改进的直接部件模态综合法

Yee和Tsuei提出的直接部件模态综合法将自由界面模态综合法与频响函数相结合 ,简化了计算过程、减少了计算量 ,并便于与实验模态分析相结合。本文在其基础上利用矩阵级数展开的方法将被截断高阶模态的贡献用保留模态和系统物理特性矩阵精确表达 ,结合分块计算方法 ,进一步减少了计算量并有效地提高了计算精度 ,且确定了模态截断准则。数值算例表明其行之有效     

复振型导数计算的移位多次模态加速法

本文研究了含粘性阻尼结构的复振型导数计算问题，将导数计算问题看成是一个简谐激振的响应计算问题，采用多次模态加速法和移位法，导出了复振型导数计算的移位多次模态加速法。该方法具有明确的数学和物理意义，可导出已有的各种计算方法。算例表明本方法计算复振型导数只需用很少几个模态即可保证精度，计算量大大减少。     

钢筋混凝土结构三维模态pushover分析方法的改进

三维多模态pushover分析方法(即三维MPA方法)为评估不规则结构的抗震性能提供了一种简便的方法.本文以与荷载相关的Ritz(LDR)向量组代替传统三维MPA方法中的振型向量组,自动滤除了对结构反应无贡献的相关模态,从而提高了结构三维MPA方法的计算效率.另一方面,本文提出了增大除"基本模态组"之外的高阶模态对应的目标位移,以考虑被截断的高阶模态对结构反应的影响,从而改善了结构三维MPA方法的计算精度.算例结果表明,改进后的结构三维MPA方法的计算效率效率大大提高,而结构反应计算结果的准确度有了显著的改善,可用于包括不规则高层建筑结构在内的各种房屋建筑结构的抗震变形验算及抗震性能评估.     

复振型导数计算的移位多次模态加速法

本文研究了含粘阻尼结构的复振型导数计算问题，将导数计算问题成是一个简谐激谐的振的响应计算问题，采用多次模态加速法和移位法，导出了复振型导数计算的移位多次模态加速法，该方法具有确的数学和物理意义，可导出了已有的各种计算方法。算例表明本方法计算复振型导数只需用很少几个模态即可保证精度，计算量大大减少。     

关于一类非线性阻尼模型控制问题的一个注记

利用模态方法进行柔性体控制，就必须作模态截断。在线性粘性阻尼模型情形，阻尼正比于频率，高阶模态会很快衰减。但对于非线性阻尼模型，情况就复杂了。结果表明，高阶模态响应不总能很快衰减，并依赖于初始条件、因为对于非线性阻尼模型的控制，如何进行模态截断，是一个需要谨慎处理的问题。     

一种中高频激励下动力系统响应的级数展开改进算法

针对频率响应函数的级数展开法在中高频激励时计算发散的问题,提出一种新的级数展开改进算法.将系统的结构模态划分为低阶和截断的高阶模态,在模态叠加分析的基础上,将频率响应函数进行泰勒级数展开.根据高低阶模态对质量矩阵和刚度矩阵的耦合特性,用低阶模态及系统矩阵表达高阶模态对响应的影响.研究结果表明,该算法将频率响应函数的级数展开法扩展到高频激励和中频激励范围阶段,在非完备模态条件下提高了频率响应函数的计算精度,数值计算检验了该方法准确可靠并有很好的收敛性.     

基于键合图法的多个子结构自由界面模态综合法研究和应用

将键合图方法用于动态子结构研究,提出了一种求解多个子结构自由界面模态综合法的新方法。通过一系列物理和数学上的分析,详细推导出多个子结构自由界面模态综合法的计算过程。在本文给出的算例中,基于键合图方法的自由界面模态综合法,通过建立各个子结构的状态空间方程,计算子结构相应的特征值矩阵和振型矩阵;这些子结构在进行模态综合后,获得整体结构的特征值矩阵和振型矩阵,该结果和原来整体结构完全相同,这进一步说明了本文提出的新方法的正确性。运用本文提出的方法建立状态空间方程,在子结构的模态综合过程中,不会产生系统特征值的增失根问题,确保了子结构在综合后其整体结构模态信息的完整性。     

简化有限元方法的波纹管模态分析

采用简化方法对单层U型波纹管进行有限元模态分析。在长度,质量,对轴线转动惯量,体积,轴向弹簧比率以及周向弹簧比率不变的前提下,把波纹管简化成直壁薄管,用直壁薄管模型代替波纹管模型进行有限元模态计算。简化了计算模型,减少了计算量,提高了计算效率。本文给出了明确的等效直壁薄管的几何参数,物性参数的求解公式,改变了Broman人为设定直壁薄管厚度的方法,取得了更加准确的计算结果,使利用直壁薄管模型求解波纹管固有频率具有通用意义。进行了直壁薄管模型得到模态振型与波纹管模型得到模态振型之间的比较,认为直壁薄管模型可以求解波纹管的振型。     

约束位置的修改对振动模态的影响

结构的约束位置对其固有振动特性有重要的影响，因此计算固有振动特性对约束位置的敏度是结构优化以及模型校正等的重要基础性工作．本文从广义变分原理出发，建立了约束位置的小修改导致的固有值和振动模态增量的方程，针对模态增量方程的求解，提出了一种处理原本征函数空间不完备的方法，从而可以在扩展的原本征函数空间内对振型增量进行展开．针对模态截断问题，本文又提出了一种“模态展开＋幂级数民开”的混合展开法，这种展开法的收敛性好，收敛程度易于估计，实用方便．     

应用结构传递函数计算高阶剩余模态

本文分析了高阶剩余模态的组成形式,提出了一种利用结构传递函数求解高阶剩余模态的方法.由于文中通过传递函数得到结构高阶剩余模态,因此实验模态技术能方便地引入模态综合之中,为模态综合技术更广泛地应用提供了方便.     

薄板弯曲分析的高阶高效无网格法

与传统有限元法相比,无网格法具有节点形函数高度光滑、易于形成高阶近似等优势,更适合于以薄板弯曲问题为代表的高阶偏微分方程的数值求解。然而,高阶无网格法的形函数是非多项式的有理函数,导致弱形式的区域积分难以得到精确计算,通常采用的高阶高斯积分方法需使用大量积分点,计算效率低且精度不高。本文针对薄板弯曲问题的高阶（三阶）无网格法分析,首次发展了与该高阶近似相一致的曲率光顺方案,并基于背景三角形积分单元建立了相应的数值积分格式,大幅度减少了所需的积分点数目。所发展方法的关键在于计算刚度阵所需的形函数的二阶导数由形函数及其一阶导数通过散度定理确定,而非对形函数直接求导获得。数值结果表明,基于标准的高斯积分方案的高阶无网格法精度不高,不能精确再现纯弯曲和线性弯曲模式,且得到的弯矩场分布存在严重的虚假数值振荡。而本文所建议的基于曲率光顺方案的高阶无网格法能够方便高效地求解薄板弯曲问题,尤其是它能精确反映纯弯曲和线性弯曲模式。与标准的高斯积分方法和目前主流的常曲率光顺方法相比,本文方法在计算效率、精度、弯矩分布等方面均展现出显著优势,因而具有较好的应用价值。     

Nonlinear interactions in a parametrically excited system with widely spaced frequencies

An investigation is presented into the transfer of energy from high- to low-frequency modes. The method of averaging is used to analyze the response of a two-degree-of-freedom system with widely spaced frequencies and cubic nonlinearities to a principal parametric resonance of the high-frequency mode. The conditions under which energy can be transferred from high- to low-frequency modes, as observed in the experiments, are determined. The interactions between the widely separated modes result in various bifurcations, the coexistence of multiple attractors, and chaotic attractors. The results show that damping may be destabilizing. The analytical results are validated by numerically solving the original system.     

波导结构频散分析的特征频率法及在板条结构中的应用

提出一种求解波导结构频散特性的有限元特征频率法,该方法基于振动问题的特征频率计算理论,根据模态振型识别波数与模态类型,建立了相速度及群速度的求解方法。该方法可适用于任意波导结构的频散关系求解。首先分析满足收敛精度要求的最大网格单元尺寸与最小模型长度,并用该方法对简支板条结构的频散特性进行了计算。结果表明,有限元特征频率法适合求解波动频散关系,板条结构中模态受边界影响会产生同阶高次模态,边界尺寸决定新模态的截止频率;随频率的增大,同阶低次反对称模态会趋于一致;对称模态能量分布受边界影响较大。本文也为板条类结构导波实验结果的分析提供了理论依据。     

Modal interactions in the vibrations of a heated annular plate

We investigate the non-linear forced vibrations of a thermally loaded annular plate with clamped–clamped immovable boundary conditions in the presence of a three-to-one internal resonance between the first and second axisymmetric modes. We consider the in-plane thermal load to be axisymmetric and excite the plate externally by a harmonic force near primary resonance of the second mode. We then use the non-linear von Kármán plate equations to model the behavior of the system and apply the method of multiple scales to investigate its responses. We found that the response can be periodic oscillations consisting of both modes, with a large component from the first mode. Moreover, the periodic solutions may undergo Hopf bifurcations, which lead to aperiodic oscillations of the plate.     

Tikhonov's regularization approach in mode shape curvature analysis applied to damage detection

In this paper an on-going research effort aimed at detecting and localizing damage in plate-like structures by using mode shape curvature based damage detection algorithm is described. The proposed damage index uses exclusively mode shape curvature data from the damaged structure. This method was originally developed for beam-like structures. In this paper, the method is generalized to plate-like structures which are characterized by two-dimensional mode shape curvature. To calculate mode shape curvatures from the measured mode shapes, three approaches are proposed: the first one is the well-known central difference approximation, the other two are classical approaches based on Tikhonov's regularization technique with smoothing functional. The applicability and effectiveness of the proposed damage detection algorithms are demonstrated experimentally on an aluminium plate containing mill-cut damage. The validity of the method is assessed by comparing the identification results of the experimental test case to the results obtained from the simulated test case. The modal frequencies and the corresponding mode shapes of the aluminium plate are obtained via finite element models for numerical simulations and by using a scanning laser vibrometer (SLV) for the experimental study.     

Dynamic near three-to-one internal resonance in composite stiffened panels

Nonlinear dynamic of composite stiffened panels to parametric and three-to-one internal resonances is investigated. The ordinary differential equation of two mode shapes is established by using Galerkin method and the condition of three-to-one internal resonance between the first mode (1,3) and the second mode (3,1) is examined near the principal resonance 2:1 of the first mode. Then, the nonlinear behavior of the two buckling mode shapes is analyzed using a perturbation analysis. We show the existence of jump phenomena for the two modes indicating a complex dynamic of the structure near the three-to-one internal resonance for the HM Graphite/epoxy materials.     

二元复合材料板的自由振动: 半解析法

二元复合材料板是超材料板结构中常见的单元之一. 针对由材料参数相差两个量级的基体和嵌入体组成的二元复合材料板, 提出结构自由振动的半解析模型, 并对其振动特性进行了研究. 基于区域分解法和二元材料的分布, 将二维平板分解成两个子区域. 通过在振型函数中附加区域试函数, 来描述复合材料板面内刚度突变引起局部位移和转角的非光滑性. 基于二元复合材料板的基本边界条件和两子区连接处的变形协调条件, 构造了新的振型函数. 基于经典薄板理论, 利用带特殊试函数的里兹法, 求得不同几何构型下二元复合材料板的固有频率和振型, 并研究了嵌入体的尺寸和位置对结构振动特性的影响规律. 通过收敛分析并与有限元仿真结果对比, 验证了本文方法的准确性. 研究结果表明: 传统的全局试函数在分析具有振动局部化的模态时会得到不准确的结果, 而附加区域试函数可以显著提高里兹法的收敛速度以及结果的准确性; 嵌入体位置对低阶固有频率的作用不明显, 却能显著改变低阶振型节线的分布和振动局部化发生的区域.      

Modal identification based on Hilbert-Huang Transform of structural response with SVD preprocessing

In recent years, Empirical mode decomposition and Hilbert spectral analysis have been combined to identify system parameters. Singular-Value Decomposition is pro- posed as a signal preprocessing technique of Hilbert-Huang Transform to extract modal parameters for closely spaced modes and low-energy components. The proposed method is applied to a simulated airplane model built in Automatic Dynamic Analysis of Mechanical Systems software. The results demonstrate that the identified modal parameters are in good agreement with the baseline model.     

Coupling between high-frequency modes and a low-frequency mode: Theory and experiment

An analytical and experimental investigation into the response of a nonlinear continuous system with widely separated natural frequencies is presented. The system investigated is a thin, slightly curved, isotropic, flexible cantilever beam mounted vertically. In the experiments, for certain vertical harmonic base excitations, we observed that the response consisted of the first, third, and fourth modes. In these cases, the modulation frequency of the amplitudes and phases of the third and fourth modes was equal to the response frequency of the first mode. Subsequently, we developed an analytical model to explain the interactions between the widely separated modes observed in the experiments. We used a three-mode Galerkin projection of the partial-differential equation governing a thin, isotropic, inextensional beam and obtained a sixth-order nonautonomous system of equations by using an unconventional coordinate transformation. In the analytical model, we used experimentally determined damping coefficients. From this nonautonomous system, we obtained a first approximation of the response by using the method of averaging. The analytically predicted responses and bifurcation diagrams show good qualitative agreement with the experimental observations. The current study brings to light a new type of nonlinear motion not reported before in the literature and should be of relevance to many structural and mechanical systems. In this motion, a static response of a low-frequency mode interacts with the dynamic response of two high-frequency modes. This motion loses stability, resulting in oscillations of the low-frequency mode accompanied by a modulation of the amplitudes and phases of the high-frequency modes.