含等档的输电线面内自由振动理论

连续档导线运动方程包含平方和立方非线性项,倍频时会产生多模态耦合的复杂响应,因此研究连续档导线模态及共振的频率分布规律尤为重要．基于模态综合法获得了具有相等档距的连续档导线模态函数,基于动刚度理论得到了不同模态对应的频率理论公式,并应用有限元方法验证了模态及频率理论公式的准确性．研究了不同模态对应频率随几何参数的变化趋势,结果表明有相等档距的连续档导线的共振条件和单档导线有明显区别,连续档导线面内对称模态之间容易产生1:1共振,面内对称与反对称模态之间易产生1:2共振．本文研究内容可用来分析连续档导线内共振及其分岔行为．     

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     

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。     

损伤效应对悬索2:1内共振响应影响分析

损伤是结构振动测试和运营维护中不可避免的问题，损伤效应会导致结构振动特性发生改变．本文以受损悬索为例，探究该非线性系统同时发生主共振和2:1内共振时，损伤效应对其面内耦合共振响应影响．首先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，建立受损悬索面内动力学模型，并推导其无穷维非线性运动微分方程．以2:1耦合共振为例，采用Galerkin法和多尺度法得到系统直角坐标形式的调谐方程．数值算例表明：损伤会导致悬索固有频率降低，使得频率间公倍关系发生改变，影响系统耦合共振响应；损伤会引发系统振动特性发生明显定量和定性改变，尤其是共振响应幅值及弹簧特性；损伤对直接激励模态响应幅值的影响比对内共振激发对响应幅值的影响要明显；损伤会导致霍普夫、鞍节点、叉形和倍周期分岔的位置发生偏移，从而影响分岔点附近系统的动力学行为；系统动态解和周期运动与损伤密切相关，损伤会导致系统展现出完全不同类型的吸引子．     

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。     

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.     

结构的模态运动的非线性调制与平均Lagrange函数（A.L.F）

本文提出以幅相表示和复幅表示两种形式的Ａ．Ｌ．Ｆ，由Ａ．Ｌ．Ｆ可直接求出模记运动的非线性调幅。设想相或复调制，集中反映一种或多种内共振存在的影响。便于分析结构模态的非线性相互作用及其规律性；对于无阻尼的结构，更可导出联系有关的各模态的若干内共振幅值关系或与一个内振幅相关系式，文中同有内共振时构成与使用Ａ．Ｌ．Ｆ的关键是正确地确定Ａ．Ｌ．Ｆ对与此内共振关联的典型组合相位γ或复幅积τ的依赖关系。     

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.     

周期激励浅拱分岔研究

研究了一阶和二阶模态在１：２内共振条件下浅拱的复杂动力学行为，指出当周期激励浅拱具有初始静变形时，系统的一阶模态和二阶模态会产生内共振，系统两共振模态之间会产生相互作用，系统的能量会在其低阶和高阶模态之间相互传递，对称破缺后的Ｈｏｐｆ分岔解会通过一系列的倍化周期分岔导致混沌，在混沌域中还会发现稳定的周期解窗口．     

外激励作用下弹性支撑浅拱的非线性动力学分析

研究了外激励下两端采用转动弹簧约束的铰支浅拱在发生1:1内共振时的非线性动力学行为。通过引入基本假定和无量纲化变量得到浅拱的动力学控制方程, 将阻尼项、外荷载项和非线性项去掉后,所得线性方程及对应边界条件即可确定考虑转动弹簧影响的频率和模态, 发现转动约束取不同刚度值时系统存在模态交叉与模态转向两种内共振形式。对动力方程进行Galerkin全离散, 并采用多尺度法对内共振进行了摄动分析, 得到了极坐标和直角坐标两种形式的平均方程, 其中平均方程系数与转动弹簧刚度一一对应。最低两阶模态之间1:1内共振的数值研究结果表明: 外激励能激发内共振模态的非线性相互作用, 参数处于某一范围时系统存在周期解、准周期解和混沌解窗口, 且通过(逆)倍周期分岔方式进入混沌。     

Bifurcation of nonlinear internal resonant normal modes of a class of multi-degree-of-freedom systems

The method of multiple scales is applied for constructing nonlinear normal modes (NNMs) of a three-degree-of-freedom system which is discretized from a two-link flexible arm connected by a nonlinear torsional spring. The discrete system is with cubic nonlinearity and 1:3 internal resonance between the second and the third modes. The approximate solution for the NNM associated with internal resonance are presented. The NNMs determined here tend to the linear modes as the nonlinearity vanishes, which is significant for one to construct NNM. Greatly different from results of those nonlinear systems without internal resonance, it is found that the NNM involved in internal resonance include coupled and uncoupled two kinds. The bifurcation analysis of the coupled NNM of the system considered is given by means of the singularity theory. The pitchfork and hysteresis bifurcation are simultaneously found. Therefore, the number of NNM arising from the internal resonance may exceed the number of linear modes, in contrast with the case of no internal resonance, where they are equal. Curves displaying variation of the coupling extent of the coupled NNM with the internal-resonance-deturing parameter are proposed for six cases.     

Transition of Energy to a Nonlinear Localized Mode in a Highly Asymmetric System of Two Oscillators

Redistribution of energy in a highly asymmetric system consisting ofcoupled linear and highly nonlinear damped oscillators isinvestigated. Special attention is paid to the excitation of a nonlinearnormal mode while the energy is initially stored in other modes of thesystem. The transition proceeds via the mechanism of subharmonicresonance which is possible because of the strong nonlinearity of thesystem. The conditions of the energy transition to NNM being effectiveare revealed and guidelines to design such a systems are formulatedin detail.     

SINGULAR ANALYSIS OF BIFURCATION OF NONLINEAR NORMAL MODES FOR A CLASS OF SYSTEMS WITH DUAL INTERNAL RESONANCES

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Non-linear normal modes and their bifurcation of a class of systems with three double of pure imaginary roots and dual internal resonances

The method of multiple scales is used to construct non-linear normal modes (NNMs) of a class of systems with three double of pure imaginary roots and 1:2:5 dual internal resonance. It is found that the three NNMs associated with dual internal resonances include two uncoupled NNMs as well as a coupled NNM. And the bifurcation problem of the coupled NNM is in two variables, which is greatly different from the bifurcation of the NNMs of systems with single internal resonance. Because no results in singularities can be straightly applied, a practical way is proposed to do singularity analysis for bifurcation of two dimensions. It is also noted that with the variation of the bifurcation parameters, the modes may convert to each other or suddenly emerge and disappear, which give rise to the number of the NNMs more or fewer than the number of the degrees of freedom.     

Non-linear normal modes and their bifurcation of a two DOF system with quadratic and cubic non-linearity

The non-linear normal modes (NNMs) and their bifurcation of a complex two DOF system are investigated systematically in this paper. The coupling and ground springs have both quadratic and cubic non-linearity simultaneously. The cases of ω₁:ω₂=1:1, 1:2 and 1:3 are discussed, respectively, as well as the case of no internal resonance. Approximate solutions for NNMs are computed by applying the method of multiple scales, which ensures that NNM solutions can asymtote to linear normal modes as the non-linearity disappears. According to the procedure, NNMs can be classified into coupled and uncoupled modes. It is found that coupled NNMs exist for systems with any kind of internal resonance, but uncoupled modes may appear or not appear, depending on the type of internal resonance. For systems with 1:1 internal resonance, uncoupled NNMs exist only when coefficients of cubic non-linear terms describing the ground springs are identical. For systems with 1:2 or 1:3 internal resonance, in additional to one uncoupled NNM, there exists one more uncoupled NNM when the coefficients of quadratic or cubic non-linear terms describing the ground springs are identical. The results for the case of internal resonance are consistent with ones for no internal resonance. For the case of 1:2 internal resonance, the bifurcation of the coupled NNM is not only affected by cubic but also by quadratic non-linearity besides detuning parameter although for the cases of 1:1 and 1:3 internal resonance, only cubic non-linearity operate. As a check of the analytical results, direct numerical integrations of the equations of motion are carried out.     

Seismic base isolation by nonlinear mode localization

Summary In this paper, the performance of a nonlinear base-isolation system, comprised of a nonlinearly sprung subfoundation tuned in a 1∶1 internal resonance to a flexible mode of the linear primary structure to be isolated, is examined. The application of nonlinear localization to seismic isolation distinguishes this study from other base-isolation studies in the literature. Under the condition of third-order smooth stiffness nonlinearity, it is shown that a localized nonlinear normal mode (NNM) is induced in the system, which confines energy to the subfoundation and away from the primary or main structure. This is followed by a numerical analysis wherein the smooth nonlinearity is replaced by clearance nonlinearity, and the system is excited by ground motions representing near-field seismic events. The performance of the nonlinear system is compared with that of the corresponding linear system through simulation, and the sensitivity of the isolation system to several design parameters is analyzed. These simulations confirm the existence of the localized NNM, and show that the introduction of simple clearance nonlinearity significantly reduces the seismic energy transmitted to the main structure, resulting in significant attenuation in the response. This work was supported in part by the National Science Foundation Grant CMS 00-00060. The authors are grateful for this support.     

Resonance captures and targeted energy transfers in an inertially-coupled rotational nonlinear energy sink

We explore the conservative and dissipative dynamics of a two-degree-of-freedom (2-DoF) system consisting of a linear oscillator and a lightweight nonlinear rotator inertially coupled to it. When the total energy of the system is large enough, the motion of the rotator is, generically, chaotic. Moreover, we show that if the damping of the rotator is sufficiently small and the damping of the linear oscillator is even smaller, then the system passes through a cascade of resonance captures (transient internal resonances) as the total energy gradually decreases. Rather unexpectedly, all these captures have the same principal frequency but correspond to different nonlinear normal modes (NNMs). In each NNM, the rotator is phase-locked into periodic motion with two frequencies. The NNMs differ by the ratio of these frequencies, which is approximately an integer for each NNM. Essentially non-integer ratios lead to incommensurate periods of ??slow?? and ??fast?? motions of the rotator and, thus, to its chaotic behavior between successive resonance captures. Furthermore, we show that these cascades of resonance captures lead to targeted energy transfer (TET) from the linear oscillator to the rotator, with the latter serving, in essence, as a nonlinear energy sink (NES). Since the inertially-coupled NES that we consider has no linearized natural frequency, it is capable of engaging in resonance with the linear oscillator over broad frequency and energy ranges. The results presented herein indicate that the proposed rotational NES appears to be a promising design for broadband shock mitigation and vibration energy harvesting.     

Nonlinear Normal Modes in a System with Nonholonomic Constraints

We investigate the dynamics of a system involving the planar motionof a rigid body which is restrained by linear springs and whichpossesses a skate-like nonholonomic constraint known as aplygin'ssleigh. It is shown that the system can be reduced to one with 2 degrees of freedom. The resulting phase flow is shownto involve a curve of nonisolated equilibria. Using second-orderaveraging, the system is shown to possess two families of nonlinearnormal modes (NNMs). Each NNM involves two amplitude parameters.The structure of the NNMs is shown to depart from the generic formin the neighborhood of a 1:1 internal resonance.     

Experimental evidence of bifurcating nonlinear normal modes in piecewise linear systems

A system with piecewise linear restoring forces, typical of damaged beams with a breathing crack, exhibits bifurcations characterized by the onset of superabundant modes in internal resonance with a significantly different shape than that of modes on a fundamental branch. A 2-DOF frame with piecewise linear stiffness is analyzed by means of an experimental investigation; the frame is forced by an harmonic base excitation and the operative modal shapes as well as the response amplitude are directly measured; the results are compared with numerical outcomes for different damping values. This study shows that the shapes and the frequencies of certain nonlinear normal modes (NNMs) of the related autonomous system strongly affect the forced response, in both the numerical and the experimental environments. Therefore, it is possible to match the NNM with the forced response of the system, leading to the prospect of identifying the severity and position of the damage from experimental tests.