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Mathematical Problems in Engineering
Volume 2015, Article ID 391452, 9 pages
Research Article

Transverse Vibration of Axially Moving Functionally Graded Materials Based on Timoshenko Beam Theory

School of Urban Rail Transportation, Soochow University, Suzhou 215006, China

Received 16 August 2014; Accepted 18 October 2014

Academic Editor: Kim M. Liew

Copyright © 2015 Suihan Sui 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 transverse free vibration of an axially moving beam made of functionally graded materials (FGM) is investigated using a Timoshenko beam theory. Natural frequencies, vibration modes, and critical speeds of such axially moving systems are determined and discussed in detail. The material properties are assumed to vary continuously through the thickness of the beam according to a power law distribution. Hamilton’s principle is employed to derive the governing equation and a complex mode approach is utilized to obtain the transverse dynamical behaviors including the vibration modes and natural frequencies. Effects of the axially moving speed and the power-law exponent on the dynamic responses are examined. Some numerical examples are presented to reveal the differences of natural frequencies for Timoshenko beam model and Euler beam model. Moreover, the critical speed is determined numerically to indicate its variation with respect to the power-law exponent, axial initial stress, and length to thickness ratio.