Lower Bounds to Eigenvalues for Vibrations and Buckling of Structures by Analytical Applications of a-priori Known Qualitative Information

Two methods to obtain lower bounds to eigenvalues are presented for cases which have equivalent minimum variational formulations. One method is an extension and elaboration of a theorem presented by the author in 1972, which affected the transfer of a weight function from one location to another over the physical system considered. The extension relies on information known a-priori about the exact solution of the problem, although the exact solution is not obtained. The other method is akin to the Rayleigh–Ritz method but yields lower bounds. The two methods are applied to various physical examples of vibrations and of buckling with rather good results. The application to other examples is direct and may be performed in a way quite similar to those examples shown.