THEORY AND COMPUTATION OF SECONDARY BIFURCATIONS NEAR A DOUBLE EIGENVALUE

This paper deals with secondary bifurcations near a double eigenvalue of a nonlinear equation with two parameters. Utilizing symmetries (or more generally, equivariances ) and introducing two new parameters, we give some extended systems so that the double singular points, secondary bifurcation points and initial secondary branches respectively become their regular solutions. The methods in this paper not only give more general conditions of secondary bifurcation but also avoid the adjacent singularities of existing extended systems for computing the simple bifurcation points on non-trivial solution branches A numerical example is presented, showing the effectiveness of our methods.     

反中心对称矩阵的广义特征值反问题

Given matrix X and diagonal matrix A , the anti-centrosymmetric solutions (A, B) and its optimal approximation of inverse generalized eigenvalue problem AX = BXA have been considered. The general form of such solutions is given and the expression of the optimal approximation solution to a given matrix is derived. The algorithm and one numerical example for solving optimal approximation solution are included.     

线性流形上双反对称矩阵的最佳逼近

本文讨论了线性流形上用双反对称矩阵构造给定矩阵的最佳逼近问题,给出问题解的表达式,最后给出求最佳逼近解的数值方法与数值算例.     

广义双随机矩阵反问题及其应用

研究广义双随机矩阵反问题.给出广义双随机矩阵的最小二乘解,得到了解的具体表达形式.并讨论了用广义双随机矩阵构造给定矩阵的最佳逼近问题,给出该问题有解的充分必要条件和解的表达形式.包括算法及数值例子.     

BIFURCATION OF SINGULAR POINTS NEAR A DOUBLE FOLD POINT INZ_2 -SYMMETRIC NONLINEAR EQUATIONS

We consider the bifurcation of singular points near a double fold point in Z2 -symmetric nonlinear equations with two parameters,where the linearization has a two dimensional null space spanned by a symmetric null vector and an ami-symmetric null vector. In particular, we show the existence of a turning point path and a pitchfork point path passing ihrough the double fold point and they are the only singular points nearby. Their nondegeneracy is confirmed. A supporting numerical example is also provided. The main tools for our analysis as well as the compulation are some extended systems.     

广义双对称矩阵反问题

利用矩阵的奇异值分解讨论了一类广义双对称矩阵反问题,得到了此类矩阵反问题有解的充要条件及通解的表达式.     

弹性基础上的杆在轴压下的二次屈曲

本文研究弹性基础上受轴向加载的两端铰支杆当其最低两屈曲荷载很近时的后屈曲行为。首先,使用Liapunov-Schmidt约化并借助稳定性分析,揭示了杆的二次屈曲现象;基于分叉方程给出了原始后屈曲分支及二次分支的渐近展开。其次,我们使用作者建立的二次分叉的计算方法对杆的二次屈曲做了数值计算,数值结果与渐近展开符合得很好     

Euler杆大挠度屈曲的解析逼近

研究Euler杆大挠度屈曲问题，将控制方程的线性化与谐波平衡法组合起来，分别建立以杆端转角形式表示的屈曲荷载及最大挠度的解析逼近公式，这些公式既适用于小变形又适用于大变形。     

基于改进灰狼优化算法的结构损伤识别

结合模态柔度矩阵、广义模态柔度矩阵和振型三个识别精度较好的指标,构造新的目标函数求解损伤识别问题。通过Nelson方法求解得到的频率与振型的导数,得到对结构刚度发生变化时更具敏感性的位置,然后在这些位置布置传感器以提取结构信息。针对原有的灰狼算法虽然全局搜索能力强,但是存在局部搜索精度差的问题,本文从初始种群和收敛因子等方面着手,改善灰狼算法的局部搜索能力及收敛速度。最后利用提出的方法,通过识别梁模型及桁架模型中的损伤单元说明本文方法的有效性。     

细长弹性杆在轴压下的二次分叉

基于[1]的弹性曲杆的平衡方程,本文研究了矩形横截面细长杆在轴压下的后屈曲行为。设横截面的边长比为 1:2δ,使用 Poincare-Keller 的打靶法并引进坐标的伸缩变换,研究了δ在 δ_0=1 附近的情形。当δ≠1 时,发现了杆平衡态的二次分叉。我们也给出了原始后屈曲解支及二次分支的渐近表示并分析了各个解支的稳定性。