A METHOD OF DETERMINING BUCKLED STATES OF THIN PLATES AT A DOUBLE EIGENVALUE

A method of determining bifurcation directions at a double eigenvalue is presented by combining the finite element method with the perturbation method. By using the present method, the buckled states of rectangular plates at a double eigenvalue are numerically analyzed. The results show that this method is effective.     

Analysis of Stability on Elastic Plates with Initial Imperfections

ＡＮＡＬＹＳＩＳＯＦＳＴＡＢＩＬＩＴＹＯＮＥＬＡＳＴＩＣＰＬＡＴＥＳＷＩＴＨＩＮＩＴＩＡＬＩＭＰＥＲＦＥＣＴＩＯＮＳＸｕＫａｉｙｕ（徐凯宇）（ＲｅｃｅｉｖｅｄＯｃｔ．５．１９９４；ＣｏｍｍｕｎｉｃａｔｅｄＰａｉＬｉｚｈｏｕ）ＡＮＡＬＹＳＩＳＯＦＳＴＡ...     

0rr－Sommerfeld方程数值解法中的复广义矩阵特征值问题

Ｏｒｒ－Ｓｏｍｍｅｒｆｅｌｄ方程的求解通常可以化为一个复广义矩阵特征值问题ＡＸ＝ωＢＸ。本文用酉变换分别约化Ａ，Ｂ为上Ｈｅｓｓｅｎｂｅｒｇ阵和上三角阵，然后利用Ｍｕｌｌｅｒ求根方法可以求出其全部特征值，其中特征多项式的值由Ｈｙｍａｎ方法给出。当仅需要判断有无不稳定模态时，利用一个简单的矩阵变换将其化为强特征值的求解问题，从而可使用最简单的幂迭代，Ｃｈｅｂｙｓｈｅｖ配置点法算例表明两种算法均快速有效。     

Time-harmonic BEM for 2-D piezoelectricity applied to eigenvalue problems

We will derive the fundamental generalized displacement solution, using the Radon transform, and present the direct formulation of the time-harmonic boundary element method (BEM) for the two-dimensional general piezoelectric solids. The fundamental solution consists of the static singular and the dynamics regular parts; the former, evaluated analytically, is the fundamental solution for the static problem and the latter is given by a line integral along the unit circle. The static BEM is a component of the time-harmonic BEM, which is formulated following the physical interpretation of Somigliana’s identity in terms of the fundamental generalized line force and dislocation solutions obtained through the Stroh–Lekhnitskii (SL) formalism. The time-harmonic BEM is obtained by adding the boundary integrals for the dynamic regular part which, from the original double integral representation over the boundary element and the unit circle, are reduced to simple line integrals along the unit circle.The BEM will be applied to the determination of the eigen frequencies of piezoelectric resonators. The eigenvalue problem deals with full non-symmetric complex-valued matrices whose components depend non-linearly on the frequency. A comparative study will be made of non-linear eigenvalue solvers: QZ algorithm and the implicitly restarted Arnoldi method (IRAM). The FEM results whose accuracy is well established serve as the basis of the comparison. It is found that the IRAM is faster and has more control over the solution procedure than the QZ algorithm. The use of the time-harmonic fundamental solution provides a clean boundary only formulation of the BEM and, when applied to the eigenvalue problems with IRAM, provides eigen frequencies accurate enough to be used for industrial applications. It supersedes the dual reciprocity BEM and challenges to replace the FEM designed for the eigenvalue problems for piezoelectricity.     

The calculation of eigenvalues for the stationary perturbation of couette-poiseuille flow

The problem considered is that of two-dimensional viscous flow in a straight channel. The decay of a stationary perturbation from the Couette-Poiseuille flow in the downstream is sought. A differential eigenvalue equation resembling the Orr-Sommerfeld equation is solved by using a spectral method and an initial-value method (the compound matrix method) for values of the Reynolds number R between 0 and 2000. The eigenvalues are presented for several of interesting cases with different measures of mass flux. These eigenvalues determine the rate of decay for the purturbation.     

On the fundamental sloshing frequency in the ice-fishing problemSur la fréquence de ballottement principale dans le problème de la glace avec trous pour pêcher

The sloshing problem is considered in a half-space covered by a rigid dock with apertures. The dependence of the fundamental sloshing frequency on the shape of the free surface region is studied. It is proved that the inequality holds between the fundamental eigenvalues corresponding to two different regions if some conditions are fulfilled. These conditions are verified for particular classes of regions of a fixed area in order to demonstrate that the disk yields the maximum of the fundamental eigenvalue for each of these classes. On the other hand, examples of regions are constructed for which the fundamental eigenfrequency is larger than that for the circular aperture of the same area and even as large as one wishes. To cite this article: V. Kozlov, N. Kuznetsov, C. R. Mecanique 330 (2002) 723–728.     

Well-posedness and stability of the repairable system with N failure modes and one standby unit

The well-posedness and stability of the repairable system with N failure modes and one standby unit were discussed by applying the c₀ semigroups theory of function analysis. Firstly, the integro-differential equations described the system were transformed into some abstract Cauchy problem of Banach space. Secondly, the system operator generates positive contractive c₀ semigroups T(t) and so the well-posedness of the system was obtained. Finally, the spectral distribution of the system operator was analyzed. It was proven that 0 is strictly dominant eigenvalue of the system operator and the dynamic solution of the system converges to the steady-state solution. The steady-state solution was shown to be the eigenvector of the system operator corresponding to the eigenvalue 0. At the same time the dynamic solution exponentially converges to the steady-state solution.     

The distance-regular graphs such that all of its second largest local eigenvalues are at most one

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the first author, who classified the distance-regular graphs with smallest eigenvalue .     

The inverse eigenvalue problem of generalized reflexive matrices and its approximation

This paper studies inverse eigenvalue problems of generalized reflexive matrices and their optimal approximations. Necessary and sufficient conditions for the solvability of the problems are derived, the solutions and their optimal approximations are provided.