Energy Methods in Self-Adjoint Eigenvalue Problems. II. Ritz-Galerkin and Related Methods

摘要

For linear self-adjoint systems with discrete eigenvalue spectra, the Galerkin, Rayleigh-Ritz and modified Rayleigh-Ritz methods are shown to yield upper bounds of the eigenvalues, and to converge, in all modes. Methods of obtaining lower bounds of the eigenvalues in all modes by means only of the above energy methods are established. The theory is illustrated by numerical examples, especially on vibrations of non-uniform beams. A simple general theorem and approximation is given for the effect of additional terms in the governing differential equations. These are then applied to vibrations of a beam on a nonuniform elastic foundation. (Author)