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Shock and Vibration
Volume 18 (2011), Issue 5, Pages 697-707

Clustered Natural Frequencies in Multi-Span Beams with Constrained Characteristic Functions

Khodabakhsh Saeedi and Rama B. Bhat

Mechanical and Industrial Engineering Department, Concordia University, Montreal, QC, H3G1M8, Canada

Received 24 December 2008; Revised 13 October 2009

Copyright © 2011 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


A study of the natural frequencies and mode shapes of a multi-span beam is carried out by introducing constrained beam characteristic functions. The conventional method used for the dynamic analysis of such a beam is to consider span-wise characteristic function solution and then to solve it by using compatibility conditions such as the continuity in the a slope and bending moment at the intermediate supports and boundary conditions at the ends. In the method proposed here, the matrix size is reduced and, if the support conditions are symmetric about the midpoint, the symmetry and anti-symmetry conditions at the midpoint can be conveniently exploited for computational economy. The natural frequencies occur in clusters, each one containing the number of natural frequencies equal to the number of spans. The results are presented and discussed.