%0 Journal Article
%T Parametric and Internal Resonances of an Axially Moving Beam with Time-Dependent Velocity
%A Bamadev Sahoo
%A L. N. Panda
%A G. Pohit
%J Modelling and Simulation in Engineering
%D 2013
%I Hindawi Publishing Corporation
%R 10.1155/2013/919517
%X 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. 1. Introduction Band saws, fibre textiles, magnetic tapes, paper sheets, aerial tramways, pipes transporting fluids, thread lines, and belts are some technological examples classified as axially moving continua. Analytical models for axially moving systems have been extensively used in the last few decades. The 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 sufficiently 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. They 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. There 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. £¿z and Pakdemirli [8] investigated principal parametric resonances and combination resonances of sum and difference types for any two modes for an axially
%U http://www.hindawi.com/journals/mse/2013/919517/