolding were studied by using the singularity theory. The method of this paper can be applied to analyze the bifurcation behavior in strong internal resonance on4-dimension center manifolds. Paper from Chen Yu-shu, Member of Editorial Commuttee, AMM Foundation item: the National Natural Science Foundation of China (1990510); the National Key Basic Research Special Fund (G1998020316); the Doctoral Point Fund of Education Committee of China (D09901) Biography: Chen Yu-shu (1931-)     

Codimension two bifurcation and chaos of a vibro-impact forming machine associated with 1：2 resonance case

A vibro-impact forming machine with double masses is considered. The components of the vibrating system collide with each other. Such models play an important role in the studies of dynamics of mechanical systems with impacting components. The Poincaré section associated with the state of the impact-forming system, just immediately after the impact, is chosen, and the period n single-impact motion and its disturbed map are derived analytically. A center manifold theorem technique is applied to reduce the Poincaré map to a two-dimensional map, and the normal form map associated with codimension two bifurcation of 1:2 resonance is obtained. Unfolding of the normal form map is analyzed. Dynamical behavior of the impact-forming system, near the point of codimension two bifurcation, is investigated by using qualitative analyses and numerical simulation. Near the point of codimension two bifurcation there exists not only Neimark-Sacker bifurcation associated with period one single-impact motion, but also Neimark-Sacker bifurcation of period two double-impact motion. Transition of different forms of fixed points of single-impact periodic orbits, near the bifurcation point, is demonstrated, and different routes from periodic impact motions to chaos are also discussed. The project supported by the National Natural Science Foundation of China (10572055, 50475109) and the Natural Science Foundation of Gansu Province Government of China (3ZS051-A25-030(key item)) The English text was polished by Keren Wang.