黏弹性圆柱形壳动力学高余维分岔、普适开折问题

讨论两端受到谐波激励的黏弹性圆柱形壳的非线性动力学行为，利用奇异性理论，研究了分岔方程的普适开折问题，严格证明了它是一个高余维分岔问题。余维数为5（含有一个模参数），给出了它的所有可能的普适开折形式。在分岔参数满足某些条件时得到该分岔问题的转迁集及分岔图，展示了一些新的动力学行为，改进和完善了奇异性分析方法。

THE HIGH CODIMENSIONAL BIFURCATIONS AND UNIVERSAL UNFOLDING PROBLEMS OF A VISCOELASTIC CIRCULAR CYLINDRICAL SHELL

The nonlinear dynamical behavior of a viscoelastic circular cylindrical shell under a harmonic excitation applied at both ends is discussed. Using singularity theory and choosing nondegenerate germ, the universal unfolding problem of the bifurcation equation is studied. The results indicate that the universal unfolding is a high codimensional bifurcation problem with codimension 5 (contains a modal parameter). Additionally, all possible forms of the universal unfoldings are given, and the transition sets in parameter plane and the bifurcation diagrams are plotted under some conditions for unfolding parameters. It is well know that the dynamical behavior of nonlinear Mathieu equation in some special forms has been studied by many authors, such as, Bogoliubov N.N., Mitropolsky Y.A., Nayfeh A.H. and Mock D.T. Especially, by introduction of symmetry group action technique, Chen Yushu and Langford W.F. established the bifurcation classification results for stable responses of non- linear Mathieu equation in general form. A sense of theoretical and experimental results with Z2-symmetric bifUrcations were obtained for a parametric excitation system with one degree of freedom. One knows that a symmetric unfolding of a bifurcation equation may describe possible dynamical behavior when original system is subjected to a small perturbation with the symmetry. However, when original system is subjected to a small perturbation which does not possess the symmetrys symmetric unfolding cann't present completely the dynamical behavior of original sys- tem. In this case, one should investigate the universal unfoldings of bifurcation equation including symmetry-breaking bifurcations. For a thin viscoelastic circular cylindrical shell with Flugge-type nonlinearity under a periodic excitation applied at axis. The present paper studies the symmetry-breaking bifurcation problem. The results obtained here indicate that, under symmetry-breaking cases, there exist richer dynamic buckling patterns than those obtained under symmetry case. Clearly, the new results provide some inspirations for the analysis and design of dynamic bulking experiments of this class of system, and improve the singularity analysis method proposed by Chen Yushu and Langford W.F.