A THEOREM OF UPPER AND LOWER BOUNDS ON EIGENVXLUES FOR STRUCTURES WITH BOUNDED TRIMETERS
有界参数结构特征值的上下界定理

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".