A Second-Order Approximation of Multi-Modal Interactions in Externally Excited Circular Cylindrical Shells

We investigate the nonlinear response of an infinitely long, circularcylindrical shell to a primary-resonance excitation of one of itsflexural modes, which is involved in a one-to-two internal resonancewith the breathing mode. The excited flexural mode is involved in aone-to-one internal resonance with its orthogonal flexural mode. Thereare two simultaneous internal (autoparametric) resonances: two-to-oneand one-to-one. The method of multiple scales is directly applied to thepartial-differential equations to obtain a system of six first-ordernonlinear ordinary-differential equations governing modulation of theamplitudes and phases of the three interacting modes. In the absence ofdamping, the modulation equations are derivable from a Lagrangian,reflecting the conservative nature of the system. The modulationequations are used to study the equilibrium and dynamic solutions andtheir stability and hence their bifurcations. The response may be eithera two-mode or a three-mode solution. For certain excitation parameters,the equilibrium three-mode solutions undergo Hopf bifurcations. Acombination of a shooting technique and Floquet theory is used tocalculate limit cycles and their stability, and hence theirbifurcations.