LOW-DIMENSIONAL CHAOTIC RESPONSE OF AXIALLY ACCELERATING CONTINUUM INTHE SUPERCRITICAL REGIME

Nonlinear dynamics of one-mode approximation of an axially moving cont inuum such as a moving magnetic tape is studied. The system is modeled as a beam moving with varying speed, and the transverse vibration of the beam is considered. The cubic stiffness term, arising out of finit e stretching of the neutral axis during vibration, is included in the analysis while deriving the equations of motion by Hamilton's principl e. One-mode approximation of the governing equation is obtained by the Galerkin's method, as the objective in this work is to examine the lo w-dimensional chaotic response. The velocity of the beam is assumed to have sinusoidal fluctuations superposed on a mean value. This approxi mation leads to a parametrically excited Duffing's oscillator. It exhi bits a symmetric pitchfork bifurcation as the axial velocity of the be am is varied beyond a critical value. In the supercritical regime, the system is described by a parametrically excited double-well potential oscillator. It is shown by numerical simulation that the oscillator h as both period-doubling and intermittent routes to chaos. Melnikov's c riterion is employed to find out the parameter regime in which chaos o ccurs. Further, it is shown that in the linear case, when the operatin g speed is supercritical, the oscillator considered is isomorphic to t he case of an inverted pendulum with an oscillating support. It is als o shown that supercritical motion can be stabilised by imposing a suit able velocity variation.