Small amplitude quasi-periodic solutions for the forced radial vibrations of cylindrical shells with incompressible materials

Abstract

In this paper, we proved the existence of small amplitude quasi-periodic solutions for the forced radial vibrations of cylindrical shells with hyperelastic, homogeneous, isotropic, and incompressible materials. The proof mainly relies on action angle variables and a series of approximately identical transformations, which transform the Hamiltonian into the near integrable one. Owing to Moser’s twist theorem, there exist infinitely many invariant curves at any sufficiently small neighborhood of the equilibrium point of free radial oscillations, some of which are corresponding to quasi-periodic solutions. Moreover, this procedure also provides a new method to calculate the approximate period of the small amplitude free radial periodic oscillation. Some numerical examples demonstrate and support our results.