Second-order sensitivity of eigenpairs in multiple parameter structures

This paper presents methods for computing a second-order sensitivity matrix and the Hessian matrix of eigenvalues and eigenvectors of multiple parameter structures. Second-order perturbations of eigenvalues and eigenvectors are transformed into multiple parameter forms,and the second-order perturbation sensitivity matrices of eigenvalues and eigenvectors are developed.With these formulations,the efficient methods based on the second-order Taylor expansion and second-order perturbation are obtained to estimate changes of eigenvalues and eigenvectors when the design parameters are changed. The presented method avoids direct differential operation,and thus reduces difficulty for computing the second-order sensitivity matrices of eigenpairs.A numerical example is given to demonstrate application and accuracy of the proposed method.     

Computing the eigenvectors of a matrix with multiplex eigenvalues by SVD method

Every matrix is similar to a matrix in Jordan canonical form, which has very important sense in the theory of linear algebra and its engineering application. For a matrix with multiplex eigenvalues , an algorithm based on the singular value decomposition ( SVD ) for computing its eigenvectors and Jordan canonical form was proposed. Numerical simulation shows that this algorithm has good effect in computing the eigenvectors and its Jordan canonical form of a matrix with multiplex eigenvalues. It is superior to MATLAB and MATHEMAtICA.     

接近亏损系统的矩阵摄动法

随着系统参数的变化，带有重频的系统可转变成带有密集频率的系统，反之亦然．本文讨论了亏损系统与接近亏损系统之间的关系．并提出了接近亏损系统的平均移位的摄动方法．算例表明了此方法的有效性．     

非对称矩阵特征值问题密集模态重分析方法

本文提出了一种非对称矩阵特征值问题的密集模态重分析方法。它将原密集特征值问题表达为与其临近的某一重特征值问题的小摄动,从而密集模态的重分析问题就转化为重频模态的重分析问题。     

亏损系统振动模态的可控可观性

利用奇异值分解方法来讨论系统广义模态的可控可观性的量度问题,得到了亏损系统广义模态可控可观性的量度指标,同时用实例说明了本文方法是有效的。     

时域模态参数识别的直接特征系统实现算法

本文提出了一种新的结构模态参数识别算法,称为直接特征系统实现算法(DERA)。直接采用时域多输入多输出数据构成线性时不变离散系统的最小实现,并利用系统矩阵的特征值和特征向量求取模态参数。输入输出数据的直接应用避免了利用脉冲响应函数作原始数据的ERA、PRCE等算法的一些局限。奇异值分解技术的引入使系统的定阶和实现更方便,并具有了更良好的数值稳定性。为验证算法,构造了一个五自由度的弹簧质量系统进行数值模拟。良好的数值结果证明了本算法的有效性,并表明本算法对密集模态仍有较高的识别精度。     

智能结构密频系统振动控制及其摄动分析

提出一种智能结构密频控制规律的设计方法。其主要工作为：将密频子空间转化为重频子空间,把密频系统作为重频系统上的摄动;为了解决重频密频子空间相对应的特征向量选取的敏感性问题,通过摄动分析得到与重频密频子空间相对应特征向量的线性组合并利用闭环系统特征值极点配置的方法得到重频密频系统的振动规律;把所设计的振动控制规律作用于原系统和摄动系统上,讨论了结构参数改变后对系统的动态特性的影响;最后通过算例证明该方法的有效性。     

大型结构特征值问题的Lanczos子结构并行算法

本文利用子结构和Lanczos方法,提出了大型结构固有频率与模态的并行解法。该方法在Lanczos方法的求解过程中,仅利用子结构刚度阵和质量阵并行进行凝聚,进而求得新的迭代矢量,最终求得三对角阵对应的特征值和特征向量。该算法在西安交通大学ELXSI-6400并行计算机上程序实现,计算结果表明能有效地节省计算时间和计算机的内存,为一种有效的大型工程结构动力问题的求解方法。     

基于牛顿-拉夫逊的拉压不同模量问题的数值求解

针对拉压模量不同引起的材料本构非线性,首先,通过引入改进的Heaviside函数将本构方程连续光滑化;然后,基于特征值与特征向量的求导策略,推导有限元求解模型中切线刚度矩阵的列式;最终,提出基于牛顿-拉夫逊迭代格式的拉压不同模量问题的数值求解算法。数值算例验证了本文算法比传统算法具有更稳定的收敛性和更高的求解精度,特别适合于工程分析中大规模计算问题的求解。     

Modal parameter identification of flexible spacecraft using the covariance-driven stochastic subspace identification (SSI-COV) method

In this paper, the on-orbit identification of modal parameters for a spacecraft is investigated. Firstly, the cou-pled dynamic equation of the system is established with the Lagrange method and the stochastic state-space model of the system is obtained. Then, the covariance-driven stochas-tic subspace identification (SSI-COV) algorithm is adopted to identify the modal parameters of the system. In this algo-rithm, it just needs the covariance of output data of the system under ambient excitation to construct a Toeplitz matrix, thus the system matrices are obtained by the singular value decom-position on the Toeplitz matrix and the modal parameters of the system can be found from the system matrices. Finally, numerical simulations are carried out to demonstrate the validity of the SSI-COV algorithm. Simulation results indi-cate that the SSI-COV algorithm is effective in identifying the modal parameters of the spacecraft only using the output data of the system under ambient excitation.     

不确定参数闭环控制系统特征值的区间分析

运用区间理论,讨论了区间参数的结构振动控制问题,给出了求解闭环系统区间特征值的一种方法．基于区间参数导出了区间刚度矩阵和质量矩阵,然后利用矩阵摄动理论和区间扩张理论,推导了复区间特征值上下界估计的算法．这些结果是从二阶系统的左右特征向量出发得到的．将该文方法应用到悬臂梁的控制问题,数值结果表明它是有效的．     

Form-finding of tensegrity structures via genetic algorithm

We propose an efficient method for the form-finding of tensegrity structures. The force densities of each tensegrity are obtained by the minimisation of a particular objective function, leading to a semi-positive definite force density matrix (a super-stable tensegrity) with a required rank deficiency. A genetic algorithm is used as a global search technique for the minimisation. The geometry of a tensegrity is subsequently formed based on those eigenvectors of the force density matrix corresponding to zero eigenvalues. Furthermore, two other methods are introduced to convert the asymmetrical geometry obtained from the main algorithm into its symmetrical counterparts. This transformation in geometry is performed by finding a suitable linear combination of the mentioned eigenvectors. Examples from well-known tensegrities including prismatic, truncated tetrahedron, expandable octahedron and truncated icosahedron tensegrities are studied using the present method, and the results obtained are compared with those documented in the literature to verify the efficiency of the present method.     

The state space reconstruction technology of different kinds of chaotic data obtained from dynamical system

Certain deterministic nonlinear systems may show chaotic behavior. We consider the motion of qualitative information and the practicalities of extracting a part from chaotic experimental data. Our approach based on a theorem of Takens draws on the ideas from the generalized theory of information known as singular system analysis. We illustrate this technique by numerical data from the chaotic region of the chaotic experimental data. The method of the singular-value decomposition is used to calculate the eigenvalues of embedding space matrix. The corresponding concrete algorithm to calculate eigenvectors and to obtain the basis of embedding vector space is proposed in this paper. The projection on the orthogonal basis generated by eigenvectors of timeseries data and concrete paradigm are also provided here. Meanwhile the state space reconstruction technology of different kinds of chaotic data obtained from dynamical system has also been discussed in detail. The project supported by the National Natural Science Foundation of China (19672043)     

Saint-Venant problem of three-dimensional linear viscoelasticity in the Hamiltonian system

The traditional Saint-Venant problem of three-dimensional viscoelasticity is discussed under the Hamiltonia system with the use of the Laplace integral transformation, and the original problem is transformed into finding eigenvalues and eigenvectors of the Hamiltonia operator matrix. Since local effect near the boundary is usually neglected, all solutions of Saint-Venant problems can be obtained directly by the combinations of zero eigenvectors. Moreover, the adjoint relationships of the symplectic orthogonality of zero eigenvectors in the Laplace domain are generalized to the time domain. Therefore the problem can be discussed directly in the eigenvector space of the time domain, and the iterative application of Laplace transformation is not needed. Simply by applying the adjoint relationships of the symplectic orthogonality, an effective method for boundary condition is given. Based on this method, some typical examples are discussed, in which the whole character of total creep and relaxation of viscoelasticity is clearly revealed.     

A Numerical Method for Wave Propagation in Viscoelastic Stratified Porous Media

This paper presents a numerical method, a transmission matrix method, for the wave propagation in viscoelastic stratified saturated porous media. The wave propagation in saturated media, based on Biot theory, is a coupled problem. In this stratified three-dimensional model we do the Laplace transform for the time variable and the Fourier transform for the horizontal space coordinate. The original problem is transformed into ordinary differential equations with six independent unknown variables, which are only the function of the coordinate of depth. Thus, we get a transmission matrix of the wave problem for each layer. In the process of solution we use numerical method to calculate the eigenvalues and the eigenvectors of the transmission matrices. In the first step of the solution process we can obtain the wave field in the transformed space. The fast Fourier transform (FFT) method is used to do the inverse Laplace and the inverse Fourier transforms to get the solution in the time space. The detailed formulae are derived and some numerical examples are given.     

求解非线性结构动力方程的预估校正-辛时间子域法

提出了求解非线性结构动力方程的预估校正-辛时间子域法。首先,将结构非线性动力方程转换为状态空间方程,在任一时间子域内利用改进的欧拉法对各离散时刻的状态变量值进行预估和校正。然后,将离散的非线性项用Lagrange插值多项式展开并视为外荷载,结合辛时间子域法即可求解非线性动力系统的响应。这种方法不必对状态矩阵求逆,无需计算高阶导数,计算简单,格式统一,易于编程。算例结果表明,本文方法具有较高的计算精度、效率和稳定性,是一种求解非线性结构动力方程的有效方法。     

Sensitivity Analysis of Multiple Eigenvalues*

ABSTRACT

This paper is devoted to sensitivity analysis of eigenvalues of nonsym-metric operators that depend on parameters. Special attention is given to the case of multiple eigenvalues. Due to the nondifferentiability (in the common sense) of multiple roots, directional derivatives of eigenvalues and eigenvectors in parametric space are obtained. Sensitivity analysis is based on the perturbation method of eigenvalues and eigenvectors. The generalized eigenvalue problem and vibrational systems are also investigated. Strong and weak interaction of eigenvalues are distinguished and interactions in two- and three-dimensional space are treated geometrically. It is shown that the strong interaction of eigenvalues is a typical catastrophe. Simple examples that illustrate the main ideas are presented. The results obtained are important for qualitative and quantitative study of mechanical systems subjected to static and dynamic instability phenomena.     

北斗双频组合动对动高精度相对定位新算法

针对动对动相对定位基线矢量实时变化导致整周模糊度浮点解在动态情况下难以快速精确求解的问题,提出了一种北斗双频动对动相对定位算法:对组合双差方程基线向量的系数矩阵进行奇异值分解,并变换组合双差方程以消除基线参量,将变参数估计问题转化为定参数估计问题,然后采用递推最小二乘算法实时推算组合模糊度的浮点解及其协方差矩阵,在此基础上,采用最小二乘模糊度降相关平差法(LAMBDA)搜索和固定组合模糊度。试验结果表明,该方法能够快速准确固定组合模糊度,与单频解算相比,初始化时间用时更短,基线误差在5 cm以内,能较好地适用于动对动相对定位。     

Generalized-order perturbation with explicit coefficient for damage detection of modular beam

A general method is formulated to estimate damage location and extent from the explicit perturbation terms in specific set of eigenvectors and eigenvalues. At first, perturbed orthonormal equation is generated from the perturbation of eigenvectors and eigenvalues to obtain the k-th explicit perturbation coefficients. At second, perturbed eigenvalue equation is generated from the perturbation of eigenvector and eigenvalue, and first-order expansion of the stiffness matrix to obtain other explicit perturbation coefficients. Stiffness parameters are computed from these equations using an optimization method. The algorithm is iterative and terminates under certain criteria. A fixed–fixed modular beam with various numbers of elements is used as test structure to investigate the applicability of the developed approach. By comparison with the Euler–Bernoulli beam, discretization errors are analyzed. In six elements beam, first-order algorithm converges faster for small percentage damage. Second-order algorithm is more efficient for medium percentage damage. For large percentage damage, the second-order algorithm converges more effectively. Meanwhile, for eight elements large percentage damage and ten elements small percentage damage, second-order algorithm converges faster to the termination criterion.     

一类结构的（n－3）重根及其非退化性

基于陀螺模态综合法，从总装阵的块式结构出发，构造性地证明了具有ｎ片桨叶的旋翼型结构陀螺特征值问题存在一系列的（ｎ－３）重特征根，得到了对应的（ｎ－３）个完备的振型．结论进而推广到有阻尼的旋翼型结构．继续研究证明过程表明：结论适用于更广泛的一类具有重复子结构的结构系统，结果表明这类结构几何上的重复性或对称性导致的重根不会引入退化性．不同类型的算例验证了所得到的解析结果．本文还试图说明动力子结构法的定性性质保持特性是值得继续探讨的课题