标准特征值问题扰动分析的精确方法  

An exact solution method for perturbation analysis of standard eigenvalue problems

作　　者：邱志平[1] 姜南 ZhiPing JIANG Nan(QIU Institute of Solid Mechanics, School of Aeronautic Science and Engineering, Beihang University, Beijing 100191, China)

机构地区：[1]北京航空航天大学航空科学与工程学院固体力学研究所,北京100191

出　　处：《科学通报》2017年第28期3400-3408,共9页Chinese Science Bulletin

基　　金：国家自然科学基金(11372025;11432002);国防基础科研计划基金(JCKY2013601B001;JCKY2016204B101;JCKY2016601B001);科技部民用飞机专项(MJ-F-2012-04);国家重点研发计划(2016YFB0200704);高等学校学科创新引智计划(B07009)资助

摘　　要：在扰动量存在的情况下,准确计算特征值的扰动量是确保结构安全性的重要问题.针对标准特征值问题扰动分析提出了一种精确方法,能够高效地计算特征值扰动量的准确值,克服了矩阵摄动级数展开法忽略高阶项导致的计算精度不足的缺点.提出的方法推导得到了标准特征值问题扰动分析求解方程.求解方程推导过程中没有经过近似处理,将求解标称系统标准特征值问题方程得到的特征值标称值代入,就能求得特征值扰动量的准确值,从而能够有效满足高精度和高效率要求.3个数值算例分别对所提出的精确方法进行了验证,与矩阵摄动级数展开法的计算结果相比,能够准确高效地计算特征值的扰动量,具有精确和高效的双重优势.In the presence of perturbation, the accurate method of calculating the perturbed values of eigenvalues is an important issue to ensure the safety in engineering structures. Based on a power series expansion for the eigenvalues and eigenvectors, all the present perturbation methods neglect the higher-order terms in the calculating process, which may result in the deterioration of accuracy. However, the present methods for improving the accuracy usually result in amazing amounts of computation. The contradiction between accuracy and efficiency has existed for a long time, which made the research stagnant in recent years. In this paper, an exact solution method for perturbation analysis of standard eigenvalue problem is proposed to calculate the exact perturbed values of eigenvalues, overcoming the drawbacks of lack of accuracy due to neglecting the higher-order terms in matrix perturbation series expansion method. In the presented method, the perturbation solution equation of the standard eigenvalue problem is derived without any approximations in the derivation process. By substituting the normal eigenvalues obtained by solving the standard eigenvalue problem equation of the normal matrix into the derived perturbation solution equation, the perturbed values of eigenvalues can be calculated efficiently, which can satisfy the requirement of high accuracy and high efficiency for standard eigenvalue analysis with perturbed parameters. In three numerical examples with perturbed parameters of the undamped spring-mass system, the Bernoulli-Euler cantilever beam and the plane frame, the accuracy and efficiency of the proposed method are verified by comparing numerical results with those obtained by matrix perturbation series expansion method. The results show： （1） The relative errors of the perturbed values of eigenvalues yielded by the proposed method and the lower-order matrix perturbation series expansion method are much bigger and can be reduced reduce remarkably with the increase of the order. The results o

关 键 词：标准特征值问题 摄动理论 精确方法 扰动分析 实特征值 摄动级数展开法 

分 类 号：O241.6[理学—计算数学]