有界参数结构特征值的上下界定理

与方法近似性的结构特征值包含定理不同,给出参数近似性的结构的特征值上下界定理．在结构刚度矩阵和质量矩阵可以利用结构参数进行非员分解的条件下,通过区间分析,将特征值的上下界分解成两个广义特征值问题进行求解．结果可以看成是胡海昌教授的特征值质量包含定理和刚度包含定理在结构参数近似性特征值问题中的一种推广和应用．

A THEOREM OF UPPER AND LOWER BOUNDS ON EIGENVXLUES FOR STRUCTURES WITH BOUNDED TRIMETERS

In structural dynamics, generally we can not determinate accurately the natural frequencies and eigenvalues due to various approximations. But we may obtain an eigenvalue interval, where the lower bound eigenvalue can be determinated by, say, Dunkerly's or Temple's method; the upper bound eigenvalue can be computed by multi-term Rayleigh-Ritz or Galerkin method. Alternatively, the upper and lower bounds and can be obtained simultaneously by means of Hu Haichang's "Mass Inclusion Theorem and Rigidity Inclusion Theorem for Eigenvalues" or Chen Shaoting's "Region Theorem for Complex Eigenvalues". But these approaches do not consider structural uncertainties. Uncertainties of structural parameters are usually analyzed via stochastic modeling through the use of the concept of probability density and total probability formula. In addition, the probabilistic information of uncertain parameters is assumed known in advance. However, in most circumstances this is not true. In the recent studies by Qiu and Elishakoff, an interval analysis model was developed to model the uncertainties, in which only the bounds on the magnitude of uncertain parameters are required whilst the probabilistic distribution densities are no longer needed. The methodology assumes that the structural characteristics fall into a multi-dimensional rectangle. This is different from the approach in conventional probabilistic studies where the most possible response region is sought. Unlike the Inclusion theorem for eigenvalues that was used to handle approximation due to computational methods, this paper presents a theorem of upper and lower bounds on eigenvalues due to approximate structural parameters. On condition of non-negative decomposition of stiffness and mass matrices by making use of structural parameters, interval analysis can transform the upper and lower bounds on eigenvalues into two generalized eigenvalue problems. Then a solution of them is followed. The theorem proposed can be considered as an extension of Hu Haichang's "Mass Inclusion Theorem and Rigidity Inclusion Theorem for Eigenvalues".