Chaotic Dynamics of Shallow Arch Structures under 1:2 Resonance

Abstract

The global dynamics of a shallow arch structure subjected to a spatially and temporally varying force is investigated under the conditions of principal subharmonic resonance and one-to-two internal resonance near single mode periodic motions. We describe the mechanism leading to chaotic behavior in the class of systems under consideration. In this paper, a higher-dimensional, Melnikov-type perturbation method is used to analytically show that the arch structure, in the absence of any dissipation mechanism, may exhibit chaotic dynamics in the sense of Smale horseshoe for the one-to-two internal resonance case. These chaotic motions result from the existence of orbits heteroclinic to a normally hyperbolic invariant torus, which corresponds to the hyperbolic periodic orbit in the averaged system. In this case, the presence of small dissipation causes the phase flow to be attracted towards the trivial solution. Numerical simulations are also performed to confirm the theoretical predictions and hence the existence of complicated dynamics in the shallow arch system.