Nonlinear dynamics of a cantilevered beam with a tip mass and elastic-damping support

The nonlinear dynamics of a tip excited cantilevered beam with an end-mass attached to elastic-damping support is considered in this paper. This prototypical system is examined using a minimal two degree of freedom (DOF) model and its bifurcation characteristics, computed by AUTO software, is compared with the analytical solutions and that of the full system using mull-body dynamic simulation. The spring-damper support at tip introduces the length dependent geometric nonlinearities in the structure. Consequently, the two-to-one internal resonance between the first lateral mode and the first vertical mode is imposed. When the excitation frequency is in the vicinity of the first vertical natural frequency (or twice the first lateral natural frequency), lateral response loses its stability after a certain force threshold. It is found (a) the minimal model is adequate to understand the first bifurcation (pitchfork) and the subsequent Hopf bifurcation and (b) the threshold force can be tuned by changing the initial length of the support elements.