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Shock and Vibration
Volume 2017 (2017), Article ID 3924921, 26 pages
Research Article

Forced Vibration of a Timoshenko Beam Subjected to Stationary and Moving Loads Using the Modal Analysis Method

Department of Mechanical Engineering, Inha University, 100 Inha-ro, Nam-gu, Incheon 402-751, Republic of Korea

Correspondence should be addressed to Usik Lee

Received 30 September 2016; Accepted 7 November 2016; Published 4 January 2017

Academic Editor: Tony Murmu

Copyright © 2017 Taehyun Kim 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 modal analysis method (MAM) is very useful for obtaining the dynamic responses of a structure in analytical closed forms. In order to use the MAM, accurate information is needed on the natural frequencies, mode shapes, and orthogonality of the mode shapes a priori. A thorough literature survey reveals that the necessary information reported in the existing literature is sometimes very limited or incomplete, even for simple beam models such as Timoshenko beams. Thus, we present complete information on the natural frequencies, three types of mode shapes, and the orthogonality of the mode shapes for simply supported Timoshenko beams. Based on this information, we use the MAM to derive the forced vibration responses of a simply supported Timoshenko beam subjected to arbitrary initial conditions and to stationary or moving loads (a point transverse force and a point bending moment) in analytical closed form. We then conduct numerical studies to investigate the effects of each type of mode shape on the long-term dynamic responses (vibrations), the short-term dynamic responses (waves), and the deformed shapes of an example Timoshenko beam subjected to stationary or moving point loads.