Nonlinear Dynamics of an Axially Translating Viscoelastic String With Parametric And Exact Internal Resonance

Abstract

The subject of axially translating string has widespread engineering and industrial newlineapplications which can be studied both theoretically and experimentally. It belongs to newlineclass of travelling continua involving inherent coriolis force which renders the system newlinegyroscopic and induces dynamic instabilities depending on the characteristics of the newlinesystem and travelling velocity including its fluctuations.The time dependency present in newlineaxial velocity causes parametric instabilities (resonances) characterized by vibration in a newlinedirection orthogonal to the direction of unsteady velocity.The study of these parametric newlineinstabilities has practical concern as well as of immense fundamental interest. The newlineaxially travelling string exhibits wide range of linear and nonlinear dynamic behaviour newlinedepending on the system parameters. Nonlinear modal interaction in the form of internal newlineresonance also influences the system behaviour. newlineIn the present dissertation, the nonlinear characterstics of a pulley supported axially newlinetranslating string subjected to parametric resonance with exact internal resonance is newlinestudied through analytical-numerical method. The translating string with initial axial newlinetension is on pulley supports with variable stiffness. The cubic geometric nonlinearity in newlinethe system is due to elongation of the string. The system has parametric excitation newlineassociated with the travelling velocity having harmonic fluctuation about a steady mean newlinevalue leading to parametric resonance. The third mode frequency is three times the newlinefundamental frequency leading to 3:1 exact internal or autoparametric resonance newlinebetween the third and first mode for this cubic nonlinear system. The multiple-scale newlineanalysis approach is employed directly on the equation of motion to obtain approximate newlinesolutions and analyse the system dynamics. The principal parametric resonance of first newlinemode, combination parametric resonance of first and third mode, principal parametric newlineresonance of third mode, combination parametric resonance of first and second mode are newlinediscusse