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Shock and Vibration
Volume 2014, Article ID 485630, 9 pages
Research Article

On the Dynamic Analysis of a Beam Carrying Multiple Mass-Spring-Mass-Damper System

1Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, ON, Canada M5S 3G8
2Department of Mechanical and Industrial Engineering, Ryerson University, Toronto, ON, Canada M5B 2K3

Received 25 July 2013; Accepted 6 October 2013; Published 23 February 2014

Academic Editor: Jeong-Hoi Koo

Copyright © 2014 O. Barry et al. 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.


The exact natural frequencies, mode shapes, and the corresponding orthogonality relations are important in forced vibration analysis via modal expansion. In the present paper, a free vibration analysis is conducted to determine the exact natural frequencies and mode shapes of an axially loaded beam carrying several absorbers. An explicit expression is presented for the generalized orthogonality relations. These generalized orthogonality conditions are employed along with the assumed modes method to perform forced vibration analysis. The present approach is compared to other approximate methods in the literature with the classical orthogonality relations and different choice of mode shapes. The results indicate that the use of the generalized orthogonality relation with the exact mode shapes is required for a precise investigation of the dynamic response of a beam with mass-spring-mass-damper system.