二类复特征值表达式的适用性研究

对于线性系统，根据其质量、刚度或阻尼等矩阵的性质了解特征值，是十分有意义的问题．对于有阻尼陀螺系统，本文首先通过模态正交性推导了有关矩阵的若干关系式．然后分析了复特征值的两类表达式的适用性，着重讨论了通过一元二次方程研究特征值性质的现行处理方法存在的困难．提出一种确定特征值的补充方法，指出可能存在虚数特征值的系统的性质．算例表明结论正确     

求解对称矩阵特征值及特征向量的初等变换法

 在力学中有一类量的求解可归结为矩阵特征值和特征向量的求解，而求解 矩阵的特征值将要求解高次方程的根，这在数学上将遇到难以克服的困难. 应用初等变 换方法，将对称矩阵的特征矩阵对角化. 利用这种方法，同时可求出该矩阵所有的特征值和 正交的特征向量，避免了求解高次方程根的困难与把各特征向量正交化的麻烦.     

随机结构系统的一般实矩阵特征值问题的概率分析

由于工程实际结构的复杂性和所用材料在统计上的离散性以及测量、加工、制造误差的存在，必然导致具有随机参数的随机结构振动系统，按结构参数的性质来划分，随机振动问题包括两方面内容：(1)确定结构问题；(2)随机结构问题。本文以现代数学理论为依托，研究了随机结构系统的一般实矩阵的特征值问题。根据Kronecker代数、向量值和矩阵值函数的灵敏度分析、一般二阶矩法和概率摄动技术给出了计算随机结构系统的一般实矩阵的特征值和特征向量的数值方法，可以有效地得出随机结构系统的一般实矩阵的特征向量的统计量，发展了2D矩阵值函数的随机结构系统的特征值问题概率分析理论。     

大型陀螺特征值问题的广义Arnoldi减缩算法

基于Ａｒｎｏｌｄｉ法，建立陀螺特征值问题的广义Ａｒｎｏｌｄｉ格式，并利用系统矩阵的反对称特性，得到极其简洁的甚至比对称矩阵Ｌａｎｃｚｏｓ法更为简单的递推格式，可称为陀螺Ａｒｎｏｌｄｉ减缩算法。     

用动态缩聚法求解结构的特征值问题

基于逐级逼近思想提出了一种新的动态缩聚法。保留和压缩的自由度通过一个缩聚矩阵联系，最初的缩聚矩阵由系统子矩阵定义。用循环的方法逐级逼近特征值及特征向量。计算实例验证了上述方法的可行性。     

对用稳定图识别真假模态的一点改进

利用单位样值响应矩阵和自由响应的特征值分解形式中的特征值前的系数矩阵的范数作稳定判据，结合稳定图识别真假模态，计算表明，本方法具有较好的分解能力。     

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．     

拱桥悬臂施工过程中面内特征值的传递矩阵法

采用传递矩阵法对拱桥悬臂施工过程中的面内特征值问题进行求解,建立了该类桥型施工过程中面内竖弯刚度的评估方法.首先,将索和拱分别视为无垂度的张紧弦和欧拉伯努利梁,基于传递矩阵法基本理论推导了系统的总传递矩阵,考虑拱和索的边界条件以及索拱节点的位移连续性条件得到系统的特征值方程,进而计算出系统的频率和模态.同时,采用有限元...     

广义特征值问题的Collatz包含定理

工程中已发展了许多矩阵特征值问题的近似求解方法,由Duncley法给出固有频率基频的下界,Rayleigh-Ritz近似法建立的方程,给出基频的上界,以及通常的矩阵迭代法给出的矩阵的固有频率程序中是以某一元素迭代前后比值确定的,这样实际上很难说是上界或下界。Collatz包含定理仅适用于对称标准特征值问题,可以给出特征值上、下界。采用矩阵Cholesky三角分解的原理,把Collatz包含定理推广到适用于具有对称矩阵的一般结构系统的广义征值问题,对于分解刚度矩阵或质量矩阵可给出基频,或最高因有频率。为了验证理论的正确性,给出了算例。     

多自由度参激系统稳定性的数值分析

提出多自由度周期参激系统稳定性的数值直接法。通过将扰动方程表示成状态方程形式,再根据Flo-quet理论将扰动解表示成指数特征分量与周期分量之积,并将其周期分量与系统周期系数展成Fourier级数,导出一系列代数方程,建立矩阵特征值问题,从而由数值求解特征值可直接确定参激系统的稳定性。该方法可用于一般周期参激阻尼系统,特征值矩阵不含逆子阵。应用于斜拉索在支座周期运动激励下的参激振动不稳定性分析,数值结果表明该方法的有效性。     

埃尔米特梁单元的分块对角与高阶质量矩阵

埃尔米特梁单元常用的集中质量矩阵,是由挠度自由度对应的一致质量矩阵元素通过行求和或节点积分构造。然而,数值结果表明该集中质量矩阵在求解包含自由端的梁振动问题时,会出现频率精度掉阶现象。本文首先从保障质量矩阵最优收敛性的数值积分精度出发,分别针对三次和五次梁单元,发展了质量矩阵的梯度增强节点积分方案。利用梯度增强节点积分方案,可以得到具有分块对角形式的单元质量矩阵,而其组装的整体质量矩阵除边界节点外仍然呈现对角形式。对于两种单元,其分块对角质量矩阵分别具有4阶最优精度和6阶次优精度。再者,将标准一致质量矩阵和具有同阶精度的梯度增强节点积分质量矩阵进行优化组合,建立了具有超收敛特性的高阶质量矩阵。最后,通过数值算例系统验证了三次和五次单元的分块对角与高阶质量矩阵的频率计算精度。     

FAST ALGORITHMS FOR DETERMINING POSITIVE DEFINITENESS OF LARGE SCALE MATRICES IN SOLID STRUCTURE ANALYSIS

对于结构稳定性分析中超大规模矩阵正定性判定,必须采用并行计算方法,传统方法如计算特征值、主子式行列式及LDLT等直接方法难以实现.本文提出了一些可适用于并行的迭代判定算法.借鉴力学系统中能量下降的思想,发展了一种判定矩阵正定性的新思路,即将矩阵的正定性判定问题转化为一个优化问题,并基于优化算法来判定矩阵的正定性.提出了基于最速下降法和共轭梯度法来进行矩阵正定性判定的算法.然后考虑到力学系统刚度矩阵的稀疏性和结构刚失稳状态的弱非正定性,提出可以先截超平面后解方程求驻值点的方法来判定弱非正定矩阵的正定性.为了保证对强非正定矩阵判定的准确性,本文提出可以高效混杂使用截平面法和共轭梯度法.数值实验结果表明,本文提出的算法具有准确性和高效性.对于强非正定矩阵而言,共轭梯度算法更加高效;而对于弱非正定矩阵,则是截平面法和混杂算法更加高效.这些算法都容易在机群上实现并行计算,能够快速判定大规模矩阵的正定性.     

Scattering of obliquely incident waves by straight features in a plate

The scattering of obliquely incident waves by straight features in a plate is solved analytically. The reflection matrix of a free straight edge and the scattering matrix of a straight thickness step are obtained respectively by modal decomposition based on a real orthogonal relation. The formulas are illustrated through numerical examples. The matrices are found to be Hermitian for the propagating modes; thus, the mode conversions are reciprocal in terms of energy. The matrices can be used to determine the scattering from more complicated straight features if the cross sections are approximated as sequences of stairs and steps.     

Classification and dynamics of stably dissipative Lotka-Volterra systems

This paper deals with classification and dynamical behaviors of stably dissipative Lotka-Volterra (LV) systems. The sufficient and necessary conditions for a matrix being stably dissipative are discussed firstly. Then, a graph-theoretic classification method for stably dissipative matrices is proposed, based on which, for all five-order stably dissipative matrices, the associated graphs are classified completely as 27 topologically different graphs and for each graph the dynamics of the corresponding LV system is discussed. Finally, the effects of removing some links from the above graphs on the dynamics of the corresponding LV systems are discussed.     

Kronecker product and a new matrix form of Lagrangian equations with multipliers for constrained multibody systems

The automatic derivation of motion equations is an important problem of multibody system dynamics. Firstly, an overview of the matrix calculus related to Kronecker product of two matrices is presented. A new matrix form of Lagrangian equations with multipliers for constrained multibody systems is then developed to demonstrate the usefulness of Kronecker product of two matrices in the study of dynamics of multibody systems. Finally, the equations of motion of mechanisms are derived using the proposed matrix form of Lagrangian equations as application examples.     

滑动轴承油膜力Jacobi矩阵的一种快速算法

基于变分不等方程理论,把滑动轴承油膜力及其Jacobi矩阵的求解转换为求解一组三对角矩阵代数方程,采用一种修正的追赶法同步快速求解。同时将系数矩阵分解为轴颈运动相关量和常矩阵乘积叠加的形式,常矩阵一次求得反复调用,大大减少冗余运算。算例表明,采用本文算法,结果精度得到保证,运算时间大幅减少,能很好地揭示滑动轴承-转子系统的非线性动力特性。     

Finding vibrations of inclined cable structures by approximately solving governing equations for exterior matrix

In this paper, how to compute the eigenfrequencies of the structures composed of a series of inclined cables is shown. The physics of an inclined cable can be complicated, so solving the differential equations even approximately is difficult. However, rather than solving the system of 4 first-order equations governing the dynamics of each cable, the governing equations are instead converted to a set of equations that the exterior matrix satisfies. Therefore, the exterior matrix method (EMM) is used without solving the original governing equations. Even though this produces a system of 6 first-order equations, the simple asymptotic techniques to find the first three terms of the perturbative solution can be used. The solutions can then be assembled to produce a 6 × 6 exterior matrix for a cable section. The matrices for each cable in the structure are multiplied together, along with the exterior matrices for each joint. The roots of the product give us the eigenfrequencies of the system.     

The Explicit Solution of the Matrix Equation AX-XB=C

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The free-interface method of component mode synthesis for systems with viscous damping

This paper presents a new free-interface method of component mode synthesis for linear systems with arbitrary viscous damping. The left and right projection matrices described by slate-variable vectors are first introduced for components with rigid-body freedom. The operator function of projection matrices for state displacement and state force is proved, and then the state residual fiexibility matrix and the state residual inertia-relief attachment mode are defined and employed. The results of three examples demonstrate that the method proposed in this paper leads to very accurate system eigenvalues and high mode-synthesis efficiency     

An inversion algorithm for general tridiagonal matrix

An algorithm for the inverse of a general tridiagonal matrix is presented. For a tridiagonal matrix having the Doolittle factorization, an inversion algorithm is established. The algorithm is then generalized to deal with a general tridiagonal matrix without any restriction. Comparison with other methods is provided, indicating low computational complexity of the proposed algorithm, and its applicability to general tridiagonal matrices.