Nonlinear Dynamics of an Euler-Bernoulli Beam with Parametric and Internal Resonances

The nonlinear transverse vibration of a simply-supported travelling Euler-Bernoulli beam subjected to principal parametric resonance in presence of internal resonance is investigated. The variable velocity of beam is assumed to be comprised of a harmonically varying component superimposed over a mean value. 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 the natural frequency of first mode for a range of mean velocity of the beam, resulting in a three-to-one internal resonance. The method of multiple scales (MMS) is directly applied to the governing nonlinear integral-partial-differential equations and the associated boundary conditions. This results in a set of first order ordinary differential equations governing the modulation of amplitude and phase of the first two modes, which are analyzed numerically for principal parametric resonance of first mode. The nonlinear steady state response along with the stability and bifurcation of the beam are investigated. The dynamical behaviour of the system is observed in terms of periodic, quasiperiodic and chaotic responses, showing the influence of internal resonance.