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

## Palavras-chave:

discontinuous elastic base, cylindrical shell, nonlinear dynamics, reduced order model## 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.