多索-单梁耦合结构的动力学建模及非线性特性研究

大跨度斜拉桥通常采用密索体系,拉索的振动频率非常接近,同时拉索之间也可能存在倍频关系.因此,可能存在索与索之间的相互耦合振动.索是斜拉桥的重要承重构件,对多索 单梁的动力学特性进行研究至关重要.本文考虑到索梁、索塔及塔梁的边界条件和连接点处的连接条件,基于Hamilton变分原理,建立了多索 单梁结构的运动微分方程.经过无量纲化处理,根据分离变量法,得到了其降维约化后的运动微分方程.本文取相邻的两根索考虑两种工况模型,对不同工况下的双索 单梁耦合结构的动力学行为进行了参数分析.研究发现,不同于以往资料中的“频率转向”现象,在本文中,两条频率曲线在频率值接近处,并未迅速分离,而是在相对一小段参数范围内继续保持平行且相互靠近,随后再迅速分离.最后,研究了双索 单梁耦合结构的非线性特性,分别对结构进行了两自由度和单自由度离散,研究发现,结构单模态假设在非共振区域能反映结构的非线性特性,但是在共振区域,结构的非线性特性会发生跳跃变化,应用2自由度模态理论进行研究更为准确.     

单侧主缆刚度损伤悬索桥的模态分析及1∶1内共振

悬索桥缆索在长期服役状态下损伤很难避免,单侧主缆刚度损伤会使系统出现主梁竖弯与扭转自由度间的耦合,这可能会进一步对悬索桥的模态和大振幅下的非线性响应造成影响.为此,建立了考虑单侧主缆刚度受损的七自由度悬索桥横截面模型,模态分析发现单侧主缆刚度损伤会使得系统前两阶固有频率曲线由交叉变为跃迁,导致两个本身相互独立的模态发生耦合.以此为基础,考虑悬索桥主缆刚度有损伤和未损伤两种工况,运用拓展的增量谐波平衡法(EIHB)计算系统在内共振条件下的非线性振动.研究结果表明：单侧主缆刚度有损伤的悬索桥在外部简谐激励下发生1:1内共振,系统能量表现出明显的转移现象;竖向和扭转简谐激励下,有损伤的悬索桥较未损伤工况响应幅值有所减小,但出现了两个共振响应峰值,对激励频率更为敏感.数值结果与利用Runge-Kutta法计算得到的结果吻合一致,验证了EIHB法的准确性.     

复模态分析超临界轴向运动梁横向非线性振动

近似解析研究了简支边界条件下超临界轴向运动梁横向非线性自由振动的固有频率和模态函数.采用复模态方法处理控制方程,一个积分偏微分方程.将Galerkin截断思想用于近似处理线性化方程,一个含空间依赖系数的常微分方程.给出了不同截断项数对固有频率的影响.基于8项截断,讨论了系统参数对模态函数的影响.     

高速切削过程中颤振现象的二自由度非光滑模型分析

研究了一类二自由度模型在高速切削过程中的颤振运动.首先建立了二自由度切削运动模型,得到了四维的非线性分段方程,然后研究切削力中的动态分量对切削颤振的影响,应用特征值法解析建立了系统发生Hopf分岔的临界条件.结果表明,当分岔参数经过某一临界值时发生Hopf分岔.最后,通过数值方法对该系统进行了数值模拟,从而验证了该临界条件的有效性.     

一类强非线性二阶微分方程的多模态近似解析解研究

利用自治力学系统的哈密顿函数为守恒量的性质,提出一种求非线性二阶微分方程多模态近似解析解的方法,称为哈密顿函数法.首先,介绍哈密顿函数法求多模态近似解的基本理论.其次,以质点在旋转的抛物线上运动为模型建立强非线性二阶微分方程.最后,用哈密顿函数法求得在给定初始条件和参数下强非线性二阶微分方程的三模态近似解析解表达式,作出三模态近似解析解的解曲线,并与直接用Mathematica软件作出的解曲线进行比较,讨论三模态近似解析解的精确性.结果表明：用哈密顿函数法求得的三模态近似解析解的解曲线与直接用Mathematica软件作出的解曲线十分吻合.     

拉索-阻尼器系统的改进实模态分析方法

阻尼减振是拉索振动控制的主要措施,目前拉索 阻尼器系统的主要分析方法为复模态法.虽然采用复模态法可以精确分析拉索 阻尼器系统的阻尼比和频率等主要特征参数,但采用复模态法求出的拉索复振型由于不包含减振所需要的参数而被研究较少,而系统的振型对研究阻尼器的减振效果又具有非常重要的意义.文中首先采用复模态法求解了拉索 阻尼器系统,基于复振型的物理意义,引入相位函数,提出了一种改进的实数分离变量法,分析了阻尼力在拉索中的传递特征;建立了拉索 阻尼器系统的改进实模态分析方法,采用该方法求解得到的非线性运动方程通解与复模态法一致.基于该新方法计算得到的实模态振型,分析了阻尼器对拉索实际振型以及相位的影响.     

含参数多自由度非线性系统的降维方法研究

研究一类具有一定结构特点的含参数多自由度非线性系统,以固定界面动态子结构方法为基础,借助于灵敏度分析的基本思路,提出一种实用的降维方法首先根据实际系统的结构特点将其划分为若干子系统,使其中部分子系统是线性的,其余子系统是非线性的基于灵敏度分析的相关理论导出线性部分子系统的频率、模态对于参数的依赖关系,采用固定界面模态综合法将线性部分子系统的截断模态与非线性子系统进行综合,最终得到具有较低维数的含参数非线性动力系统结合具体算例,将降维前后的非线性动力系统随参数值变化时的对应的幅频特性曲线进行了比较算例表明,采用这里提出的降维方法所得到的低维系统可以较为准确地反映原系统的动力学特性．     

1:1内共振环形桁架天线的稳定性分析

复杂的太空环境易导致环形桁架天线产生大幅非线性振动,严重影响天线的稳定性和结构性能.在大幅的非线性振动中不同模态之间的能量可以相互传递和转换,将引起天线产生内振动.因此内共振在天线的大幅振动中起着重要的作用.本文所研究的环形桁架天线被简化成等效圆柱壳模型,建立其非线性动力学方程.根据有限元模态分析,得出环形桁架天线第四阶模态和第五阶模态的振动频率接近1:1.所以本文利用理论分析和数值计算研究1:1内共振情形下环形桁架天线的局部动力学性质.即系统受到小扰动后平衡点的稳定性.详细研究了两种不同形式下的平衡点的稳定性,即系统特征根为双零和两个负数时的稳定性以及系统特征根为双零和一对纯虚复数时的稳定性.利用中心流形理论、非线性变换、Routh-Hurwitz判据得到平衡点的稳定区域、不稳定区域和临界分岔曲线.最后通过数值模拟验证理论分析.     

基于DMD方法的矩形断面涡振模态分析

为探究涡激振动下,矩形断面绕流场的模态演化规律及涡振发生机理,基于雷诺平均 SST k ω模型对宽高比为5:1的矩形断面的涡激振动进行了二维数值模拟研究.通过CFD/CSD交错迭代求解,结合动网格技术实现了二维流固耦合计算,并与风洞试验结果进行对比分析,验证了数值计算的准确性.随后,对流场的瞬时状态进行可视化分析,以z方向涡量值为基本物理量,基于动力学模态分解(DMD)方法对其静态与涡振态流场模态进行了对比分析.结果表明：在涡振发展的不同阶段,主导流场的模态频率不同,主模态频率会由静态涡脱频率向结构固有频率转变,此主模态的转变表征了流固耦合的内在机制；与静态矩形相比,涡振态矩形由于运动诱导涡的存在,导致其前缘涡与尾缘涡的相位差减小,前缘涡与尾涡而合并时前缘涡占据主导,使得在尾缘脱落的Karman涡相关性减弱,形成了涡振锁频现象.     

工程结构多刚度尺度分析与模态理论

当前工程结构设计大多基于成熟的线性振动理论,未综合考虑非线性、多尺度等特性的影响,导致各类工程振动问题频发,减振措施失效.首先,本文以质量弹簧系统为例,对系统刚度比和质量比等关键参数开展分析,指出刚度比对系统模态具有显著影响；再者,简述大跨桥梁动力学研究现状,从系统全局动力学角度,根据非线性动力学分析和有限元分析,提出工程结构多刚度尺度概念,分析并指出多刚度尺度耦合系统的全局模态、局部模态和混合模态基于不同刚度尺度的定义.为建立桥梁全局动力学模型和理论,桥梁非线性动力学研究奠定基础.     

Two methods for the computation of nonlinear modes of vibrating systems at large amplitudes

The aim of this paper is to present two methods for the calculation of the nonlinear normal modes of vibration for undamped nonlinear mechanical systems: the time integration periodic orbit method and the modal representation method. In the periodic orbit method, the nonlinear normal mode is obtained by making the continuation of branches of periodic orbits of the equation of motion. The terms “periodic orbits” means a closed trajectory in the phase space, which is obtained by time integration. In the modal representation method, the nonlinear normal mode is constructed in terms of amplitude, phase, mode shape, and frequency, with the distinctive feature that the last two quantities are amplitude and total phase dependent. The methods are compared on two DOF strongly nonlinear systems.     

Hybrid component mode synthesis based on test derived data

A method of component mode synthesis in which a majority of the dynamic characteristics of individual components are determined by modal testing is described. For the purpose of testing, the component can be supported on a suitable number of rigid supports. Alternatively, the test can be carried out in a free–free configuration, in which very flexible cables are used to hold the component. In either case, a number of practical difficulties arise. A new approach to modal testing in which the fixed–fixed modes of the component are obtained from a test configuration in which the component is held by supports of arbitrary characteristics is described. A complete theoretical formulation is developed. Computer simulation studies are carried out to determine the sensitivity of the calculated mode shapes and frequencies of the assembled structure to the various measurement errors in modal testing. Some results from these studies are presented.     

Asymptotic non-linear normal modes for large-amplitude vibrations of continuous structures

Non-linear normal modes (NNMs) are used in order to derive accurate reduced-order models for large amplitude vibrations of structural systems displaying geometrical non-linearities. This is achieved through real normal form theory, recovering the definition of a NNM as an invariant manifold in phase space, and allowing definition of new co-ordinates non-linearly related to the initial, modal ones. Two examples are studied: a linear beam resting on a non-linear elastic foundation, and a non-linear clamped–clamped beam. Throughout these examples, the main features of the NNM formulation will be illustrated: prediction of the correct trend of non-linearity for the amplitude-frequency relationship, as well as amplitude-dependent mode shapes. Comparisons between different models—using linear and non-linear modes, different number of degrees of freedom, increasing accuracy in the asymptotic developments—are also provided, in order to quantify the gain in using NNMs instead of linear modes.     

A comparison of analytical–numerical models for nonlinear vibrations of cylindrical shells

The nonlinear flexural vibration behaviour of cylindrical shells has received considerable attention to date. It is pointed out that, although in a well-known reference case there seems to be a reasonable agreement, there are unresolved discrepancies between the results obtained by different authors. In the present paper, the problem is studied using various analytical–numerical models with different levels of accuracy and complexity. The frequency–amplitude curves from the different analysis models developed are compared both for isotropic shells and for an orthotropic composite shell. Secondary modes can play an important role. In more complicated cases modal interactions may significantly influence the nonlinear vibration behaviour, and the results obtained strongly depend on the analysis model chosen.     

Nonlinear dynamics by mode superposition

A mode superposition technique for approximately solving nonlinear initial-boundary-value problems of structural dynamics is discussed, and results for examples involving large deformation are compared to those obtained with implicit direct integration methods such as the Newmark generalized acceleration and Houbolt backward-difference operators. The initial natural frequencies and mode shapes are found by inverse power iteration with the trial vectors for successively higher modes being swept by Gram—Schmidt orthonormalization at each iteration. The subsequent modal spectrum for nonlinear states is based upon the tangent stiffness of the structure and is calculated by a subspace iteration procedure that involves matrix multiplication only, using the most recently computed spectrum as an initial estimate. Then, a precise time integration algorithm that has no artificial damping or phase velocity error for linear problems is applied to the uncoupled modal equations of motion. Squared-frequency extrapolation is examined for nonlinear problems as a means by which these qualities of accuracy and precision can be maintained when the state of the system (and, thus, the modal spectrum) is changing rapidly.The results indicate that a number of important advantages accrue to nonlinear mode superposition: (a) there is no significant difference in total solution time between mode superposition and implicit direct integration analyses for problems having narrow matrix half-bandwidth (in fact, as bandwidth increases, mode superposition becomes more economical), (b) solution accuracy is under better control since the analyst has ready access to modal participation factors and the ratios of time step size to modal period, and (c) physical understanding of nonlinear dynamic response is improved since the analyst is able to observe the changes in the modal spectrum as deformation proceeds.     

Model reduction tools for nonlinear structural dynamics

Three mode types are proposed for reducing nonlinear dynamical system equations, resulting from finite element discretizations: tangent modes, modal derivatives, and newly added static modes. Tangent modes are obtained from an eigenvalue problem with a momentary tangent stiffness matrix. Their derivatives with respect to modal coordinates contain much beneficial reduction information. Three approaches to obtain modal derivatives are presented, including a newly introduced numerical way. Direct and reduced integration results of truss examples show that tangent modes do not describe the nonlinear system sufficiently well, whereas combining tangent modes with modal derivatives and/or static modes provides much better reduction results.     

Structural damage detection using imperialist competitive algorithm and damage function

In practical damage detection problems, experimental modal data is only available for a limited number of modes and in each mode, only a limited number of nodal points are recorded. In using modal data, the majority of the available damage detection solution techniques either require data for all the modes, or all the nodal data for a number of modes; neither of which may be practically available through experiments. In the present study, damage identification is carried out using only a limited number of nodal data of a limited number of modes. The proposed method uses the imperialist competitive optimization algorithm and damage functions. To decrease the number of design variables, several bilinear damage functions are defined to model the damage distribution. Damage functions with both variable widths and variable weights are proposed for increased accurately. Four different types of objective functions which use modal responses of damaged structure are investigated with the aim of finding the most suitable function. The efficiency of the proposed method is investigated using three benchmark numerical examples using both clean and noisy modal data. It is shown that by only using a limited number of modal data, the proposed method is capable of accurately detecting damage locations and reasonably accurately evaluate their extents. The proposed algorithm is most effective with noisy modal data, compared to other available solutions.     

Recursive identification of modal participation in substructuring applications

The utility of modal models in substructuring applications is well known. A recursive scheme for identifying the contributions of subsystem modes to the coupled system is presented. The modal contributions are characterised by evaluating an index based on the difference in the frequency response functions (FRFs), at arbitrarily selected frequencies of interest, between the coupled system containing all available modes and a series of coupled systems with reduced modes. It is shown that the contribution of the subsystem modes can be quantified by eliminating one mode at a time, performing coupling, and evaluating the FRFs at the frequencies of interest.     

Improved subspace modal identification of industrial robots

Industrial robots have become key components for manufacturing automations due to their larger workspaces and flexibility. However, low stiffness and high compliance of industrial robots may inevitably lead to vibration by self-excitation or periodic force dependent on workspace configuration. Therefore, the knowledge of the robot's modal properties should be accurately required to enhance the operation accuracy of industrial robots. To improve the identification accuracy of experimental modal parameters of field industrial robots, an improved subspace identification method is proposed to perform nonlinear iterative optimization for updating the state parameters of industrial robots. Experimental response measurement of a six-degrees-of-freedom industrial robot is carried out to obtain modal parameters under various poses. The identification results of the improved subspace modal method are preferable to that of the traditional method. Moreover, the reconstructed three-dimension working frequency space is presented to exactly characterize experimental modal frequencies throughout its workspace. The proposed method effectively improves the identification accuracy of modal parameters when compared with the traditional algorithms and the influence of robots' pose change on modal parameters is also investigated by experimental modal measurements.     

State Observation for Nonlinear Switched Systems Using Nonhomogeneous High‐Order Sliding Mode Observers

This article presents the problem of finite time reconstruction of the continuous state and operating mode for a class of nonlinear switched systems. The proposed method is based on the nonhomogeneous high‐order sliding mode approach. It is able to reconstruct both the state and operating mode of a switched system based only on its measurable outputs and through the use of the features of the equivalent output injection. The observability is derived in terms of certain geometric restrictions on the vector fields of the switched system that require the availability of all its modes. The method does not require the system to be transformed into any normal form. Simulation results support the proposed method.