NONLINEAR-ANALYSIS OF THE BUCKLING AND VIBRATION OF A ROTATING ELASTICUM

We investigate the static and dynamic behaviour of an elasticum which is clamped radially to the inside of a rotating ring and carries a mas s at the free end. A nonlinear model for buckling and vibration of the structure in the plane of the ring is established and a subsequent Ga lerkin procedure leads to a system of two first order ODEs, accessible to a bifurcation and phase plane analysis. It will be shown that thes e results can only claim limited validity. Afterwards, more adequate r esults are obtained by a Poincare-Lindstedt perturbation method. This approach is possible, since the linearized model equation can be solve d analytically. Regarding the angular velocity of the ring as a bifurc ation parameter. the analysis shows the existence of a supercritical b ifurcation for any length of the elasticum. Finally, the case of out-o f-plane buckling is shortly investigated, with the result that no buck ling occurs for a sufficiently long elasticum.