Bifurcation and universal unfolding for a rotating shaft with unsymmetrical stiffness

The 1/2 subharmonic resonance bifurcation and universal unfolding are studied for a rotating shaft with unsymmetrical stiffness. The bifurcation behavior of the response amplitude with respect to the detuning parameter was studied for this class of problems by Xiao et al. Obviously, it is highly important to research the bifurcation behavior of the response amplitude with respect to the unsymmetry of stiffness for this problem. Here, by means of the singularity theory, the bifurcation and universal unfolding of amplitude with respect to the unsymmetrical stiffness parameter are discussed. The results indicate that it is a high codimensional bifurcation problem with codimension 5, and the universal unfolding is given. From the mechanical background, we study four forms of two parameter unfoldings contained in the universal unfolding. The transition sets in the parameter plane and the bifurcation diagrams are plotted. The results obtained in this paper show rich bifurcation phenomena and provide some guidance for the analysis and design of dynamic buckling experiments of this class of system, especially, for the choice of system parameters. The project supported by the National Natural Science Foundation of China (19990510), the National Key Basic Research Special Foundation (G1998020316) and Liuhui Center for Applied Mathematics, Nankai University and Tianjin University