Low-dimensional chaotic response of axially accelerating continuum in the supercritical regime

Summary Nonlinear dynamics of one-mode approximation of an axially moving continuum 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 finite stretching of the neutral axis during vibration, is included in the analysis while deriving the equations of motion by Hamilton's principle. One-mode approximation of the governing equation is obtained by the Galerkin's method, as the objective in this work is to examine the low-dimensional chaotic response. The velocity of the beam is assumed to have sinusoidal fluctuations superposed on a mean value. This approximation leads to a parametrically excited Duffing's oscillator. It exhibits a symmetric pitchfork bifurcation as the axial velocity of the beam 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 has both period-doubling and intermittent routes to chaos. Melnikov's criterion is employed to find out the parameter regime in which chaos occurs. Further, it is shown that in the linear case, when the operating speed is supercritical, the oscillator considered is isomorphic to the case of an inverted pendulum with an oscillating support. It is also shown that supercritical motion can be stabilised by imposing a suitable velocity variation. Received 13 February 1997; accepted for publication 29 July 1997