Forced nonlinear oscillations of a cylindrical panel resting on a discontinuous unilateral elastic base

Abstract

The aim of this work is to analytically investigate the influence of a discontinuous unilateral elastic base on the nonlinear resonance curves of imperfect cylindrical panels. To achieve this, the set of partial differential nonlinear equilibrium equations representing the imperfect cylindrical panel is complemented with a transversal equilibrium equation derived from Donnell’s nonlinear shallow shell theory and Airy’s stress function. To discretize this set of equations, a standard Galerkin method is applied, using a particular modal solution for the transversal displacement field that includes simply supported boundary conditions, which can represent the nonlinear modal coupling commonly present in the dynamic behavior of cylindrical panels. The effects of the discontinuous unilateral elastic base are directly inserted into the equilibrium equation by using the Heaviside and signum functions to represent the reaction of the elastic base. From the set of discretized equilibrium equations, the nonlinear resonance curves, phase portraits and Poincaré section mapping are obtained via several numerical strategies. The results presented here demonstrate that the unilateral contact of the elastic base significantly changes the dynamic behavior of the imperfect cylindrical panel by creating several bifurcation points that give rise to quasiperiodic responses. These also modify the stability along the resonance curves compared with the case where the elastic base is considered to be bilateral. This work contributes to the nonlinear dynamic analysis of imperfect cylindrical panels with discontinuous unilateral elastic foundations, and introduces a novel evaluation of bifurcation sequences and stability of resonance curves using a non-smooth contact model.