BIFURCATION AND CHAOS IN EXTERNALLY EXCITED CIRCULAR CYLINDRICAL-SHELLS

The nonlinear response of an infinitely long cylindrical shell to a pr imary excitation of one of its two orthogonal flexural modes is invest igated The method of multiple scales is used to derive four ordinary d ifferential equations describing the amplitudes and phases of the two orthogonal modes by (a) attacking a two-mode discretization of the gov erning partial differential equations and (b) directly attacking the p artial differential equations. The two-mode discretization results in erroneous solutions because it does not account for the effects of the quadratic nonlinearities. The resulting two sets of modulation equati ons are used to study the equilibrium and dynamic solutions and their stability and hence show the different bifurcations, The response coul d be a single-mode solution or a two-mode solution. The equilibrium so lutions of the two orthogonal third flexural modes undergo a Hopf bifu rcation. A combination of a shooting technique and Floquet theory is u sed to calculate limit cycles and their stability. The numerical resul ts indicate the existence of a sequence of period-doubling bifurcation s that culminates in chaos, multiple attractors, explosive bifurcation s, and crises.