M. Brons et W. Kliem, NONLINEAR-ANALYSIS OF THE BUCKLING AND VIBRATION OF A ROTATING ELASTICUM, International journal of mechanical sciences, 36(7), 1994, pp. 673-681
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.