Hindawi Publishing Corporation Modelling and Simulation in Engineering Volume 2013, Article ID 919517, 18 pages http://dx.doi.org/10.1155/2013/919517 Research Article Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity Bamadev Sahoo, 1 L. N. Panda, 2 and G. Pohit 3 1 Department of Mechanical Engineering, International Institute of Information Technology, Bhubaneswar 751003, India 2 Department of Mechanical Engineering, College of Engineering and Technology, Bhubaneswar 751003, India 3 Department of Mechanical Engineering, Jadavpur University, Kolkata 700032, India Correspondence should be addressed to G. Pohit; gpohit@gmail.com Received 10 May 2013; Accepted 27 August 2013 Academic Editor: Abdelali El Aroudi Copyright © 2013 Bamadev Sahoo et al. his 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. he nonlinear vibration of a travelling beam subjected to principal parametric resonance in presence of internal resonance is investigated. he beam velocity is assumed to be comprised of a constant mean value along with a harmonically varying component. he stretching of neutral axis introduces geometric cubic nonlinearity in the equation of motion of the beam. he natural frequency of second mode is approximately three times that of irst mode; a three-to-one internal resonance is possible. he method of multiple scales (MMS) is directly applied to the governing nonlinear equations and the associated boundary conditions. he nonlinear steady state response along with the stability and bifurcation of the beam is investigated. he system exhibits pitchfork, Hopf, and saddle node bifurcations under diferent control parameters. he dynamic solutions in the periodic, quasiperiodic, and chaotic forms are captured with the help of time history, phase portraits, and Poincare maps showing the inluence of internal resonance. 1. Introduction Band saws, ibre textiles, magnetic tapes, paper sheets, aerial tramways, pipes transporting luids, thread lines, and belts are some technological examples classiied as axially moving continua. Analytical models for axially moving systems have been extensively used in the last few decades. he vast literature on axially moving continua vibration has been reviewed by Wickert and Mote Jr. [1] up to 1988. While a linear analysis provides natural frequencies, mode shapes, and critical speeds, its validity regarding the response of the system diminishes as the vibration amplitude becomes suiciently large or as the critical speed is approached [2]. In these cases one must resort to a nonlinear analysis. Wickert and Mote Jr. [3, 4] studied the transverse vibration of axially moving strings and beams using an eigenfunction method. hey also studied the dynamic response of an axially moving string loaded suspended mass. Wickert [5] presented a detailed study of the nonlinear vibrations and bifurcations of moving beams using the Krylov-Bogoliubov-Mitropolsky asymptotic method. Chakraborty et al. [6, 7] investigated both free and forced vibration of the nonlinear traveling beam using complex normal modes. here are papers devoted to the analysis of the dynamic behavior of traveling systems with time-dependent axial velocity or with time-dependent axial tension force. ¨ Oz and Pakdemirli [8] investigated principal parametric resonances and combination resonances of sum and diference types for any two modes for an axially accelerating beam using the method of multiple scales. hey found that for combination resonances, instabilities occurred only for additive type but not for diference type. ¨ Oz et al. [9] extended the work to nonlinear transverse vibration and stability analysis. Com- prehensive review of nonlinear modal interactions is there in [10–12]. Using method of multiple scales Riedel and Tan [13] studied the coupled and forced behavior of an axially moving strip with internal resonance. ¨ Ozkaya et al. [14] investigated nonlinear transverse vibrations and 3 : 1 internal resonances of a beam with multiple supports and plotted frequency response curves for diferent support numbers. Bagdatli et al. [15] extended this work to ind existence of internal resonance cases between diferent modes. Chin and