Nonlinear vibrations of a FG cylindrical shell on a circumferential discontinuous elastic foundation

Resumo

The nonlinear vibrations of a simply supported cylindrical shell made by a functionally graded material
with a circumferentially discontinuous elastic base is analyzed. The equilibrium equations are obtained from
Donnell's nonlinear shallow shell theory. The modal solution to the transversal displacement field, used to
discretize the equilibrium equations, is obtained by perturbation techniques. The discretized equations are
analyzed, considering a harmonic excitation in the form of the combination of the lowest vibration modes of
cylindrical shell. The chosen geometry of the cylindrical shell presents natural frequencies nearly commensurate
to an internal resonance 1:1:1:1, due to the discontinuity of the elastic base in the circumferential direction. The
nonlinear dynamic behavior is analyzed from the resonance curves that they are obtained by the continuation
method and the basins of attraction. Several resonance peak regions are observed, due to the interaction between
the modes of the transversal displacement field, showing the competition of multiple stable, quasi -periodic and
chaotic solutions. Time responses, phase portraits and Poincaré sections are also used to understand the nonlinear
dynamic behavior of the cylindrical shell.