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Modelling and Simulation in Engineering
Volume 2013, Article ID 919517, 18 pages
Research Article

Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity

1Department of Mechanical Engineering, International Institute of Information Technology, Bhubaneswar 751003, India
2Department of Mechanical Engineering, College of Engineering and Technology, Bhubaneswar 751003, India
3Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India

Received 10 May 2013; Accepted 27 August 2013

Academic Editor: Abdelali El Aroudi

Copyright © 2013 Bamadev Sahoo 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 nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. The beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. The stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. The natural frequency of second mode is approximately three times that of first mode; a three-to-one internal resonance is possible. The method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. The nonlinear steady state response along with the stability and bifurcation of the beam is investigated. The system exhibits pitchfork, Hopf, and saddle node bifurcations under different control parameters. The dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the influence of internal resonance.