Higher order eigensensitivity analysis of damped systems with repeated eigenvalues

A simplified method for the computation of first-, second- and higher-order derivatives of eigenvalues and eigenvectors associated with repeated eigenvalues is presented. Adjacent eigenvectors and orthonormal conditions are used to compose an algebraic equation. The algebraic equation which is developed can be used to compute derivatives of eigenvalues and eigenvectors simultaneously. Since the coefficient matrix in the proposed algebraic equation is non-singular, symmetric and based on N-space, it is numerically stable and very efficient compared to previous methods. To verify the efficiency of the proposed method, the finite element model of the cantilever beam and a mechanical system in the case of a non-proportionally damped system are considered.     

Generation and properties of banded viscous damping matrices

Numerical analysis of discrete vibratory systems subjected to broad band excitation usually requires specification of a viscous damping matrix. A method of determining the entries in such matrices directly is presented in this paper. The modal damping ratios, which can normally be estimated, are used to produce a fully populated damping matrix. To facilitate numerical integration of the equations of motion, a procedure termed stripping has been established for reducing the matrix to banded form. Additionally, using a procedure termed replacement, small element matrices were identified, extracted from the fully populated damping matrix and utilized (much as element mass and stiffness matrices are) to construct the global damping matrix. The effects of stripping or replacement were evaluated for a number of boundary conditions on plates, beams, and shells. Errors caused by variation of the number of system degrees of freedom, the number of diagonal rows remaining in the stripped matrix, and the magnitude and mo dal distribution of damping ratios were studied. Results were, in general, good. Damping ratios, damped natural frequencies, and eigenvectors were closely approximated by both type of systems.     

Coupling measures between modes and state variables in power-system dynamics

A new method for model reduction called Selective Modal Analysis uses the eigenvectors and reciprocal eigenvectors of the system matrix A to calculate participation factors associating modes to certain state variables. This paper investigates the idea in depth. The novel aspect of this paper is the use of MacFarlane's concept that the system matrix A represents an energy transformation map to show that the participation factors are actually modal energies. This interpretation has some advantage because of its direct link to stability. It is shown that the participation factors, or modal energies, can be taken to be coupling measures between modes and state variables. An application to a single-machine infinite-busbar system with and without controllers is given. The properties and limitations of these coupling measures are investigated.     

Spectral determination for a cantilever beam

The eigenvalue specification problem is discussed for a beam with one end clamped and the other end free, which is allowed to have some structural damping, and has a moment force at the free end acting as a control. The spectrum and eigenvectors of the uncontrolled system are first analyzed and are then used to construct the feedback element which solves the problem. It is shown how this feedback element can be numerically approximated, and a spillover result is proved for the approximation. In the case where there is no damping, this method yields a bounded feedback law which exponentially stabilizes the beam     

A practical algorithm for the efficient computation of eigenvector sensitivities

Derivatives of eigenvalues and eigenvectors have become increasingly important in the development of modern numerical methods for areas such as structural design optimization, dynamic system identification and dynamic control, and the development of effective and efficient methods for the calculation of such derivatives has remained to be an active research area for several decades. In this paper, a practical algorithm has been developed for efficiently computing eigenvector derivatives of generalized symmetric eigenvalue problems. For eigenvector derivative of a separate mode, the computation only requires the knowledge of eigenvalue and eigenvector of the mode itself and an inverse of system matrix accounts for most computation cost involved. In the case of two close modes, the modal information of both modes is required and the eigenvector derivatives can be accurately determined simultaneously at minor additional computational cost. Further, the proposed method has been extended to the case of practical structural design where structural modifications are made locally and the eigenderivatives of the modes concerned before are still of interest. By combining the proposed algorithm together with the proposed inverse iteration technique and singular value decomposition theory, eigenproperties and their derivatives can be very efficiently computed. Numerical results from a practical finite element model have demonstrated the practicality of the proposed method. The proposed method can be easily incorporated into commercial finite element packages to improve the computational efficiency of eigenderivatives needed for practical applications.     

On topology optimization of damping layer in shell structures under harmonic excitations

This paper investigates the optimal distribution of damping material in vibrating structures subject to harmonic excitations by using topology optimization method. Therein, the design objective is to minimize the structural vibration level at specified positions by distributing a given amount of damping material. An artificial damping material model that has a similar form as in the SIMP approach is suggested and the relative densities of the damping material are taken as design variables. The vibration equation of the structure has a non-proportional damping matrix. A system reduction procedure is first performed by using the eigenmodes of the undamped system. The complex mode superposition method in the state space, which can deal with the non-proportional damping, is then employed to calculate the steady-state response of the vibrating structure. In this context, an adjoint variable scheme for the response sensitivity analysis is developed. Numerical examples are presented for illustrating validity and efficiency of this approach. Impacts of the excitation frequency as well as the damping coefficients on topology optimization results are also discussed.     

基于小机组详细模型的安徽电网小干扰稳定分析

本文从小机组对220kV主网动态稳定性影响的角度,对基于小机组详细模型的2006年安徽电网进行了小干扰稳定性深入研究,具体采用频域法及PSD-SSAP程序对全系统的特征根进行仿真,分析安徽电网加入地区电网小机组后的阻尼特性,找出动态稳定的薄弱环节,用时域曲线仿真计算校核时域计算结果,并对全网弱阻尼特性机组提出改善与控制措施。     

Applications of dual MRM for determining the natural frequencies and natural modes of a rod using the singular value decomposition method

In this article, the dual multiple reciprocity method is employed to solve the natural frequencies and natural modes for a rod. The conventional approach using dual MRM is not qualified as a systematic method because of the following two reasons: (1) it needs to distinguish the spurious eigenvalue only after the corresponding eigenmode is obtained; (2) the possible indeterminancy of eigenvector may be encountered when the constraint equations chosen are highly dependent such that the rank of the leading coefficient matrix is insufficient. To construct a systematic way, we propose to consider all constraint equations together instead of using the singular or hypersingular equation alone as the conventional MRM uses. The singular value decomposition method is, then, used to solve the eigenproblem after combining the singular and hypersingular equations. This method can avoid the spurious eigenvalue problem and the possible indeterminancy of boundary eigenvectors at the same time. Three numerical examples are given to verify the validity of the present method.     

Multivariate Polynomial System Solving Using Intersections of Eigenspaces

The solutions of a polynomial system can be computed using eigenvalues and eigenvectors of certain endomorphisms. There are two different approaches, one by using the (right) eigenvectors of the representation matrices, one by using the (right) eigenvectors of their transposed ones, i.e. their left eigenvectors. For both approaches, we describe the common eigenspaces and give an algorithm for computing the solution of the algebraic system. As a byproduct, we present a new method for computing radicals of zero-dimensional ideals.     

双回路复杂波网络的镇定

本文研究双回路的复杂波网络系统,主要探索带回路系统的控制问题,通过设计反馈控制器使得闭环系统稳定.同时借助于谱分析方法讨论了相应闭环系统算子谱的分布情况并证明了系统算子的(广义)本征向量构成状态空间上的一组加括号Riesz基,得到了闭环系统特征模态的展开.     

基于复模态方法的二维夹层壁板颤振分析

本文研究二维夹层壁板在一侧受超音速气动力的情况下的颤振现象.利用复模态方法和伽辽金方法分析颤振临界马赫数以及夹芯粘性阻尼对颤振的影响.结果发现考虑前四阶模态时,由于一二阶频率重合而使振动能量积聚发生颤振.考虑中间层的粘弹性时,发现随着粘性阻尼的增加,颤振临界马赫数和临界颤振频率均呈现先降低后升高的现象,其原因是粘弹性一方面降低系统固有频率使得临界马赫数降低,另一方面又使能量耗散使得临界马赫数升高,在这两种作用的影响下出现了上述复杂的现象.本文的研究结果有利于颤振抑制时的设计优化.     

基于等效模态阻尼的复合隔振系统振动计算方法

基于线性黏弹性假设,将应变能阻尼理论推广到复合隔振系统的等效模态阻尼计算中,运用Python和Abaqus编制相应的计算程序,该程序可考虑材料阻尼的频变特性。以多种材料组成的船舶双层复合隔振系统为算例,计算其等效模态阻尼和隔振器等效阻尼系数。分别采用直接积分法和模态叠加法计算系统振动响应,对比设备、筏架、船底壳的振动加速度响应,验证基于等效模态阻尼的模态叠加法的准确性。结果表明,该方法可以准确计算复杂组合模型的模态阻尼,算例的振动响应计算结果一致性较好,用模态叠加法可以大幅提高复合隔振系统稳态振动响应的计算效率。     

Calculation of derivatives of multiple eigenpairs of unsymmetrical quadratic eigenvalue problems

New methods are presented for computing the derivatives of multiple eigenvalues and the corresponding eigenvectors of unsymmetrical quadratic eigenvalue problems. The expressions of eigenpair derivatives are derived in terms of the eigenvalues and eigenvectors of quadratic eigenvalue problems, and the use of rather undesirable state-space representation is avoided. Hence the cost of computation is greatly reduced. The proposed methods are valid for both the case of distinct eigenvalue derivatives and the case of equal eigenvalue derivatives. Numerical results show that the proposed methods are efficient.     

Engineering MEMS Resonators With Low Thermoelastic Damping

This paper presents two approaches to analyzing and calculating thermoelastic damping in micromechanical resonators. The first approach solves the fully coupled thermomechanical equations that capture the physics of thermoelastic damping in both two and three dimensions for arbitrary structures. The second approach uses the eigenvalues and eigenvectors of the uncoupled thermal and mechanical dynamics equations to calculate damping. We demonstrate the use of the latter approach to identify the thermal modes that contribute most to damping, and present an example that illustrates how this information may be used to design devices with higher quality factors. Both approaches are numerically implemented using a finite-element solver (Comsol Multiphysics). We calculate damping in typical micromechanical resonator structures using Comsol Multiphysics and compare the results with experimental data reported in literature for these devices     

四元数矩阵正交特征向量系的求解方法及其在彩色人脸识别中的应用

针对四元数矩阵正交特征矢量系求解困难的缺点, 本文提出一种获取四元数矩阵正交特征矢量集等效、便捷的方法, 其基本思路为: 首先, 构造四元数矩阵定义于复数域的导出阵, 并利用该导出阵特征矢量空间的一种特殊的等价空间间接获取相应特征值所对应的特征矢量. 然后, 将复数矢量转换为四元数矢量, 按如此方式获取的对应所有特征值的非零特征矢量则构成原始四元数矩阵的正交特征矢量系. 同时, 本文将定义于实数域的主成分分析方法 (Principal component algorithm, PCA) 向四元数体作合理的推广, 给出详细的数学推导过程, 证明该方法的合理性及其在统计模式识别领域得以应用的可能性. 最后, 作者将彩色图像像素的R、G、B三分量作为四元数的三个虚数部分, 首次在人脸识别中引入基于四元数的彩色人脸识别方法. 较传统的基于灰度图像的识别方法, 本文方法不仅利用了人脸图像灰度值的空间分布信息, 而且充分利用不同人脸之间的色彩差异, 从而得到更多的鉴别信息.在四川大学人工智能研究所的彩色人脸库上进行的实验表明, 所提出的基于四元数的识别方法不仅大幅度提高了识别率, 而且具有较高的鲁棒性.     

一种求解复Hermite矩阵特征值的方法

介绍几种求解矩阵特征值和特征向量的经典算法及各自优缺点,通过理论推导,提出了一种性能稳健的方法,可以求解信号处理中常见的复Hermite阵.将对复Hermite矩阵求特征值和特征向量的问题转化为求解实对称阵的特征值和特征向量,而实对称阵的求解采用一种改进的三对角Householder法.最后把结果与Matlab仿真结果比较,可以看出该方法有很高的精确度.     

Calculating modes of quantum wire and dot systems using a finite differencing technique

In this paper the Schrödinger equation of both a quantum wire and a quantum dot are solved using a finite difference approach. It is demonstrated that the method is valid for the simple case of an infinitely deep quantum wire, where the solutions obtained are within 0.25 meV of the analytical solutions. The method is then used to calculate the eigenenergies of a triangular wire with finite barriers. The eigenenergies of the more complex case of a pyramidal quantum dot were then calculated using this method. The method is compared to an eigenvalue method in terms of memory usage, time requirements and the numerical solutions. It is shown that this method has the advantages of being relatively fast, usable with any wire geometry and any potential profile. In addition, the demand on computer memory varies linearly with the size of the system under investigation.     

A generalized complex symmetric eigensolver

For a heavily damped system, either viscous or hysteresis or both, the homogeneous solution constitutes a generalized complex symmetric eigenproblem [A]{x}= λ [B]{x}, where [A] and [B]are complex symmetric matrices. The general complex method to solve the transformed eigenproblem [B] su−1 [A]{x} = λ {x} is very demanding in computation. A new method of Jacobi rotation is introduced to solve the complex symmetry eigenproblem completely. Full advantages of the symmetry are taken. The complex eigenvalues can be computed directly when both matrices are diagonalized. The complex eigenvectors are obtained as the products of the complex plane rotation. A Fortran subroutine and examples are given.     

A new noniterative method for pattern synthesis of unequally spaced linear arrays

In this article, a new noniterative beam shaping method is introduced to synthesise array factor (AF) of an unequally spaced linear array (UESLA). The proposed method is based on eigenvector decomposition of sampled data matrix of a given pattern. Using matrix analysis, the eigenvalues and their corresponding eigenvectors of the sampled matrix are determined. It is shown that the eigenvalues and eigenvectors of the sampled matrix are related to the locations and complex excitation coefficients of the array elements. According to the concept of generalized eigenvalue concept, the solution of locations and excitation coefficients is derived using least square method. In order to reduce the number of array elements, singular value decomposition is applied to obtain a low ranked matrix using rejection of nonzero eigenvalues. Using the approximated sampled matrix, the excitation and locations of the optimized array elements are calculated. A few comprehensive examples are investigated to verify the accuracy of the proposed method and the obtained results are compared with those of an equally spaced linear array (ESLA). It is shown that the total number of array elements in an UESLA is less than that of ESLA, which is the most advantage of the introduced method in AF synthesis.