Parametric Resonance of Hopf Bifurcation in a Generalized Beck’s Column

Abstract

A generalized damped Beck’s column under pulsating actions is considered. The nonlinear partial integrodifferential equations of motion and the associated boundary conditions, expanded up to cubic terms, are tackled through a perturbation approach. The multiple scales method is applied to the continuous model in order to obtain the bifurcation equations in the neighborhood of a Hopf bifurcation point in primary parametric resonance. This codimension-2 bifurcation entails two control variables, namely, the amplitude of the static and dynamic components of the follower force, playing the role of detuning and bifurcation parameters, respectively. In the postcritical analysis bifurcation diagrams and relevant phase portraits are examined. Two bifurcation paths associated with specific values of the follower force static component are discussed and the birth of new stable period-2 subharmonic motion is observed.