Nonlinear dynamic response of a fractionally damped cylindrical shell with a three-to-one internal resonance

Nonlinear vibrations of a cylindrical shell embedded in a fractional derivative viscoelastic medium and subjected to the different conditions of the internal resonance are investigated. The viscous properties of the surrounding medium are described by Riemann-Liouville fractional derivative. The displacement functions are determined in terms of eigenfunctions of linear vibrations of a free-simply supported shell. A novel procedure resulting in decoupling linear parts of equations is proposed with the further utilization of the method of multiple scales for solving nonlinear governing equations of motion by expanding the amplitude functions into power series in terms of the small parameter and different time scales. It is shown that the phenomenon of internal resonance can be very critical, since in a circular cylindrical shell the one-to-one, two-to-one, and three-to-one internal resonances, as well as additive and difference combinational resonances are always present. The three-to-one internal resonance is analyzed in detail. Since the internal resonances belong to the resonances of the constructive type, i.e., all of them depend on the geometrical dimensions of the shell under consideration and its mechanical characteristics, that is why such resonances could not be ignored and eliminated for a particularly designed shell. It is shown that the energy exchange could occur between two or three subsystems at a time: normal vibrations of the shell, its torsional vibrations and shear vibrations along the shell axis. Such an energy exchange, if it takes place for a rather long time, could result in crack formation in the shell, and finally to its failure. The energy exchange is illustrated pictorially by the phase portraits, wherein the phase trajectories of the phase fluid motion are depicted.