Convergence of the Ritz Method

The Ritz method is widely used for the solution of problems in structural mechanics, especially eigenvalue problems where the free vibration frequencies orbuckling loads are sought. It is well.known that he method yields upper bounds for these igenvalues, and that convergence to exact eigenvalues will occur ff proper admissible fimctions are used to represent the displacements (eigenfunctions). However, tittle is known about he convergence of the derivatives of the eigenfimctions. In this paper the method is studied for the problem of free vibrations of a cantilever beam. Convergences of the eigenfunctions and their second and third derivatives (i.e., bending moments and shear forces) are examined, as well as convergence to satisfy the differential equation of motion (which involves fourth ~fivativcs). Another study examines the effects of deliberately omitting one term fi'om the set of admissible displacement functions which would otherwise be complete. It is found that in such cases, when orthogonal polynomials are used to represent the displac, en'vmt, one cigenvaluc (such as the lowest frequency) may be completely missed, and that the others will converge incorrectly. When ordinary polynomials are used, correct convergence is obtained even 'with a missing term