Stability and Bifurcation Analysis of an Axially Accelerating Beam

The present work deals with nonlinear transverse vibration of a beam moving with a harmonically fluctuating velocity and subjected to parametric excitation at a frequency close to twice the natural frequency in presence of internal resonance. The geometric cubic nonlinearity in the equation of motion is due to stretching effect of the beam. The analysis is carried out using the method of multiple scales (MMS) by directly attacking the governing nonlinear integral-partial-differential equations and the associated boundary conditions. The resulting set of first- order ordinary differential equations governing the modulation of amplitude and phase of the first two modes are analyzed numerically to obtain steady state and dynamic bevaviour along with stability as well as bifurcation of the travelling system. The system exhibits trivial, single mode and two mode solutions with pitchfork, saddle-node and Hopf bifurcations. The sensitivity of the system towards the variation in frequency and amplitude of fluctuating velocity component, variation in damping, flexural stiffness and initial points are studied.