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.     

基于积分本构黏弹性带的横向非线性动力学

研究了黏弹性传动带在1:1内共振时的横向非平面非线性动力学特性. 首先,利用Hamilton原理建立了黏弹性传动带横向非平面非线性动力学方程. 然后综合应用多尺度法和Galerkin离散法对偏微分形式的动力学方程进行摄动分析,得到了四维平均方程. 对平均方程的稳定性进行了分析,从理论上讨论了动力系统解的稳定性变化情况. 最后数值模拟结果表明黏弹性传动带系统存在混沌运动、概周期运动和周期运动.      

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.     

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。     

小参数法在悬垂索共振中的应用

当面外横向振动和面内横向振动频率的比接近1:2时,悬索会出现面内和面外耦合共振现象。为了研究悬索这种复杂独特的非线性特性,利用多尺度法对谐波激励的悬索动力学方程进行求解,得到对应于不同阶小量的偏微分方程组,其中二阶小量偏微分方程中的久期项不为0;采用提出的小参数法可以得到由久期项引起的悬索振动形态,解决久期项频率与系统频率相同但不能直接求解的问题;为了证明小参数法的准确性,采用Galerkin方法离散悬垂索的运动方程,然后利用多尺度法求解离散的运动方程,得到采用基函数描述的由久期项引起的连续系统的振动形态,与小参数法结论一致。     

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.     

周期荷载激励下覆冰输电线的主共振及谐波共振分析

为了研究动态风对覆冰输电线非线性舞动特征的影响,在原有稳定风作用下覆冰输电线舞动控制方程中添加周期激励载荷,并建立了新的受迫-自激振动控制方程,该控制方程也适用于描述相邻档导线对舞动档导线运动特征的影响.运用多尺度法分别对弱激励和强激励下的受迫-自激振动求解,得到主共振和谐波共振的幅频响应函数,分析了受迫-自激系统的主...     

悬索非线性动力学中的直接法与离散法

以悬索为例对结构非线性动力学中直接法与离散法的应用进行了研究. 针对悬索面内运动的第$n$阶模态的主共振，分别利用这两种方法对悬索的非线性响应进行求解，得到悬索非线性响应的二次近似解以及相应的幅频响应曲线，并比较和讨论了这两种方法得到的结果及其差异. 通过分析得知：离散法在用于非对称结构非线性动力学的求解时可能导致错误的结果.     

悬索在外激励作用下的1:3内共振分析(Ⅱ):数值结果

本研究的第一部分已经推导了悬索在第一阶面内对称模态主共振和第三阶面内对称模态主共振下的平均方程,其中考虑了这两阶模态之间1∶3内共振.本文对平均方程的稳态解,周期解以及混沌解进行了研究.利用 Newton-Naphson 方法和拟弧长的延拓算法确定了主共振情况下的幅频响应曲线,通过利用 Jacobian 矩阵的特征值判断幅频响应曲线中解的稳定性.在这些幅频响应曲线中.都存在超临界 Hopf 分叉,导致平均方程的周期解.以这些超临界 Hopf 分叉为起点.利用打靶法和拟弧长的延拓算法确定了两种主共振情况下的周期解分支,同时通过利用 Floquet 理论判断这些周期解的稳定性.然后利用数值结果研究了两种主共振情况下的厨期解经过倍周期分叉通向混沌的过程.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.     

Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate

Global bifurcations and multi-pulse chaotic dynamics are studied for a four-edge simply supported composite laminated piezoelectric rectangular plate under combined in-plane, transverse, and dynamic electrical excitations. Based on the von Karman type equations for the geometric nonlinearity and Reddy’s third-order shear deformation theory, the governing equations of motion for a composite laminated piezoelectric rectangular plate are derived. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional non-autonomous nonlinear system is simplified to a three-order standard form by using the method of normal form. The extended Melnikov method is improved to investigate the six-dimensional non-autonomous nonlinear dynamical system in mixed coordinate. The global bifurcations and multi-pulse chaotic dynamics of the composite laminated piezoelectric rectangular plate are studied by using the improved extended Melnikov method. The multi-pulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.     

Bifurcations and chaos of an inclined cable

The nonlinear behavior of an inclined cable subjected to a harmonic excitation is investigated in this paper. The Galerkin’s method is applied to the partial differential governing equations to obtain a two-degree-of-freedom nonlinear system subjected to harmonic excitation. The nonlinear systems in the presence of both external and 1:1 internal resonances are transformed to the averaged equations by using the method of averaging. The averaged equations are numerically examined to obtain the steady-state responses and chaotic solutions. Five cascades of period-doubling bifurcations leading to chaotic solutions, 3-periodic solutions leading to chaotic solution, boundary crisis phenomena, as well as the Shilnikov mechanism for chaos, are observed. In order to study the global dynamics of an inclined cable, after determining the averaged equations of motion in a suitable form, a new global perturbation technique developed by Kova?i? and Wiggins is used. This technique provides analytical results for the critical parameter values at which the dynamical system, through the Shilnikov type homoclinic orbits, possesses a Smale horseshoe type of chaos.     

Nonlinear dynamic response and active vibration control of the viscoelastic cable with small sag

IntroductionCablesareveryefficientstructuralmembersandhencehavebeenwidelyusedinmanylong_spanstructures,includingcable_supportbridges,guyedtowersandcable_supportroofs.Sincecablesarelight,veryflexibleandlightlydamped ,structuresutilizingcables,i.e .,cable_structuresystems,usuallyhavevariousdynamicproblems.Theirmodelsarethereforeverimportantinpredictingandcontrollingtheirresponses.Inthelastdecade,thenonlineardynamicvibrationandstabilitybehaviorofcablesandcable_structureshavedrawntheattentionofman…     

Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory

In this paper, an analysis on the nonlinear dynamics and chaos of a simply supported orthotropic functionally graded material (FGM) rectangular plate in thermal environment and subjected to parametric and external excitations is presented. Heat conduction and temperature-dependent material properties are both taken into account. The material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on the Reddy’s third-order share deformation plate theory, the governing equations of motion for the orthotropic FGM rectangular plate are derived by using the Hamilton’s principle. The Galerkin procedure is applied to the partial differential governing equations of motion to obtain a three-degree-of-freedom nonlinear system. The resonant case considered here is 1:2:4 internal resonance, principal parametric resonance-subharmonic resonance of order 1/2. Based on the averaged equation obtained by the method of multiple scales, the phase portrait, waveform and Poincare map are used to analyze the periodic and chaotic motions of the orthotropic FGM rectangular plate. It is found that the motions of the orthotropic FGM plate are chaotic under certain conditions.     

斜拉桥拉索-阻尼器系统非线性响应分析

考虑索的抗弯刚度、垂度及几何非线性的影响，得出了索一阻尼器系统的空间非线性振动偏微分方程，用中心差分法将微分方程在空间内离散，导出了系统的非线性振动常微分方程组。结合Newmark法及虚拟力法提出了一种用于求解非线性振动瞬态响应的杂交分析算法。并以典型的斜拉桥拉索为研究对象，给出了数值算例，并与Runge—Kutta直接积分法进行了比较，说明了杂交算法的准确性及有效性。     

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.     

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I: Theoretical formulation and model validation

This paper is first of the two papers dealing with analytical investigation of resonant multi-modal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables – which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations – are presented. A multi-dimensional Galerkin expansion of the solution of nonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effects of quadratic/cubic nonlinearities, approximate closed-form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation.     

两种离散方法对覆冰四分裂导线舞动特征的影响

首先基于Hamilton变分准则推导了覆冰导线三自由度耦合动力学方程组,接着采用两种离散方法离散动张力应变,一种是直接使用Galerkin法离散,另一种是先将动张力应变等效处理,然后再使用Galerkin法离散.由风洞试验测得新月形覆冰四分裂导线的气动力系数,并将各子导线的气动力系数等效,接着利用泰勒展开式拟合气动力系数,选取攻角55°进行舞动分析.基于Runge-Kutta法得到两种离散方法下的位移响应曲线,通过对比这两者位移响应曲线的差别,发现不同的离散方法对系统的相位、频率以及振幅皆有一定程度的影响.本文研究有助于理论建模的完善,并能给予实际工程一些指导.     

Fluid Flow-Induced Nonlinear Vibration of Suspended Cables

The nonlinear interaction of the first two in-plane modes of a suspended cable with a moving fluid along the plane of the cable is studied. The governing equations of motion for two-mode interaction are derived on the basis of a general continuum model. The interaction causes the modal differential equations of the cable to be non-self-adjoint. As the flow speed increases above a certain critical value, the cable experiences oscillatory motion similar to the flutter of aeroelastic structures. A co-ordinate transformation in terms of the transverse and stretching motions of the cable is introduced to reduce the two nonlinearly coupled differential equations into a linear ordinary differential equation governing the stretching motion, and a strongly nonlinear differential equation for the transverse motion. For small values of the gravity-to-stiffness ratio the dynamics of the cable is examined using a two-time-scale approach. Numerical integration of the modal equations shows that the cable experiences stretching oscillations only when the flow speed exceeds a certain level. Above this level both stretching and transverse motions take place. The influences of system parameters such as gravity-to-stiffness ratio and density ratio on the response characteristics are also reported.     

Nonlinear vibration analysis of viscoelastic cable with small sag

Both the inplane and out-of-plane transverse vibrations of a viscoelastic cable subjected to an initial stress distributing uniform on the cross section are studied. The constitution of the cable material is assumed to be of the hereditary integral type. The partial differential-integral equations of motion are derived first. Then by applying Galerkin's method, the governing equations are reduced to a set of second-order nonlinear differential-integral equations which are solved by finite difference numerical integration procedures. Finally, the effects of the viscosity parameter and the elastic parameter on the transient amplitudes of the first mode are investigated by numerical simulation. Project supported by the National Natural Science Foundation of China (No. 59635140) and the National Postdoctoral Foundation of China.