Auto-parametric resonance in thin cylindrical shells using the slow fluctuation method

Internal auto-parametric instabilities in the non-linear vibrations of an undamped and unloaded cylindrical shell are studied. The focus is on the coupling between two modes that can combine to break the in–out symmetry and give an energetically favourable pattern of deformation. When the ratio of the natural frequencies is close to 2, internal resonance triggers significant energy transfer between the modes. This energy exchange has both regular and chaotic features, which were previously investigated by other methods of analysis in [J. Sound Vibrat. 227 (1999) 65; J. Sound Vibrat. 248 (2001) 395]. A Rayleigh-Ritz discretization of the von Kármán–Donnell equations is employed to derive the Hamiltonian equations in R4. A perturbation analysis using action-angle coordinates gives slow fluctuation equations that can be solved by quadrature in Jacobian elliptic functions: and, a simple analytical criterion for the parametric instability is presented. This analytical solution gives a good approximation to the regular motions, and provides a framework in which to examine the chaotic motions that are studied in [J. Sound Vibrat. 248 (2001) 395].