The method of multiple scales is used to analyze the nonlinear response of infinitely long, circular cylindrical shells (thin circular rings) in the presence of a two-to-one internal (autoparametric) resonance to a subharmonic excitation of order one-half of the higher mode. Four autonomous first-order ordinary differential equations are derived for the modulation of the amplitudes and phases of the interacting modes. These modulation equations are used to determine the fixed points and their stability. The fixed points correspond to periodic oscillations of the shell, whereas the limit-cycle solutions of the modulation equations correspond to amplitude and phase-modulated oscillations of the shell. The force response curves exhibit saturation, jumps, and Hopf bifurcations. Moreover, the frequency response curves exhibit Hopf bifurcations. For certain parameters and excitation frequencies between the Hopf values, limit-cycle solutions of the modulation equations are found. As the excitation frequency changes, all limit cycles deform and lose stability through either pitchfork or cyclic-fold (saddle-node) bifurcations. Some of these saddlenode bifurcations cause a transition to chaos. The pitchfork bifurcations break the symmetry of the limit cycles.
Nonlinear Response of Infinitely Long Circular Cylindrical Shells to Subharmonic Radial Loads
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Nayfeh, A. H., Raouf, R. A., and Nayfeh, J. F. (December 1, 1991). "Nonlinear Response of Infinitely Long Circular Cylindrical Shells to Subharmonic Radial Loads." ASME. J. Appl. Mech. December 1991; 58(4): 1033–1041. https://doi.org/10.1115/1.2897679
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