Eigenvector error bounds and their applications to structural modification

Abstract

We have shown previously that some natural frequencies and mode shapes of a modified vibrating structure may be approximated from the incremental mass and stiffness matrices, and from an incomplete set of measured natural frequencies and mode shapes for the unmodified system. In addition, error bounds were obtained for the natural frequencies of the modified structure. We bound here the error in the approximated mode shapes. portion. Nevertheless, we have developed in Ref. 5 a method for approximating the modal parameters of a structure based on such data and have shown that the approximation is opti- mal in a Rayleigh-Ritz sense. The associated inverse problem, in which the incremental matrices are derived to meet prescribed modal data, is posed and also partially solved in Ref. 6. Naturally, it is of impor- tance to investigate the question How good are these approx- imations? Clearly, several error mechanisms contribute to the discrepancy between the modal parameters of the real structure and their analytical approximation. First, the real structure is a nonlinear continuous system, whereas the model is linear and discrete. Second, there are uncertainties regarding the physical properties and boundary conditions of the vibrat- ing structure. Third, the approximation is based on incom- plete modal data for the unmodified structure, e.g., only n natural frequencies and mode shapes are assumed to be known. We have named this last effect, the modal truncation error. The analytical approximation is based mainly on data from experiments, and we assume that the mechanisms of error resulting from nonlinearity, discretization, and uncer- tainties in the physical parameters are negligible with respect to the dominant error caused by the modal truncation. Under such an assumption the possible errors of the natural frequen- cies of the modified structure have been bounded, using vari- ous methods, in a series of papers. 710 The estimation of natural frequences and their range of variation is a main objective in the design and analysis of vibratory systems. For some applications it is equally important to be able to predict mode shapes and error bounds for the modified structure. This information is essential for the analysis of the dynamic stresses and strains. Bounding the nodal points and the ex- treme values of the fundamental mode shapes is also of great