Convergence of the Ritz Method

[+] Author and Article Information
Arthur W. Leissa, Samir M. Shihada

Applied Mechanics Program, Ohio State University, Columbus OH 43210 USA

Appl. Mech. Rev 48(11S), S90-S95 (Nov 01, 1995) doi:10.1115/1.3005088 History: Online April 29, 2009


The Ritz method is widely used for the solution of problems in structural mechanics, especially eigenvalue problems where the free vibration frequencies or buckling loads are sought. It is well-known that the method yields upper bounds for these eigenvalues, and that convergence to exact eigenvalues will occur if proper admissible functions are used to represent the displacements (eigenfunctions). However, little is known about the convergence of the derivatives of the eigenfunctions. 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 derivatives). Another study examines the effects of deliberately omitting one term from 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 displacement, one eigenvalue (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.

Copyright © 1995 by The American Society of Mechanical Engineers
Your Session has timed out. Please sign back in to continue.





Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In